Subdiagonal Subalgebras and Commutative Banach Algebras by Toeplitz Operators

Sudan University of Science and Technology College of Graduate Studies

الجبريات الجزئية القطرية الجزئية و جبريات باناخ التبديلية بواسطة مؤثرات تبوليتز

A thesis Submitted in Fulfillment of the requirements for the degree of Ph.D in Mathematics

Siddig Ibrahim HassabAlla Gomer

Supervisor

Prof. Dr. Shawgy Hussein Abdalla

12017

Dedication

To my Family

Acknowledgements

I wish to acknowledge valuable support of my supervisor Prof. Dr. Shawgy Hussein Abd All for his guidance and help during writing the dissertation. My thanks are also due to the staff of Mathematics Department, College of sciences, Sudan University of Sciences and Technology. I would like to extend to thank to Dean of the College of Sciences, Sudan University of Sciences and Technology for him helpful advice & encouragement throughout this research. I would like my thanks to my wife Sulafa Abdel Rahim Mohammed. My thanks are due to Mohammed Ali at Bahri University. Finally I would like to thank my Colleagues.

Abstract

We determine the eigenvalues inequalities, sums of hermitian and normal matrices, Schubert calculus, Wielant’s theorem with spectral sets and Banach algebra. The principal submatrices with noncommutative function theory and unique extensions was shown. We give applications of the Fuglede-Kadison determinant, Riesz and Szegö type factorizations theorem for noncommutative Hardy spaces and for a Helson-Szegö theorem noncommutative Hardy- Lorentz spaces. We also give a Helson-Szegö subdiagonal subalgebras with applications to Toeplitz operators. The algebraic structure of non-commutative analytic with quasi-radial quasi-homogeneous symbols and commutative Banach algebra of Toeplitz algebra and operators are presented, the structure of a commutative Banach algebra on the unit ball and quasi-nilpotent group action, generated by Toeplitz operators with quasi-radial quasi- homogeneous symbols are discussed.

III

الخالصة حددنا متباينات القيم الذاتية والمجاميع الهيرميتية والمصفوفات الناظمة وحسبان شيبورت ومبرهنة ويالنت مع الفئات الطيفية وجبر باناخ. تم ايضاح المصفوفات الجزئية االساسية مع نظرية الدالة غير التبديلية والتمديدات الوحيدة. اعطينا تطبيقات لمحددة فيقليد−كادسون ومبرهنة التحليل الى عوامل نوع ريس−سيزيقو ألجل فضاءات هاردي−لورنتز. أيضا اعطينا الجبريات الجزئية القطرية جزئية هيلسون−سيزيقو مع التطبيقات الى مؤثرات تبوليتز. تم احضار التشييد الجبري للتحليل غير التبديلي مع الرموز شبه− المتجانسة شبه− نصف القطرية وجبر باناخ التبديلي لجبر مؤثرات تبوليتز. تم مناقشة تشييد جبرباناخ التبديلي على كرة الوحدة وفعل زمرة شبه−متالشية القوى والمولدة بواسطة مؤثرات تبوليتز مع الرموز شبه−المتجانسة شبه−نصف القطرية.

Introduction

We examine, simultaneously, all of the 푘-square principal submatrices of an n-square matrix 퐴. Usually 퐴 has been symmetric or Hermitian, and much of our effort has centered around the well-known fact asserting that the eigenvalues of an (푛 − 퐼)-square principal submatrix of Hermitian 퐴 always interlace the eigenvalues of 퐴.

We generalize to the setting of Arveson’s maximal subdiagonal subalgebras of finite von Neumann algebras, the Szegö 퐿푝-distance estimate, and classical theorems of F. and M. Riesz, Gleason and Whitney, and Kolmogorov. We first use properties of the Fuglede-Kadison determinant on 퐿푝(푀), for a finite von Neumann algebra 푀, to give several useful variants of the noncommutative Szegö theorem of 퐿푝(푀), including the one usually attributed to Kolmogorov and Krein.

The non-commutative analytic Toeplitz algebra is the WOT-closed algebra generated by the left regular representation of the free semigroup on 푛 generators. We develop a detailed picture of the algebraic structure of this algebra. In particalur, we show that there is a canonical homomorphism of group of the automorphism group onto the of conformal automorphisms of the complex 푛-ball. We present here a quite unexpected result: Apart from already known commutative 퐶∗- algebras generated by Toeplitz operators on the unit ball, there are many other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergaman space.

We extend eigenvalue inequalities due of Freede-Thompson and Horn for sums of eignevalues of two Hermitian matrices.Let A be a complex unital Banach algebra and let 푎, 푏 ∈ 퐴. We give regions of the complex plane which contain the spectrum of 푎 + 푏 or ab using von Neumann spectral set theory.

Let A be a finite subdiagonal algebra in Arveson’s sense. Let Hp(A) be the associated noncommutative Hardy spaces, 0 < 푝 ≤ ∞. We extend to the case of all positive indices most recent results about these spaces, which include notably the Riesz, Szegö and inner-outer type factorizations. We formulate and establish a noncommutative version of the well-known Helson- Szegö theorem about the angle between past and future for subdiagonal subalgebras.

Studying commutative C∗-algebras generated by Toeplitz operators on the unit ball it was proved that, given a maximal commutative subgroup of biholomorphisms of the unit ball, the C∗-algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each standard weighted Bergman space. There are five different pairwise non-conjugate model classes of such subgroups: quasi-elliptic, quasi- parabolic, quasi-hyperbolic, nilpotent, and quasi-nilpotent. It was observed in Vasilevski that V there are many other, not geometrically defined, classes of symbols which generate commutative Toeplitz operator algebras on each weighted Bergman space. These classes of symbols were subordinated to the quasi-elliptic group. The corresponding commutative operator algebras were Banach, and being extended to 퐶∗-algebras they became noncommutative. These result were extended then to the classes of symbols, subordinated to the 2 n quasi-hyperbolic and quasi-parabolic groups. Let 𝒜λ (𝔹 ) denote the standard weighted Bergman space over the unit ball 𝔹n in ℂn. New classes of commutative Banach 2 n algebras 풯(λ) which are generated by Toeplitz operators on 𝒜λ (𝔹 ) have been recently discovered in Vasilevski. ). These algebras are induced by the action of the quasi-elliptic group of biholomorphisms of 𝔹n. we analyze in detail the internal structure of such an algebra in the lowest dimensional case 푛 = 2. Extending recent results to the higher dimensional setting 푛 ≥ 3 we provide a futher step in the structural analysis of a class of commutative Banach algebras generated by Toeplitz operators on the standard weighted Bergman space over the 푛- dimensional complex unit ball. The algebras ℬ푘(ℎ) under study are subordinated to the qausi- elliptic group of automorphisms of 𝔹푛 and in term of their generators they were described.

The contents

Subject Page on

Dedication I

Acknowledgements II

Abstract III

Abstract (Arabic) IV

Introduction V

The contents VII

Chapter 1

Eigenvalues of Sums and Principal Submatrices

Section (1.1) Hermitian Matrices 1

Section (1.2) Interlacing Inequalities for Singular Values Of Submatrices 13

Chapter 2

Noncommutative Function Theory and Applications of The Fuglede-Kadison Determinant

Section (2.1) Unique Extensions 22

Section (2.2) Szegö's Theorem and Outers for Noncommutative Hp 34

Chapter 3

The Algebraic Structure and Quasi-Radial Quasi-Homogeneous Symbols

Section (3.1) Non-commutative Analytic Toeplitz Algebras 47

Section (3.2) Commutative Banach A lgebras of Teoplitz O perators 70

VII

Chapter 4

Eigenvalue Inequalities and on the Eigenvalues of Normal Matrices

Section (4.1) Schubert Calculus 79

Section (4.2) Wielant’s Theorem with Spectral sets and Banach Algebra 93

Chapter 5

Riesz and Szegö Type Szegö Factorization Theorem for a Helson-Szegö Theorem

Section (5.1) Factorizations for Noncommutative Hardy Spaces 100

Section (5.2) Noncommutative Hardy-Lorentz Spaces 111

Section (5.3) Subdiagonal Subalgebras with Applications to Toeplitz 126 Operators

Chapter 6

Structure of a Commutative Banach Algebra

Section (6.1) Quasi-Nilpotent Group Action 138

Section (6.2) Toeplitz Operators with Quasi-Radial Quasi-Homogeneous 163 Symbols

Section (6.3) Toeplitz Operators on the Unit Baal 189 list of symbols 218

References 220

VIII

Chapter 1 Eigenvalues of Sums and Principal Submatrices We study the singular values of the submatrices (not necessarily principal submatrices) of an arbitrary matrix 퐴. Although we study not necessarily principal submatrices, we Principal Submatrices series because the singular values of an arbitrary submatrix of matrix 퐴 may be approached through an examination of the principal submatrices of 퐴퐴*. Section (1.1) Hermitian Matrices Let 푎 = 훼(훼1, … 훼푛) and 훽 = (훽1 … , 훽푛) be arbitrary nonincreasing sequences of real numbers. We consider the question: for which nonincreasing sequences훾 = (훾1, … 훾푛)do there exist Hermitian matrices A and B such that 퐴, 퐵 and 퐴 + 퐵 have α, β and γ respectively as their sequences of eigenvalues. Necessary conditions have been obtained by Weyl [108], Lidskii [292], Wielandt [312, 263, 278, 289], and Amir-Moez [18], Besides the obvious condition 훾1 + ⋯ + 훾푛 = 훼1 + ⋯ +훼푛 + 훽1 + ⋯ + 훽푛, (1) these conditions are linear inequalities of the form

훾푘푖 + ⋯ + 훾푘푟 ≤ 훼푖푖 + ⋯ +훼푖푟 + 훽푗푖 + ⋯ + 훽푗푟, (2) where 푖, 푗 and k are increasing sequences of integers. As far as we know all other known necessary conditions are consequences of these inequalities. It is therefore natural to conjecture that the set E of all possible γ forms a convex subset of the hyperplane (1). The set E has hitherto not been determined except in the simple cases 푛 = 1, 2, and will not be determined in general here. We give a method of finding conditions of the form (2) which will yield many new ones. We shall find all possible inequalities (2) for 푟 = 1, 2, and arbitrary 푛, and establish a large class of such inequalities for 푟=3. We use Lidskii's method to find a necessary condition on the boundary points of a subset 퐸' of 퐸. These results are used to determine the set 퐸 for n = 3, 4. In addition a conjecture is given for E in general. 푡ℎ ∗ If 푥 is a sequence, 퓍p denotes the 푝 component of 퓍. If 퐴 is a matrix, 퐴 and 퐴퐴′ denote the conjugate transpose and transpose of 퐴. If 푖 is a sequence of integers such that 1 ≦ 푖1 < ⋯ < 푖푟 ≦ 푛, by the complement of 푖 with respect to 푛 we mean the sequence obtained by deleting the terms of i from the sequence 1,2, … , 푛. If a is a sequence of numbers, diag (훼1, … 훼푛) denotes the diagonal matrix with diagonal a. If 푀 and 푁 are matrices, diag (푀, 푁) denotes the direct sum matrix 푀 0 ( ) 0 푁 The inner product of the vectors x and y is written (푥. 푦).퐼푟 is the unit matrix of order 푟. ∞ 푛 Finally 푒푥푝B denotes the ∑푏=0 퐵 /푛!. We are going to use methods introduced by Lidskii [292,31.1]. Lidskii gave sketchy proofs of his results and it is not obvious how to reconstruct his argument, see [312]. we will therefore derive the results of Lidskii which are needed. The set 퐸 referred to in the introduction is the set of points γ such that 훾1 ≧ ⋯ ≧ 훾푟 and ∗ 훾 is the sequence of eigenvalues of diag (훼1, … , 훼푛) + 푈 diag (훽1, … , 훽푛)푈, where 푈 ranges over all unitary matrices. Fix 훼, 훽, with 훼1 > ⋯ > 훼푛, and 훽1 > ⋯ > 훽푛. let 퐸′ be the subset

1 of 퐸 obtained by letting U range over real orthogonal matrices. To indicate the dependence of E' on α and β we write 퐸′(훼1, … 훼푛; 훽1, … 훽푛). Boundary points and interior points of 퐸′ are always taken with respect to the relative topology of the hyperplane (1). Theorem (1.1.1)[10]:If 훾 is a boundary point of 퐸′ with distinct coordinates then there exist a positive integer 푟 < 푛 and increasing sequences 푖, 푗, and 푘 of order r such that

(훾푘푖. … . 훾푘푟) ∈ 퐸′(훼푖1, … 훼푘푟 ; 훽푖1, … 훽푗푟) And

(훾푘′푖 . … . 훾푘′푛−푟) ∈ 퐸′(훼푖′1, … 훼푖′푛−푟; 훽푗′1, … 훽푗′푟−푛) where 푖′, 푗′ and 푘′are the complements of i, j and k with respect to n. ′ Proof: Let U0 be a real orthogonal matrix such that diag (훼1, … 훼푛) + 푈0diag (훽1, … , 훽푛)푈0 has eigenvalues 훾. If 푇 = (푡푝푞) is a real anti-symmetric matrix, 푒푥푝푇 is orthogonal. For sufficiently small values of 푡푝푞, the eigenvalues 휆1 > ⋯ > 휆푛of ′ diag (훼1, … 훼푛) + 푈0푒푥푝(−푇)퐵푒푥푝 (푇)푈0 where 퐵 = 푑푖푎푔(훽1, … 훽푛), are distinct and determine a point of 퐸'. Let 퐴 = 푈표diag ′ (훼1, … 훼푛)푈0, and let 퓍ι be a unit eigenvector of 퐴 + 푒푥푝(−푇)퐵푒푥푝푇 corresponding to the eigenvector λι which varies continuously with T. We have 퐴퓍푖 + 푒푥푝(−푇)퐵 푒푥푝(푇)퓍푖 = 휆푖퓍푖. (3) Using superscripts to denote derivatives with respect to 푡푝푞푝 < 푞, it follows that 푝푞 푝푞 푝푞 퐴퓍휄 + 푒푥푝(−푇)퐵 푒푥푝(푇)퓍휄 + (푒푥푝(−푇) 퐵 푒푥푝(푇)) 퓍휄 푝푞 푝푞 = 휆휄 퓍휄 + 휆휄퓍휄 . (4) It is easily seen that (푒푥푝푇) 푝푞 reduces to 푇푝푞 when 푇 = 0. Hence when 푇 = 푝푞 푝푞 푝푞 0, (푒푥푝( −푇)퐵푒푥푝푇) = (훽푝 − 훽푞)푍 where 푍 is the matrix whose (푝, 푞) and (푞, 푝) 퐼푞 entries are 1 and whose other entries are 0. .Since 퓍Jis a unit vector,(퓍퐽, 퓍퐽 ) = 0. Therefore by (3), 퐼푞 퐼푞 (퐴퓍휄, 퓍휄 ) + (푒푥푝(−푇)퐵 푒푥푝(푇)퓍𝚤, 퓍휄 = 0 (5) Taking the inner product of (4) with 퓍𝚤we find by (5) and the symmetry of A and B, 푝푞 푝푞 휆휄 = ((푒푥푝(−푇) 퐵 푒푥푝 푇) 퓍휄, 퓍휄) Setting T = 0, 푝푞 훾퐽 = 2(훽 − 훽)휔휄푝휔휄푞, (6) pq 퐼푞 where 퓌J and γJ denote the values of 퓍J and 퓍휄휄 when 푇 = 0. If γ is not an interior point pq of 퐸′ the rank of the 푛 by 푛(푛 — 푙)/2 matrix G = (γι ) must be less than 푛 − 1. Now let 퐷 = (휔휄푝휔휄푞) be the n by 푛(푛 − 1) matrix whose rows are indexed by J, where 1 ≦ 푝 ≦ 푛, and whose columns are indexed by (푝, 푞), where 푝 and 푞 vary over the range 1 ≦ 푝 ≦ 푛, and p ≠ q rather than p < q. Clearly D, and hence DD′ has the same rank as G. If F is the square 2 matrix (ωιm) of order n, then 퐷퐷′ = 퐼 − 퐹퐹′. Thus if rank 퐷 < 푛 − 1, 퐹퐹' has 1 as a multiple eigenvalue. Since 퐹퐹′ is stochastic, it follows that FF′is decomposable [91,158,310,122]. That is to say, FF' = P diag (M, N)P', where 푀 and 푁are square matrices and 푃 is a permutation matrix. Let 퐺 퐻 퐹 = 푃 ( ) 푃′ 퐽 퐾 2 be the decomposition of 퐹 corresponding to that of 퐹퐹′. Then 퐺퐽′ + 퐻퐾′ = 0. Since the entries of 퐹 are nonnegative, we have 퐺퐽′ = 퐻퐾′ = 0. It follows that if a column of 퐺 contains a nonzero term then all terms of the corresponding column of 퐽 vanish, and similarly for 퐻 and 퐾. Moving all nonzero columns of G and H to the left, we find S 0 F = P ( 1 ) P′ 0 S2 where 푅 is another permutation matrix. Since 퐹 is doubly stochastic, 푆1and 푆2 must be square 푊1 0 matrices. If W = (ωιm) then 푊 = 푃 ( ) 푅, where 푊 1and 푊2 are square. Setting 0 푊2 ′ ′ ′ ′ 퐼 =diag(훾1, … , 훾푛), we have 퐴 + 퐵 = 푊 퐼′푊. Therefore 푅퐴푅 + 푅퐵푅 = 퐺, where퐶 = ( ′ ′) ′ ′ 푑푖푎푔 푊1 , 푊2 푃 퐼′푃 diag(푊1, 푊2) Let j and k be such that 푅퐵푅 = diag (훽푗1, … , 훽퐽푛) and ( ) 푃′ 퐼′푃 = 푑푖푎푔 (훾푘1, … , 훾푘푛 ). If 푊1 is of order 푟, then 퐶 =diag 퐶1, 퐶2 where 퐶1 has eigenvalues 훾푘1 , … , 훾푘푛and 퐶2 has eigenvalues 훾푘푟+1, … , 훾푘푛 . Therefore 푅퐴푅′ =

푑푖푎푔 (퐴1, 퐴2), where 퐴1 + diag (훽푗1, … , 훽퐽푟) = 퐶1 and 퐴2 +diag (훽푗푟+1, … , 훽퐽푛) = 퐶2. This completes the proof. If 푀 = (푚푖,푗) 1 ≦ 푖 ≦ 푟, 1 ≦ 푗 ≦ 푟 a matrix of order 푟 and 푁 = (푛푘휄), 푟 + 1 ≦ 푘 ≦ 푛, 푟 + 1 ≦ 휄 ≦ 푛 is a matrix of order 푛 − 푟, we define M × N to be the matrix (푚푖퐽, 푛푘퐽) of order 푟(푛 − 푟) whose rows are indexed by pairs (푖, 푘)and whose columns are indexed by pairs (j, l). This product is left and right distributive and (푀 × 푁)′ = 푀′ × 푁′. Also (푀1 × 푁1)(푀2 × 푁2) = (푀1푀2 × 푁1푁2). We set 푀 ⊝ 푁 = (푀 × 퐼푛−푟) − (퐼푟 − 푁). It follows from these remarks that if 푊1 and 푊2 are orthogonal then so is 푊1 × 푊2 and ′ ′ (푊1 푀푊) ⊝ (푊1 푁푊2) = (푊1 × 푊2)(푀 ⊝ 푁)(푊1 × 푊2) (7) The index of a real symmetric matrix is the number of its positive eigenvalues. Lemma (1.1.2)[10]: If 푀, 푁, and 푀 + 푁 are nonsingular real symmetric matrices then index M + index N = index (M + N) + index (푀−1 + 푁−1). Proof: We have 푀−1 + 푁−1 = 푁−1(푁 + 푀)푀−1so that 푀−1 + 푁−1 is nonsingular. Also 1 1 푀 0 −1 푀 + 푁 0 ( ) ( ) (1 푀 ) ( ) 푀−1 −푁−1 0 푁 1 −푁−1 0 푀−1 + 푁−1 The result now follows by the Law of Inertia. Theorem (1.1.3)[10]:Let 훾 be a boundary point of 퐸' with distinct coordinates. Then there exist sequences 푖, 푗 and 푘 satisfying the conclusion of Theorem (1.1.1) and such that 푖1 + ⋯ + 푖푟 + 푗1 + ⋯ + 푗푟 = 푘1 + ⋯ + 푘푟 + 푟(푟 + 1)/2. Proof: Using a slight change of notation, we have seen that there exist permutations 푖, 푗 and 푘 of (1,…,n)and real symmetric matrices 퐴1, 퐴2, 퐵1, 퐵2, 퐶1 , 퐶2 such that 퐴1 has eigenvalues

훼푖1, … 훼푖푟,퐴2 has. eigenvalues 훼푖푟+1, … 훼푖푛, 퐵1 = 푑푖푎푔(훽푖1, … 훽푖푟), 퐵 = 푑푖푎푔(훽푖푟+1, … 훽푖푛), 퐶1 has eigenvalues 훾푘1, … , 훾푘푟, 퐶2 has eigenvalues 훾푘푟+1, … , 훾푘푛, and퐴 + 퐵 = 퐶, where 퐴 = 푑푖푎푔(퐴1, 퐴2), 퐵 = 푑푖푎푔(퐵1, 퐵2) 퐶 = 푑푖푎푔(퐶1, 퐶2 ) . We also assume 푖1 < ⋯ < 푖푟, and ̅ 푖푟+1 < ⋯ < 푖푛 and similarly for the 푗′푠 and 푘′푠. We set 훼̅𝚤 = 훼푖𝚤, 훽𝚤 = 훽푖𝚤 and 훾𝚤̅ = 훾푘𝚤, 1 ≦ 횤 ≦ 푛. Let 푇 = (푡푝푞) be a real anti-symmetric matrix and let 휆1 > ⋯ > 휆푛be the eigenvalues of 퐴 + 푒푥푝(−푇)퐵푒푥푝푇. If x1 , … , xn is a real orthonormal system of corresponding 푝푞 푝푞 푝푞 eigenvectors, we let 퓌푙 and 휔푙 be the values of 푥푘 and 푥푘푙 when 푇 = 0, where 푥푙 denotes 3 the derivative of 푥푙 with respect to 푡푝푞, 푝 < 푞. If 푊 is the matrix whose rows are 휔1, … , 휔푛, ′ ′ then 푊 = 푑푖푎푔(푊1, 푊2)and 퐶1 = 푊1 훤1 푊1, 퐶2 = 푊2 훤2 푊2where훤1 = 푑푖푎푔(훾푘1 , … , 훾푘푟 )훤2 = 푝푞 푑푖푎푔 (훾푘푟+1, … , 훾푘푛). Clearly 휆푘푙 reduces to 훾푙̅ when T = 0, and we let 훾푙̅ be the value of 푝푞 휆푘 = 휕휆푘푙/휕푡푝푞 when T = 0. Starting from the equation 퐴휆푝푞 + (푒푥푝(−푡) 퐵푒푥푝푇)퓍푘푙 = 휆푘푙퓍푘푙 (8) We find 푝푞 푝푞 푝푞 A휆푘푙 + (푒푥푝(−푡) 퐵푒푥푝푇)퓍푘푙 + (푒푥푝(−푇) 퐵 푒푥푝푇) 퓍푘푙 푝푞 푝푞 = 휆푘푙 퓍푘푙 + 휆푘푙퓍푘푙 (9) As in Theorem (1.1.1) is follows that 푝푞 푝푞 휆푙 = ((푒푥푝(−푇) 퐵 푒푥푝푇) 퓍푘푙, 퓍푘푙) (10) and therefore 푝푞 ̅ ̅ 훾푙̅ = 2(훽푝 − 훽푞)휔푙푝휔푙푞 (11)

We are going to test 휎 = 휆푙푙 + ⋯ + 휆푘푟 for a local extreme at T = 0. If p and q are ≦ r, then expT has the form 푑푖푎푔 (푒푥푝 푇1,, 0) when 푇푢푣 = 0for (푢, 푣) ≠ (푝, 푞), and hence σ remains constant for 푇푝푞 in a neighborhood of 0. Therefore all partial derivatives of σ with respect to pq tpq vanish at the origin when 푝 < 푞 ≦ 푟, and similarly when r < 푝 < 푞. by (11), σ = 0 at T=0 when 푝 ≦ 푟 < 푞, since that last 푛 − 푟 components or ωl are 0 when 1 ≦ I ≦ r. we now calculate λpq.uv at T=0 when kl 1 ≦ p ≦ r < 푞 ≦ 푛, 1 ≦ 푢 ≦ 푟 < 푣 ≦ 푛, 1 ≦ 퐼 ≦ 푟. (12) Differentiation of (10) yields 휆푝푞.푢푣 = ((푒푥푝(−푇) 퐵 푒푥푝푇)푝푞퓍푢푣, 퓍 + 2((푒푥푝(−푇) 퐵 푒푥푝푇)푝푞퓍푢푣, 퓍 ) (13) 푘푙 푘푙 푘푙 푘푙 푘푙 It is easily seen that when 푇 = 0 (푒푥푝(−푇)퐵푒푥푝 푇)푝푞.푢푣 1 = − (푇푝푞 퐵푇푢휐 + 푇푢푣퐵푇푝푞) + 퐵(푇푝푞푇푢휐 + 푇푢푣푇푝푞) 2 1 + (푇푝푞푇푢휐 + 푇푢푣푇푝푞)퐵 2 Considering only the cases (12), a straightforward calculation shows that when T = 0, ((푒푥푝(−푇) 퐵 푒푥푝푇)푝푞퓍푢푣, 퓍 = 0 for 푝 ≠ 푢, 푞 ≠ 푣 푘푙 푘푙 ̅ ̅ ̅ = (2훽푞 − 훽푝 − 훽푢)휔푙푝휔푙푢 for 푝 ≠ 푢, 푞 = 푣 ̅ ̅ ̅ = (2훽푝 − 훽푞 − 훽푣)휔푙푝휔푙푣 for 푝 = 푢, 푞 ≠ 푣 ̅ ̅ 2 2 = −2(훽푝 − 훽푝)(휔푙푝 − 휔푙푝)for 푝 = 푢, 푞 = 푣 Recalling that 휔푡푞 = 0for 퐼 ≦ 푟 < 푞, we find that when 훤 = 0, 푟 푝푞.푢푣 ̅ ̅ ∑((푒푥푝(−푇) 퐵 푒푥푝 푇) 퓍푘푙, 퓍푘푙) = −2 (훽푝 − 훽푝) for 푝 = 푢, 푞 = 푣 퐽=1 = 0 otherwise. (14) The second term on the right of (13) reduces when 푇 = 0 to 4

̅ ̅ 푢푣 2(훽푝 − 훽푝)휔푙푝 휔푙푝 (15) 푢푣 To compute 휔푙푝 , rewrite (9) in the form ( ) 푢푣 (퐴 + 푒푥푝 −푇 퐵 푒푥푝 푇 − 휆푙푙 퐼푛)퓍푘푙 =−(푒푥푝(−푇)퐵 푒푥푝 푇)푢푣퓍 + 휆푢푣 퓍 푘푙 푘푙 푘푙 Setting T = 0 and using (11), we find, since ω = 0, 푢푣 ̅ ̅ (푐 − 훾푙̅ 퐼푛)퓍푙 = −(훽푢 − 훽푣)푦, where 푦 is the vector such that 푦푢 = 휔푙휐 = 0, 푦푣 = 휔푙휐 and 푦푣 = 0 for 푚 ≠ 푢, 푚 ≠ 푣. Therefore 푢푣 ̅ ̅ −1 휔푙푝 = (훽푢 − 훽푣)((훾푙푙 퐼푛 − 퐶) 푦)푞 Since 푞 > 푟, and 퐶 = 푑푖푎푔 (퐶1, 퐶2), we may replace C by C2 and In by 퐼푛−푟,Thus 푢푣 ̅ ̅ 휔푙푝 = (훽푢 − 훽푣)푑푞푢휔푙푢 (16) −1 where 푑푞푢 is the (푞, 푣) entry of ((훾푙̅ 퐼푛−푟 − 퐶2) . Now −1 ′ −1 −1 (훾푙̅ 퐼푛−푟 − 퐶2) = (푊2((훾푙̅ 퐼푛−푟 − 훤2) 푊2) Therefore 푛 휔푚푞휔푚푣 푑푞푢 = ∑ (17) 훾푙̅ − 훾푚̅ 푚=푟+1 Combining (13), (14), (15), (16), and (17), we find at T = 0 푟 푛 휔 휔 휔 푝푞.푢푣 ̅ ̅ ̅ ̅ 푙푝 푙푢 푚푞휔푚푣 푝푞 ̅ ̅ 휎 = 2(훽푝 − 훽푞)(훽푢 − 훽푣) ∑ ∑ − 2훿푢푣 (훽푝 − 훽푞), (18) 훾푙̅ − 훾푚̅ 퐼=1 푚=푟+1 푝푞 where 훿푢푣 = 1 when (푝, 푞) = (푢, 푣), and = 0 otherwise. 푝푞.푢휐 We must now determine the index of the matrix 퐺 = (휎 )푟=0 of order 푟(푛 − 푟) whose rows and columns are indexed by pairs (푝, 푞) and (푢, 푣) satisfying (12). The double sum on the right of (18) is the (푝푞, 푢푣) entry of ′ −1 ′ −1 (푊1 × 푊2) (훤1 ⊝ 훤2) (푊1 × 푊2) = ((푊1 × 푊2) (훤1 ⊝ 훤2)(푊1 × 푊2)) By (7) this reduces to −1 −1 (퐶1 ⊝ 퐶2) = (퐴1 + 퐵2) ⊝ (퐴1 + 퐵2) = ((퐴1 ⊝ 퐴2) + (퐵1 ⊝ 퐵2) Therefore by (18) 1 퐺 = (퐵 ⊝ 퐵 )((퐴 ⊝ 퐴 ) + (퐵 ⊝ 퐵 ))−1 (퐵 − 퐵 ) ⊝ (퐵 ⊝ 퐵 ) 2 1 2 1 2 1 2 1 2 1 2 −1 −1 = (퐵1 ⊝ 퐵2(((퐴1 ⊝ 퐴2) + (퐵1 ⊝ 퐵2)) − (퐵1 ⊝ 퐵2) )(퐵1 ⊝ 퐵2) −1 −1 Thus 퐺 has the same index as((퐴1 ⊝ 퐴2) + (퐵1 ⊝ 퐵2)) − (퐵1 ⊝ 퐵2) . Applying Lemma (1.1.2) with 푀 = ((퐴1 ⊝ 퐴2) + (퐵1 ⊝ 퐵2)), 푁 = −(퐵1 ⊝ 퐵2), −1 −1 index 퐺 = 푖푛푑푒푥 ((퐶1 ⊝ 퐶2) − (퐵1 ⊝ 퐵2) ) = 푖푛푑푒푥(퐶1 ⊝ 퐶2) + 푖푛푑푒푥 − (퐵1 ⊝ 퐵2) = 푖푛푑푒푥 (퐴1 ⊝ 퐴2) = 푟(푛 − 푟) + 푖푛푑푒푥(퐶1 ⊝ 퐶2) – 푖푛푑푒푥 (퐵1 ⊝ 퐵2) − 푖푛푑푒푥 (퐴1 ⊝ 퐴2).

Thus G is positive definite if and only if index (퐶1 ⊝ 퐶2) = index (퐴1 ⊝ 퐴2) index (퐵1 ⊝ 퐵2), and 퐺 is negative definite if and only if neg (퐶1 ⊝ 퐶2)= neg (퐴1 ⊝ 퐴2)+ neg (퐵1 ⊝ 퐵2), where neg H is the number of negative eigenvalues of H. Next we determine 푖1 + ⋯ + 푖푟 + 푗1 + ⋯ + 푗푟 = 푘1 + ⋯ + 푘푟 + 푟(푟 + 1)/2, (19) and G is positive definite if and only if 푖푟+1 + ⋯ + 푖푛 + 푗푟+1 + ⋯ + 푗푛 = 푘푟+1 + ⋯ + 푘푛 + (푛 − 푟)(푛 − 푟 + 1)/2. (20) By Theorem (1.1.1) the boundary points of 퐸" lie on a finite number of hyperplanes of the form

훾푘1 + ⋯ + 훾푘푟 = 훼푖1 + ⋯ + 훼푖푟 + 훽푗1 + ⋯ + 훽푗푟 (21) The hyperplane 훾 ′ + ⋯ + 훾 ′ = 훼 ′ + ⋯ + 훼 ′ + 훽 ′ + ⋯ + 훽 ′ 푘 1 푘 푛 푘 1 푖 푛−푟 푗 1 푗 푛−푟 intersects the hyperplane (1) in the same set. If γ lies on only one of these hyperplanes (21) and does not satisfy (19) or (20), then in every small sphere about γ there exist points of 퐸' on both sides of the hyperplane (21). Therefore 퐸' must fill the sphere, for otherwise there would be boundary points of 퐸' inside the sphere and off the hyperplane (21). This being impossible,휆 must satisfy (19) or (20). Now suppose 훾 lies on several hyperplanes (21), and (19) and (20) both fail for each of these hyperplanes. By continuity the quadratic form G is not definite for all points near 훾 which satisfy the conclusion of Theorem (1.1.1). Therefore in a neighborhood of γ all points of 퐸′ lying on only one hyperplane (21) are interior points of 퐸'. Therefore γ cannot be a boundary point of 퐸′. since E' is the closure of its interior, and a finite union of linear varieties of deficienc푦 ≧ 2 cannot separate the interior of a sphere. The proof is complete. If i, j and k are increasing sequences of integers of order r and (2) holds for the eigenvalues of 퐴 + 퐵 for any Hermitian 퐴, 퐵 with arbitrary eigenvalues 훼1 ≧ ⋯ ≧ 훼푛and 훽1 ≧ ⋯ ≧ 훽푛we 푛 write (푖; 푗 ; 푘) ∈ 푆푟 . If

훾푘1 + ⋯ + 훾푘푛 ≥ 훼푖1 + ⋯ + 훼푖푛 + 훽푗1 + ⋯ + 훽푗푛 푛 for any such 퐴, 퐵 we write(푖; 푗 ; 푘) ∈ 푆̃푟 Theorem (1.1.4)[10]: The following conditions are equivalent: 푛 (i) (푖; 푗 ; 푘) ∈ 푆̃푟 (ii) (푛 − 푖, +1, … , 푛 − 푖1 + 1; 푛 − 푗푟 + 1, … 푛 − 푗1 + 1; 푛 − 푘푟 + 1, … 푛, −푘1 + 1) ∈ 푛 푆̃푟 푛 (iii) (푘1, … , 푘푟; 푛 − 푗푟 + 1, … , 푛 − 푗1 + 1; 푖1, … 푖푟)) ∈ 푆̃푟 ′ ′ ′ 푛 (iv) (푖 , 푗 , 푘 )) ∈ 푆̃푛−푟′ where 푖′, 푗′, 푘′ are the compelements of 푖, 푗, 푘 with respect to 푛. Proof: The equation 퐴 + 퐵 = 퐶 may be written −퐴 − 퐵 = −퐶 or 퐴 = 퐶 − 퐵. This proves the equivalence of (i) with (ii) and (iii). The equivalence of (i) and (iv) is immediate by the trace Condition (1). If 퐴 is a Hermitian matrix with eigenvalues 훼1 ≧ ⋯ ≧ 훼푛 and M is a linear subspace of dimension 푛 − 1, let 퐴푀 be the transformation 푃퐴 with domain restricted to 푀, where 푃 is the orthogonal projection on 푀. 퐴푀 is a Hermitian transformation on 푀 to 푀and (퐴 + 퐵)푀 = ′ 퐴푀 + 퐵푀. It is well known that the eigenvalues 훼푝 of 퐴푀 separate those of 퐴, that is 훼푝+1 ≦ ′ 훼푝 ≦ 훼푝. For 1 ≦ 푝 ≦ 푛 − 1. If (퓍p) is an orthonormal sequence of eigenvectors 6

′ corresponding to (αp) and if 푀 contains x1, … , xm then αp = αp for 1 ≦ 푝 ≦ 푚. This is an immediate consequence of the minimax principle, since (퐴푀푥, 푥) = (퐴푥, 푥)for 푥 ∈ 푀. Dually ′ if 푀 contains 푥푚+1, … , 푥푛, then 훼푝 = 훼푝+1 for 푚 ≦ 푝 ≦ 푛 − 1. The next theorem shows that S is essentially independent of 푛. 푛 Theorem (1.1.5)[10]: If (푖; 푗; 푘) ∈ 푆푟 for some 푛 then 푖푝 ≦ 푘푝and 푗푝 ≦ 푘푝 for all p, and 푛 (푖; 푗; 푘) ∈ 푆푟 ? For all 푛 ≥ 푘푟 푛 Proof: Suppose (푖; 푗; 푘) ∈ 푆푟 for some 푛. Considering the case 훽 = 0, it is clear that 푖푝 ≦ 푘푝and 푗푝 ≦ 푘푝 for all 푝. If 퐴 and 퐵 are of order kr, the identity 푑푖푎푔 (퐴 − 휆퐼) + 푑푖푎푔 (퐵 − 푘푟 휆퐼) 푑푖푎푔 (퐴 + 퐵 − 2휆푙) for large λ shows that (푖; 푗; 푘) ∈ 푆푟 . It remains to prove (푖; 푗; 푘) ∈ 푛+1 푆푟 . Let 퐴 and B be of order 푛 + 1 with eigenvalues (훼푝), (훽푝), and let (퓏p) be an orthonormal sequence of eigenvectors of 퐴 + 퐵 corresponding to the eigenvalues (훾푝). Let 푀 ′ ′ ′ be the subspace spanned by 퓏1, … , 퓏푛. Letting (훼푝), (훽푝) and (훾푝) be the eigenvalues of 퐴푀, 퐵푀 and (A + B)M, we have by hypothesis 훾′ +. . +훾′ ≦ 훼′ + ⋯ + 훼′ + ⋯ + 훽′ + ⋯ + 훽′ . 푗1 푗1 푖1 푖푟 푗1 푗푟 But 훾′ = 훾 , 훼′ ≤ 훼 and β′ ≤ β for 1 ≦ p ≦ r. Therefore (푖; 푗; 푘) ∈ 푆푛+1 푘푝 푘푝 푖푝 푖푝 jp jp 푟 푛 Theorem (1.1.6)[10]:If (푖; 푗; 푘) ∈ 푆푟 and u, v and w are integers such that 푟 + 1 ≧ 푢 ≧ 1, 푟 + 1 ≧ 푣 ≧ 1 and 푟 ≧ 푤 ≧ 1, and if 푖푢 + 푗푣 ≧ 푘푤−1 + 푘푟 + 2 then(푖1, … , 푖푢−1,, 푖푢 + 푛+1 1, … , 푖푟 + 1; 푗1 … , 푗푣−1, 푗푣 + 1, … , 푗푟 + 1; 푘1, … , 푘휔−1, 푘휔 + 1, … , 푘푟 + 1) ∈ 푆푟 . Here 푘0 = 0 and ir+1 = jr+1 = kr + 1 by definition. In particular, (푖1 + 1, … , 푖푟 + 1; 푗1 … , 푗푟; 푘1 + 1) ∈ 푛+1 푆푟 . Proof: By Theorem (1.1.5) we may assume 푛 = 푘푟. Let (푥푝), (푦푝) and (푧푝), 1 ≦ 푃 ≦ 푛 + 1, be orthonormal sequences of eigenvectors corresponding to the eigenvalues (훼푝), (훽푝) and (γP) of 퐴, 퐵 and 퐴 + 퐵. Since 푥푝, 푖푣 ≧ 푘푤−1 + 푛 + 2, there exists an n dimensional subspace 푀 containing the vectors 푥푝, 푖푢 ≧ 푘휔−1 + 푛 + 1, the vectors 푦푝, 푖푣 + 1 ≦ 푝 ≦ 푛 + 1, and the ′ ′ ′ vectors 푧푝, 1 ≦ 푝 ≦ 푘휔−1 + 1. Let (훼푝), (훽푝) and (훾푝) be the eigenvalues of 퐴푀, 퐵푀, and(퐴 + 퐵). By hypothesis 훾′ +. . . +훾′ ≦ 훼′ + ⋯ + 훼′ + ⋯ 훽′ + ⋯ +훽′ . 푘1 푘푟 푖1 푖푟 푗1 푗푟 ′ The theorem now follows because γ′푝 = γ푝 for 1 ≦ 푝 ≦ 푘푤−1, 훾푝+1 ≦ 훾푝 for 푘푤 ≦ 푝 ≦ ′ ′ ′ ′ 푛, 훼푝 ≦ 훼푝 for 1 ≦ 푝 ≦ 푖푢−1, 훼푝 = 훼푝+1for 푖푢 ≦ 푝 ≦ 푛, 훽푝 ≦ 훽푝 for 1 ≦ 푝 ≦ 푗푢−1 and 훽푝 = 훽푝+1 for 푗푣 ≦ 푝 ≦ 푛. Theorem (1.1.6) yields a simple proof of the following theorem due to Lidskii. 푛 Theorem (1.1.7)[10]:[292]. If 1 ≦ 푝 < ⋯ < 푝푟 ≦ 푛, then (푝1, … , 푝푟; 1, … , 푟; ) ∈ 푆푟 . 푛 Proof: Obviously (11, … , 푟; 1, … , 푟; 1, … , 푟) ∈ 푆푟 . Using Theorem (1.1.6) 푝1 − 1 times with 푢 = 휔 = 1, 푣 = 푟 + 1, we find (푝1, 푝1 + 1, … , 푝1 + 푟 − 1, ; 1, … , 푟: 푝1, 푝1 + 1, … , 푝1 + 푟 − 푛 1) ∈ 푆푟 . Such use of Theorem (1.1.6) is justified since 푖1 + 푗푟+1 = 푖1 + 푘푟 + 1 ≧ 푘푟 + 2 = 푘0 + 푘푟 + 2 at each stage. We may now apply Theorem (1.1.6) 푝2 − (푝1 + 1) times with 푢 = 휔 = 2. 푣 = 푟 + 1since at each stage 푖1 + 푗푟+1 = 푖1 + 푘푟 + 1 ≧ 푖1 + 푘푟 + 2 = 푘1 + 푘푟 + 2. The result is (푝1, 푝2, 푝2 + 1, … , 푝2 + 푟 − 2: 1, … , 푟; 푝1, 푝2, 푝2 7

푝2+푟−2 + 1, … , 푝2 + 푟 − 2) ∈ 푆푟 Continuing in this way we find 푝푟 (푝1, … , 푝푟; 1, … , 푟; 푝1, … , 푝푟) ∈ 푆푟 By Theorem (1.1.5) the proof is complete. 푛 Theorem (1.1.8)[10]:(푖1; 푗1; 푘1) ∈ 푆1 for 푛 ≧ 푘1 and only if 1 ≦ 푖1 ≦ 푘1, 1 ≦ 푗1 ≦ 푘1, and 푖1 + 푗1 = 푘1 + 1. Proof: The sufficiency of the conditions, due to Weyl, is usually proved by the minimax principle. It can also be proved using Theorem (1.1.6) . We have already seen the necessity of 푖1 ≦ 푘1 and 푗1 ≦ 푘1 in the proof of Theorem (1.1.5). Now suppose 푖1 + 푗1 ≦ 푘1 + 2. Let 퐴 = diag (1, … ,1,0, … , 0) with 푖1 − 1 ones, and 퐵 = 푑푖푎푔(0, … , 0,1, … , 1) with 푗1 − 1 ones, where the orders of 퐴 and 퐵 are k1. Since 푘1 − 푗1 + 1 ≦ 푖1 −1, all the eigenvalues of 퐴 + 퐵 ( ) 푘 are ≧ 1 . Therefore 훾푘1 ≧ 1, while 훼푖1 = 훽푗1 = 0, contradicting 푖1, 푗1, 푘1 ∈ 푆1 Theorem (1.1.9)[10]: If i, j and k are ordered pairs of integers satisfying 1 ≦ 푖1 ≦ 푖2 ≦ 푛, 1 ≦ 푗1 ≦ 푗2 ≦ 푛, 1 < 푘1 < 푘2 ≦ 푛, (22) 푖1 + 푗1 ≦ 푘1 + 1 (23) 푖1 + 푗2 } ≦ 푘2 + 1 (24) 푖2 + 푗1 푖1 + 푖2 + 푗1 + 푗2 = 푘1 + 푘2 + 3 (25) 푛 then (푖; 푗; 푘) ∈ 푆2 Proof: By Theorem (1.1.5) we may assume 푛 = 푘2. We proceed by induction on n. If n = 2 the theorem follows from (1). Suppose the theorem holds for all 푛 < 푁, where 푁 > 2. By (22), (23) and (24), 푖푝 ≦ 푘푝 and 푗푝 ≦ 푘푝, 푝 = 1,2. Suppose 푖1 > 1. Then the pairs (푖1 − 1, 푖2 − 1),(푗1, 푗2)and(푘1 − 1, 푘2 − 1)satisfy (22)-(25). Therefore by the induction hypothesis 푁−1 (푖1 − 1, 푖2 − 1; 푗1, 푗2; 푘1 − 1, 푘2 − 1) ∈ 푆2 If we apply Theorem (1.1.6) with 푢 = 푤 = 푁 1, 푣 = 3, we find (푖; 푗; 푘) ∈ 푆2 . A similar method takes care of the case 푗1 > 1. Therefore we may assume 푖1 = 푗1 = 1 (26) If 푁−1 (푖1 , 푖2 − 1; , 푗1, 푗2 − 1; 푘1, 푘2 − 1, 푘2 − 1) ∈ 푆2 (27) and if 푖2 + 푗2 ≧ +3 + 푘2 (28) 푁 then Theorem (1.1.6) with 푢 = 푣 = 2, 휔 = 1 allows us to conclude (푖; 푗; 푘) ∈ 푆2 . But the Condition (28) which is needed for the application of Theorem (1.1.6) will also guarantee (27). To see this, first note that (27) can fail only when (i) 푖2 = 푖1 + 1 = 2 or (ii) 푗2 = 푗1 + 1 = 2 or (iii) 푘1 = 1 or (iv) 푖1 + 푗1 = 푘1 + 1. If (i) holds then 푖1 + 푗2 = 2 + 푗2 ≦ 2 + 푘2, contradicting (28). Similarly 8

(ii) cannot hold. If (iii) holds, then by (26), 푖1 + 푖2 + 푗1 + 푗2 = 2 + 푖1 + 푗2 = 푘1 + 푘2 + 3 = 푘2 + 4, 표푟 푖1 + 푗2 − 푘2 + 2, contradicting (28). Condition (iv) implies (iii) by (26). Therefore we may assume 푖1 + 푗2 ≦ 2 + 푘2 (29) If 푖2 ≦ 푘1 + 2, it is easy to show by the induction hypothesis that (푖1, 푖2 − 1; 푗1, 푗2; 푘1 , 푘2 − 푁−1 푁 1) ∈ 푆2 and Theorem (1.1.6)with 푢 = 휔 = 2, 푣 = 3implies (푖; 푗; 푘) ∈ 푆2 . Hence we assume 푖2 ≦ 푘1 + 1 and 푗2 ≦ 푘1 + 1. (30) Now (25) and (26) imply 푖2 + 푗2 = 푘1 + 푘2 + 1, which with (29) implies 푘2 = 1. Therefore by (30) and (22), 푖2 + 푗2 = 2 and hence 푖1 + 푗1 = 1. Using (25) we find 푘2 = 2, contradicting 푁 > 2. The proof is complete. If in (25) we replace the equality sign by ≦, Theorem (1.1.9) remains true. For if 푖,j and k ′ ′ ′ ′ satisfy (22)-(24) and the modified (25), there exists a pair 푘′ = {푘1, 푘2) such that 푘1 ≦ 푘1, 푘2 ≤ 푘2 and i, j , k′ satisfy (22)-(25). However Theorem (1.1.3) suggests that we consider only cases where (19) holds. Conditions (23) and (24) combined may be expressed as follows: 푖푢 + 푗푣 ≦ 푘휔 + 1 whenever1 ≦ 푖 ≦ 2, 1 ≦ 푗 ≦ 2, 1 ≦ 휔 ≦ 2, and 푢 + 푣 = 휔 + 1. This suggests the following conjecture. Let us define inductively the following sequence 푛 of sets of triples of sequences of integers:Let (푖1, 푗1 , 퐾1) ∈ 푇1 if 1 ≦ 푖1 ≦ 푛, 1 ≦ 푗1 ≦ 푛 푛, 1 ≦ 푘1 ≦ 푛, and i1 + j1 = K1 + 1 and let (푖1, … , 푖푟; 푗1 , … , 푗푟; 퐾1, … , 푘푟) ∈ 푇푟 , if 1 ≦ 푖1 < ⋯ < 푖푟 ≦ 푛 ,1 ≦ 푗1 ≦ 푛, 1 ≦ 푘1 < ⋯ < 푘푟 ≦ 푛, and 푟(푟+1) 푖 + ⋯ + 푖 + 푗 + ⋯ + 푗 ≦ 푘 + ⋯ + 푘 + (31) 1 푟 1 푟 1 푟 2 And 푠(푠+1) 푖 + ⋯ + 푖 + 푗 + ⋯ + 푗 ≦ 푘 + ⋯ + 푘 + (32) 푢1 푢푠 푣1 푣푠 휔1 휔푠 2 Whenever 푟 (푢; 푣; 휔) ∈ 푇푠 , 1 ≤ 푠 ≤ 푟 − 1. 푛 푛 Theorem (1.1.8) and (1.1.9) show that 푇푟 ⊂ 푆푟 for 푟 = 1, 2. It seems reasonable to conjecture 푛 푛 푇푟 ⊂ 푆푟 for all 푟. The case 푟 = 3is the following. Theorem (1.1.10)[10]: If i, j and k are ordered triples of integers such that 1 ≦ 푖1 < 푖2 < 푖2 ≦ 푛, 1 ≦ 푗1 < 푗2 < 푗2 ≦ 푛, 1 ≦ 푘1 < 푘2 < 푘3 ≦ 푛 (33) 푖1 + 푗1 ≦ 푘1 + 1 (34) 푖1 + 푗2 } ≦ 푘2 + 1 (35) 푖2 + 푗1 푖1 + 푗3 푖2 + 푗2} ≦ 푘3 + 1 (36) 푖3 + 푗1 푖1 + 푖2 + 푗1 + 푗2 ≦ 푘1 + 푘2 + 3 (37) 푖1 + 푖2 + 푗1 + 푗3 } ≦ 푘1 + 푘3 + 3 (38) 푖1 + 푖3 + 푗1 + 푗2

푖1 + 푖2 + 푗2 + 푗3 푖2 + 푖3 + 푗1 + 푗2} ≦ 푘2 + 푘3 + 3 (39) 푖1 + 푖3 + 푗1 + 푗3 푖1 + 푖2 + 푖3 + 푗1 + 푗2 + 푗3 = 푘1 + 푘2 + 푘3 + 6, (40) 푛 Then (푖, 푗, 푘) ⊂ 푆3 Proof: The proof begins along the same lines as the proof of Theorem (1.1.9) and will only be sketched. We may assume n = k3, and proceed by induction on n. When 푛 = 3, 푖1 = 푗1 = 푘1 = 1, 푖2 = 푗2 = 푘2 2, 푖3 = 푗3 = 푘3 = 3, and the result follows from (1). Assume the theorem for all 푛 < 푁, where 푁 > 3. As in Theorem (1.1.9), we may assume 푖1 = 푗1 = 1 (41) If (푖1, 푖2 − 1. 푖3 − 1), (푗1, 푗2, 푗3 − 1), (푘1 − 1, 푘2 − 2, 푘3 − 1) (42) Satisfies (33)- (40) and if 푖1 + 푗3 ≧ 푘3 + 3 (43) then the induction hypothesis and Theorem (1.1.6)with 푢 = 2, 푣 = 3, 푤 = 1 yield the theorem. Again the condition (43) which is needed for the application of Theorem (1.1.6) will guarantee (42). For example 푘1 — 1 ≧ 1, because if 푘1 = 1, then by (38) and (41), 푖1 + 푗3 ≦ 푘3 + 2, contradicting (43). The second inequality of (36) together with (43) and 푗3 ≦ 푘3 (which follows from (36)) ensure 푗3 − 1 > 푗2. We may therefore assume i2 + j3 } ≦ k3 + 2. (44) i3 + j2 Next we show that we may assume 푖2 ≦ 푘1 + 1 and 푗2 ≦ 푘1 + 1 (45) 푁 by showing that if 푖2 ≦ 푘1 + 2, then(푖1, 푖2 − 1, 푖3 − 1; 푗1, 푗2, 푗3, 푘1, 푘2 − 1, 푘3 − 1) ∈ 푆3 and 푁 Theorem (1.1.6) with 푢 = 2, 푣 = 3, 푤 = 2gives (푖; 푗 ; 푘) ∈ 푆3 In a similar manner we may assume 푖3 + 푗3 ≦ 푘1 + 푘3 + 2 (46) 푖3 ≦ 푘2 + 1, 푗3 ≦ 푘3 + 1 (47) Now (33)-(41) together with (44)-(47) are easily seen to imply 푘1 + 푘2 − 푘3, 푖2 = 푖2 = 푘3 + 1, 푖3 = 푖3 = 푘2 + 1 and 푘1 + 1 ≦ 푘2 ≦ 2푘1. Therefor the theorem will be proved if we can show that 푛 (1, 푝 + 1, 푝 + 푞 + 1; 1, 푝 + 1, 푝 + 푞 + 1; 푝, 푝 + 푞, 2푝 + 푞) ∈ 푆3 whenever1 ≦ 푞 ≦ 푝and 2푝 + 푞 = 푛. Let 퐴, 퐵 and 퐴 + 퐵 be of order n with eigenvalues (훼푝), (훽푝) and (훾푝). We have 푞 훾푝 ≦ 훾푝 + 훾푝+1 + ⋯ + Ύ푃−푟+푖, 푞Ύ푃+푞, + ⋯ + Ύ푃+푞 ≤ 훾푝+푞 + ⋯ + 훾푝+1and 푞Ύ푃+푞 ≦ 훾2푝+푞 + ⋯ + 훾2푝+1. Hence 푞(훾푝 + Ύ푃+푞 + Ύ2푃+푞) ≦ 푡푟푎푐푒 (퐴 + 퐵) − (훾1 + ⋯ + 훾푝+푞 + 훾푝+푞+1 + ⋯ + 훾2푝) . Similarly 푞(훼1 + 훼푝+1 + 훼푝+푞+1) ≧ 푡푟푎푐푒 퐴 − (훼푞+1 + ⋯ + 훼푝 + 훼푝+2푞+1 + ⋯ + 훼2푝+푞)

10 and we have a similar statement for the 훽′s. Therefore we need only prove (푞 + 1, … , 푝, 푝 + 2푝 + 1, … , 2푝 + 푞; 푞 + 1, … , 푝, 푝 + 2푞 푛 + 1, … ,2푝 + 푞; 1, … , 푝 − 푞, 푝 + 1, … , 2푝) ∈ 푆2̅ 푝−2푞 This will follows from Theorem (1.1. 4) (ii) if we can show (1, … , 푝 − 푞, 푝 + 푞 + 1, … , 2푝; 1, , 푝 − 푞, 푝 + 푞 푛 + 1, . . . , 2 푝 ; 푞 + 푙, . . . , 푝 , 푝 + 2푞 + 푙, , 2푝 + 푞) ∈ 푆2푝−2푞 . (48) By Theorem (1.1.7) we have 푝2 (1, . . . , 2푝 − 2푞 ; 1, … , 푝 − 푞, 푝 + 1, … , 2푝) ∈ 푆2푝−2푞 2푝 푝 + 1, … , 푃 − 푞, 푃 + 1, … , 2푝 − 푞) ∈ 푆2푝−2푞 We may apply Theorem (1.1.6) q times with 푢 = 푤 = 푝 − 푞 + 1, 푣 = 2푝 − 2푞 + 1 to obtain (1, , p − q, ί > + 1, , 2푝 − 푞; 1, , 푝 − 푞, 푝2 푝 + 1, , 2푝 − 푞; 1, , 푝 − 푞, 푝 + 푞 + 1, , 2푝) ∈ 푆2푝−2푞 . Theorem (1.1.6) applied q times with 푢 = 푣 = 푝 − 푞 + 1, 휔 = 1yields (48).The proof is now complete. 푛 푛 A proof of 푇4 ⊂ 푆4 along the same lines runs into the following difficulty. The first half of the proof, that is, the application of Theorem (1.1.6) in all possible ways, carries through. However the cases left untouched turn out to be too numerous to handle by the methods of the second 푛 푛 half of the proof of Theorem (1.1.10). We have verified 푇4 ⊂ 푆4 for 푛 ≤ 8. 푛 푛 As for the statement 푆푟 ⊂ 푇푟 , it is possible to show by a consideration of diagonal 푛 matrices that if (푖; 푗; 푘) ∈ 푆푟 then (32) holds for 푠 = 1,2. This together with the remark 푛 푛 푛 following Theorem (1.1,9) determines 푆푟 . But the general statement 푆푟 ⊂ 푇푟 is false even if 푛 we weaken the definition of 푇푟 by replacing the equality sign in (31) by≦. For example a 16 consideration of the trace condition shows that (1, 5, 9, 12; 1, 5, 9, 12; 4, 8, 12, 16) ∈ 푆4 푛 Guided by Theorem (1.1.4) (ii), the dual set 푇̃푟 may be defined inductively as follows: 푛 푛 (푖1, 푗1, 푘1) ∈ 푇̃푟 if푖1 + 푗1 + 푘1 + 푛, and (푖, 푗, 푘) ∈ 푇̃푟 if 푖1 + ⋯ + 푖푟 + 푗1 + ⋯ + 푗푟 = 푘1 + ⋯ . +푘푟 + 푛푟 − 푟(푟 − 1)/2 and

푖푢1 + ⋯ + 푖푢푠 + 푗푣1 + ⋯ + 푗푣푠 ≦ 푘푤1 + ⋯ + 푘푤푠 + 푠(푠 + 1)/2 푛 푛 Wherever (푢; 푣; 푤) ∈ 푇̃푟 . It is easily seen that (푖, 푗, 푘) ∈ 푇̃푟 if and only if (푛 − 푖푟 + 1 … . , 푛 − 푛 푖1 + 1; 푛 − 푗푟 + 1 … . , 푛 − 푗1− + 1; 푛 − 푘푟 + 1, … , 푛 − 푘1 + 1) ∈ 푇̃푟 Hence by Theorem n n 푛 푛 (1.1,4), Tr ⊂ Sr is equivalent to 푇푟 ⊂ 푆̃푟 . we have been unable to prove the analogue of the 푛 last transformation rule of Theorem (1.1.4). However we can prove that if (푖; 푗; 푘) ∈ 푇̃푟 then 푛 (푖′; 푗′; 푘′) ∈ 푇푛−1, where i′, j ′ , and k' are the complements with respect to n. We return to the problem of determining the set E that being defined. Let 퐹 be the set of points γ defined by γ1 ≧ ⋯ ≧ γn, 훾1 + ⋯ + 훾푛 = 훼1 + ⋯ + 훼푛 + 훽1 + ⋯ + 훽푛 And

훾푘1 + ⋯ + 훾푘푟 ≦ 훼푖1 + ⋯ + 훼푘푟 + 훽푖 + ⋯ + 훽푖푟 Wherever

푛 (푖; 푗; 푘) ∈ 푇̃푟 , 1 ≦ 푟 ≦ 푛 − 1. We have shown that 퐸 ⊂ 퐹 for 푛 ≦ 4. We will prove that 퐸 = 퐹 for 푛 ≦ 4. We assuming 훼1 > ⋯ > 훼푛 and 훽1 > ⋯ > 훽푛. The set 퐸′ defined is a closed subset of E. Since F is closed and convex, it will follow that 퐸′ = 퐹, and therefore 퐸 = 퐹, if the boundary of 퐸′ is contained in the boundary of 퐹. To see this, let 훾′ be an interior point of 퐸′ and suppose훾′ is any point of F. If 훾′ is not in 퐸′ there must be a boundary point of 퐸′ in the open segment joining 훾 and 훾′. But all points of this open segment are interior points of 퐹. A boundary points of 퐸′ with at least two equal coordinates is obviously a boundary point of 퐹. If 훾 is a boundary point of 퐸′ with distinct coordinates, there is associated with 훾 a triple (푖; 푗; 푘)satisfying the conditions of Theorem (1.1.3). All that remains to prove is that 푛 (푖; 푗; 푘) ∈ 푇푟 To this end we first prove the following theorem. Theorem (1.1.11)[10]:If훾 is a boundary point of 퐸′ with associated sequences (푖; 푗; 푘) of order ̃푟 푛−푟 푟, then for any (푥; 푦; 푧) ∈ 푆푚, there cannot exist a triple (푢; 푣; 휔) ∈ 푇푚 such that 푖푥푝 ≦

퓍푝 + 푢푝 − 1, 푗푣푝 ≦ 푦푝 + 푣푝 − 1, and 푘푧푝 ≧ 푧푝 + 휔푝, for 1 ≦ 푝 ≦ 푚. Proof: For convenience, we write 훼(푝) instead of 훼푝. By hypothesis there exist Hermitian matrices 퐴1, 퐵1 and 퐴1 + 퐵1 with eigenvalues (훼(푖푝)) , (훽(푗푝)), and(훾(푘푝)) , 푝 = 1, 푟,and ′ ′ ′ Hermitian matrices 퐴2, 퐵2 and 퐴2 + 퐵2 with eigenvalues (훼(푖푝), (훽(푗푃)), and (훾(푘푝)) , 푝 = 1, … , 푛 − 푟, where 푖′ is complement of 푖 with respect to 푛. If there exists a triple (푢; 푣, 푤) ∈ 푛−푟 ′ ′ ′ ′ 푆푚 such that 푖푋푝 < 푖푢푝, 푗푣푝 < 푗 푣푝, and 푘푧푝 > 푘푤푝, 1 ≦ 푝 ≦ 푚, ,then we have 푚 푚 푚 푚 푚 푚 ′ ( ′ ) ( ′ ) ∑ 훼 (푖푧푝) + ∑ 훽 (푗푣푝) ≦ ∑ 훾 (푘푧푝) < ∑ 훾 (푘푤푝) ≦ ∑ 훼 푖푢 + ∑ 훽 푗푣 푝=1 푝=1 푝=1 푝=1 푝=1 푝=1 ′ ′ This is impossible since 훼 (푖푧푝) > 훼 (푖푢푝) and 훽 (푗푣푝 ) < 훽 (푗푣푝 ). Therefore it remains only ′ to show that 푖푝 < 푖푞 is implied by 푖푝 ≦ 푝 + 푞 — 1. If 푖푝 ≦ 푝 + 푞 − 1, then at least 푝 terms of the sequence 푖 are ≦ 푝 + 푞 − 1. Therefore at most 푞 − 1 positive integers ≦ 푝 + 푞 − 1 ′ are not in i. Hence 푖푝 > 푝 + 푞 − 1 ≧ 푖푝 Theorem (1.1.12)[10]:If 훾is a boundary point of 퐸′ with associated sequences 푖, 푗 , 푘 of order 푟 r, then 푖푥 + 푗푦 ≧ 푘푧 + r whenever (푥, 푦, 푧) ∈ 푇̃1 . More generally,if 퓍 + 푦 ≧ 푧 + 푟, the 푖퓍 − 퓍 + 푗푦 − 푦 ≧ 푘푧 − 푧. 푟 푟 Proof: We have 푛 ≧ 푟 + 1 ≧ 2. Since (퓍; 푦; 퓍 + 푦 − 푟) ∈ 푇̃1 ⊂ 푆̃1 , it follows that 푟 (푥; 푦; 푧) ∈ 푆̃1 . Let 푢 = 푖푥 − 푥 + 1, 푣 = 푗 푦 − 푦 + 1, and 휔 = 푘푧 − 푧. Clearly, 푢 ≧ 1, 푣 ≧ 1, and 푤 ≦ 푛 − 푟 since 푘1 − 1 ≦ 푘2 − 2 ≦ ⋯ ≦ 푘푟 − 푟 ≦ 푛 − 푟. We must prove 푢 + 푣 ≦ 푛−푟 휔 + 2. It 푢 + 푣 ≦ 휔 + 1, then 휔 ≧ 푙, 푢 ≦ 휔, and 푣 ≦ 휔. Therefore (푢; 푣; 휔 ) ∈ 푇̃1 This contradicts Theorem (1.1.11)

Theorem (1.1.13)[10]:Under the same hypothesis as Theorem (1.1.12),if n ≧ 푟 + 2, then 푖푧1 + ̃ 푟 푖푥2 + 푗푣1 + 푖푦2 ≧ 푘푧1 + 푘푧2 + 2푟 − 1 whenever (푥, 푦, 푧) ∈ 푇푧 .

Proof: We are given 푥1 + 푦2 ≧ 푧1 + 푟, 푥2 + 푦1 ≧ 푧1 + 푟, 푥2 + 푦2 ≧ 푧2 + 푟. and푥1 + 푥2 +

푦1 + 푦2 + 푧1 + 푧2 + 2푟 − 1.Let 훼푝 = 푖푥푝 − 푥푝 + 1, 푏푝 = 푖푣푝 − 푦푝 + 1. and 퐶푝 = 휔푧푝 − 푧푝, 푝 = 1.2. By Theorem (1.1.12), 훼1 + 푏2 ≧ 푐1 + 2, 훼2+≧ 푐1 + 푐2 + 3. Therefore 푎1 + 푏2 ≧ 푐1 + 1 (49) 푎1 + 푏2 ≧ 푐2 + 1 (50) 푎2 + 푏1 ≧ 푐2 + 1. (51) Also 1 ≦ 푎1 ≦ 푎2, 1 ≦ 푏1 ≦ 푏2 and 푐2 ≦ 푛 − 푟. By (49), c1 ≧ 1. Moreover 푐1 + 2 ≦ 푎1 + 푏2 ≦ 푐2 + 1, so that c1 + 1 ≦ c2. Now let 푢1 = 훼1, 푢2 =max (훼2, 훼1 + 1), 푣1 = 푏1, 푣2 = max (푏2, 푏1 + 1), 휔1 = 푐1, and 푤2 = 푐2. It is easy to see that, and 푢1 + 푣1 ≦ 휔1 + 1, 푢1 + 푣2 ≦ 휔2 + 1, 푢2 + 푣1 ≦ 휔2 + 1 and 푢1 + 푣2 + 푣1 + 푣2 ≦ 휔1 + 휔2 + 3. As previously ′ ′ ′ ′ 푛−푟 remarked there exists a pair (휔1, 휔2) such thatω1 ≦ ω1ω2 ≦ ω2, and (푢; 푣; 휔)) ∈ 푇̃푧 . This contradicts Theorem (1.1.11). Using a generalized version of Theorem (1.1.13), it is possible to show that

푖푥1 + 푖푥2 + 푗푣1 + 푗푣2 ≦ 푘푧1 + 푘푧2 + 푘푧3 + 푟 + 푟 − 1 + 푟 − 2 푛−푟 Whenever(푥; 푦; 푧) ∈ 푇̃푧 , 푛 ≧ 푟 + 2 Theorem (1.1.14)[10]: If 훾 is a boundary point of E′ with associated sequences i, j , k of order 푛 푟 = 1, 2, 3 or n − 1, then (푖; 푗; 푘) ∈ 푇̃푧 . Proof: For푟 = 1this is obvious. For 푟 = 푛 − 1, the complementary sequences with respect to ′ ′ ′ ′ 푛 푛 are of order 1 and satisfy 푖1 + 푗1 = 푘1 + 푛. Therefore (푖 ; 푗′; 푘′) ∈ 푇̃1 . By the last ′ 푛 sentence, it follows that(푖 푗; 푘) ∈ 푇̃1 . For the cases 푛 = 3,4 this can be easily verified by listing cases. Now suppose 푟 = 2. We must prove that (23) and (24) hold. In view of (25), this 2 means we must show that 푖푥 + 푗푣 ≧ 푘푧 + 2 whenever (푥; 푦; 푧) ∈ 푇̃1 . But this follows from Theorem (1.1.12). Suppose 푟 = 3. We may assume 푛 ≧ 5. By (40) and Theorems(1.1.12) 2 ′ ′ ′ 3 and(1.1.13) we have (34)-(39), since if (푥; 푦; 푧) ∈ 푇̃푝 then(푥 ; 푦 ; 푧 ) ∈ 푇3−푞, 푝 = 1, 2. Theorem (1.1.14) completes the proof that 퐸 = 퐹 for 푛 ≦ 5. It is possible to extend the proof to 푛 ≦ 8. Section (1.2) Interlacing Inequalities for Singular Values of Submatrices We give a brief summary of certain particular cases of our results that merit special attention. Let A bean 푛 × 푛 real or complex matrix, and let 훼1 ≧ 훼2 ≧ ⋯ ≧ 훼푛 be the singular values of A. (They are defined to be the eigenvalues of the positive semidefinite matrix ∗ 1/2 (퐴퐴 ) . ) Let 퐵 = 퐴푖푗 be the (푛 − 1)-square submatrix of A obtained by deleting row i and column 푗, and let 훽1 ≧ 훽2 ≧ ⋯ ≧ 훽푛−1 be the singular values of 퐵. Our first theorem yields, as a special case, these interlacing inequalities: 훼1 ≧ 훽1 ≧ 훼3 훼2 ≧ 훽2 ≧ 훼4 … 훼푡 ≧ 훽푡 ≧ 훼푡+2,1 ≦ 푡 ≦ 푛 − 2, (52) … 훼푛−2 ≧ 훽푛−2 ≧ 훼푛, 훼푛−2 ≧ 훽푛−1.

That inequalities (52) are the best that can be asserted is shown by (a special case of) Theorem (1.2.3). It follows from Theorem (1.2.3) that, if arbitrary nonnegative numbers 훽1 ≧ ⋯ ≧ 훽푛−1 are given satisfying (52), there will always exist unitary matrices 푈 and 푉such that the singular values of (푈퐴 푉)푖푗 are 훽1, … , 훽푛−1. (Of course, 퐴 and 푈퐴 푉 always have the same singular values 훼1, … , 훼푛.)Thus nothing more than (52) can hold in general, when looking at a fixed submatrix. Further results can be obtained, however, by examining all the submatrices of 퐴 of fixed degree. Now let 훽푖푗,1 ≧ ⋯ 훽푖푗,푛−1denote the singular values of 퐴푖푗.We obtain the following estimates on the mean square of the tth singular value of all the(푛 − 1)-square submatrices 퐴푖푗of 퐴: 푛 1 2 2(푛 − 1) 푛 − 1 2 1 ( ) 훼2 + 훼2 + ( ) 훼2 ≦ ∑ 훽2 2 푡 푛2 푡 2 푡+2 푛2 푖푗,푡 푖,푗=1 푛 − 1 2 2(푛 − 1) 1 2 ≦ ( ) 훼2 + 훼2 + ( ) 훼2 , 2 푡 푛2 푡+1 2 푡+2 1 ≦ 푡 ≦ 푛 − 2, (53) 푛 1 2 푛 − 1 1 푛 − 1 2 푛 − 1 1 ( ) 훼2 + 훼2 ≦ ∑ 훽2 ≦ ( ) 훼2 + 훼2 (54) 2 푛−1 푛2 푛 푛 푖푗,푛−1 2 푛−1 푛2 푛 푛. 푖,푗=1 2 2 2 In (53) we have displayed convex combinations of 훼푡 , 훼푡+1, 훼푡+2 which serve as upper and lower bounds for the mean square of the tth singular value (푡 ≦ 푛 − 2) of the different (푛 − 2 2 1)-square submatrices 퐴푖푗 of A. (By (52), this mean lies between 훼푡 and 훼푡+2.) In (54), we 2 2 have similar, though not precisely the same, convex combinations of 훼푛−1,, 훼푛, and 0 yielding bounds for the mean square of the 훽푖푗,푛−푙. These results, (53) and (54), will appear as special cases of Theorem (1.2.5). Let 푓푖푗(휆) = (휆 − 훽푖푗,) … (휆 − 훽푖푗,푛−푙) (55) be the singular value polynomial of퐴푖푗. This is the polynomial whose roots are the squares of the singular values of Aij. Let 2 2 푓(휆) = (휆 − 훼1 ) … (휆 − 훼푛) (56) be the corresponding polynomial for A. As a particular instance of Theorem (1.2.6), we obtain 푛 푛 푑 휆푑 ∑ ∑ 푓 (휆) = 휆 푓(휆) (57) 푖푗 푑휆 푑휆 푖=1 푗=1 It is interesting to contrast formula (57) with the well-known result asserting that the sum of the characteristic polynomials of all the principal (푛 − 1) − quare submatrices of 퐴 is just the derivative of the characteristic polynomial of 퐴. We give first the definition of the singular values of a rectangular matrix. Definition (1.2.1)[238]: Let 퐴 be an m x n matrix. The singular values 훼1 ≧ 훼2 ≧ ⋯ ≧ 훼푚푖푛 (푚,푛) (58) of A are the common eigenvalues of the positive semidefinite matrices (퐴퐴∗)1/2 and (퐴∗퐴)1/2.

Since AA* is m-square and A*A is n-square, the eigenvalues of (퐴퐴∗)1/2and ((퐴퐴∗)1/2do not coincide in full. However, it is well known that the nonzero eigenvalues (including multiplicities) of these two matrices always coincide. It is frequently convenient to define 푥푡 2 2 to be zero for 푚푖푛(푚, 푛) < 푡 ≦ 푚푎푥(푚, 푛). Then 훼1 ≧ ⋯ ≧ 훼푚푎푥 (푚,푛) and the roots of AA* (respectively A*A) are the first m (respectively n) of these numbers. We are the following Theorem (1.2.2)[238]:Let 퐴 be an 푚 × 푛 matrix with singular values (58). Let 퐵 be a 푝 × 푞 submatrix of 퐴, with singular values 훽1 ≧ 훽2 ≧ ⋯ ≧ 훽푚푖푛 (푝,푞)(59) Then 훼푖 ≧ 훽푖, 푓표푟 푖 = 1, 2, … . , 푚푖푛(푝, 푞). (60) 훽푖, ≧ 훼푖+(푚−푝)+(푛−푞), 푓표푟 푖 ≦ 푚푖푛(푝 + 푞 − 푚, 푝 + 푞 − 푛). (61) Proof: For an arbitrary matrix 푀, let 푀[푖1, … , 푖푝: 푗1, … , 푗푝]denote the submatrix of 푀 lying at the intersection of rows 푖1, . . , 푖푝, and columns 푗1, … , 푗푞. Suppose that 퐵 = 퐴 [푖1, … , 푖푝: 푗1, … , 푗푞]. To simplify notation let 휔 = {푖1, … , 푖푝}and 휏 = {푗1, … , 푗푞} denote the sets of integers giving the rows and columns of 퐴 used to form 퐵, and denote 퐵 by 퐵 = 퐴 [휔, 휏]. Let us view 퐵 as a submatrix of 푈퐴푉, where 푈 is an 푚-square unitary matrix and 푉 is an 푛- square unitary matrix. In this proof we may take 푈 = 퐼푚, and 푉 = 퐼푛,. (In the next theorem, 푈 and 푉 will become variable.) Then 퐵 = 푈[푖1, … , 푖푝: 1, … , 푚] 퐴푉[1, … , 푛: 푗1, … , 푗푞]. (62) Thus ∗ 퐵퐵 = 푈[푖1, … , 푖푝: 1, … , 푚] 퐴푉[1, … , 푛: 푗1, … , 푗푞]. (63) Where 푋 = 퐴푉[1, … , 푛: 푗1, … , 푗푞] . (64) is 푚 × 푞. Thus 퐵퐵∗ is a principal 푝 −square submatrix of the m-square Hermitian matrix 푈푋푋∗푈∗. Let 2 2 2 2 2 퓍1 ≧ 퓍2 ≧ ⋯ ≧ 퓍푚푖푛(푚,푞) ≧ 퓍푚푖푛(푚,푞)+1 = ⋯ 퓍푚 = 0, (65) ∗ denote the eigenvalues of 푋푋 . Thus 퓍1, … . , 퓍min(m,q) are the singular values of X. From the well-known formulas linking the eigenvalues of a Hermitian matrix with the eigenvalues of a principal submatrix, we obtain 2 2 2 퓍푖 ≧ 훽푖 ≧ 퓍푖+푚−푝, 푓표푟 푖 = 1, 2, … , 푝. (66) 2 2 Now 퓍i , … , 퓍min(m,q), 0(q − min(m, q) times) are the eigenvalues of ∗ ∗ ∗ 푋 푋 = 푉 [푗1, … , 푗푞: 1, … , 푛]퐴퐴 푉[1, … , 푛: 푗1, … , 푗푞] (67) Thus 푋∗푋 is a principal q-square submatrix of the 푛 −square Hermitian matrix V∗A∗A V. Hence 2 2 2 훼푖 ≧ 퓍푖 ≧ 훼 푖+푛−푞 for 푖 = 1,2, … , 푝. (68) 2 2 2 Thus for 푖 ≦ 푚푖푛(푝, 푞) we have 훼푖 ≧ 퓍푖 ≧ 훽푖 , yielding (60). And for 푖 ≦ 푚푖푛 (푝 + 푞 − 2 2 2 푚, 푞 + 푞 − 푛) we have 훽푖 ≧ 훽푖+푚−푝 ≧ 훼푖+(푛−푞)+(푚−푝) yielding (61).

The proof of Theorem (1.2.1) is now complete. We shall present a second proof of Theorem (1.2.1) at the later. Theorem (1.2.3)[238]:Let 퐴 be an 푚 × 푛 matrix with singular values (58). Let arbitrary nonnegative numbers (59) be given, satisfying both (60) and (61). Then 푚-square unitary matrix 푈 and 푛-square unitary matrix 푉 exist such that the singular values of the 푝 × 푞 submatrix (푈퐴퐶)[푖1, … , 푖푝] 퐴푉[푗1, … , 푗푞] of 푈퐴푉 are the numbers (59). Proof: Define 훽푖 to be zero if 푖 > 푚푖푛(푝, 푞), and define 훼푖 to be zero if i > 푚푖푛(푚, 푛). Now define inductively nonnegative numbers 푥1, . . . , 푋푛푖푛(푚,푞) by 훼1 푥1 = 푚푖푛 { if 푚 − 푝 < 1 (69) 훽1−푚+푝 And 훼1 푥푖 = 푚푖푛 {훽푖−푚+푝 if 푚 − 푝 < 푖 푥푖−1 For 2 ≤ 푖 ≤ 푚푖푛(푚, 푞). (70) (We include 훽푖−푚+푝in (69) and (70) only if i satisfies the indicated condition.) For all 푡 > 푚푖푛(푚, 푞), define 푥푡by 푥푡 = 0. It is plain that x1 ≧ ⋯ ≧ xmin (m,q). We claim that inequalities (68) are satisfied. Plainly, xi ≦ αifor 푖 ≦ 푚푖푛(푚, 푞), and this also holds for min(푚, 푞) < 푖 ≦ 푞 since then 푥푖 = 0. We show by induction on 푖 that the lower bounds in (68) are satisfied. To show that 푥1 ≧ 훼1+푛−푞,, we must show that both of the quantities entering into the minimum in (68) exceed α1+n−q,. Plainly, by (58), 훼1, ≧ 훼1+푛−푞,. If 푚 − 푝 < 푖 we obtain from (thus 푚 = 푝), (61) tells us that 훽1 ≧ 훼1+푛−푞,, provided 푖 ≦ 푚푖푛(푞, 푚 + 푞 − 푛).However,if 푚 + 푞 − 푛 ≦0, we have 푚 + 1 ≦ 1 + 푛 − 푞 and thus automatically0 = 훼1+푛−푞, ≦ 훽1.Hence 푚 + 1 ≦ 1 + 푛 − 푞 . Suppose (induction) 푥푖−1 ≧ 훼푖−1+푛−푞,. Let 푖 ≦ 푚푖푛(푚, 푞). If we show that each of the three quantities entering into the minimum in (70) exceeds 훼푖+푛−푞,, it will follow that 푥푖 ≧ 훼푖−1+푛−푞, Plainly, by (58), 훼푖 ≧ 훼푖−1+푛−푞,. If 푚 − 푝 < 푖, we obtain from (61) that 훽푖−푚+푝 ≧ 훼푖−1+푛−푞, provided 푖 ≦ 푚푖푛(푞, 푚 + 푞 − 푛).By induction, 푥푖 ≧ 훼푖−1+푛−푞, ≧ 훼푖+푛−푞,(by (58)).Thus 푥푖 ≧ 훼푖−1+푛−푞,, except perhaps if 푖 > 푚푖푛(푞, 푚 + 푞 − 푛). However, if 푖 > 푚푖푛(푞, 푚 + 푞 − 푛), then 푖 + 푛 − 푞 > 푚푖푛(푛, 푚), so that 훼푖+푛−푞, = 0 and hence automatically 푥푖 ≧ 훼푖−1+푛−푞,.Therefore 푥푖 ≧ 훼푖+푛−푞,is established if 푖 ≦ 푚푖푛(푚, 푞).If 푖 > 푚푖푛(푚, 푞), then 푖 + 푛 − 푞 > 푚푖푛(푛 − 푞 + 푚, 푛) ≧ 푚푖푛(푚, 푛), so that automatically 0 = αi+n−q, ≦ xi. Therefore inequalities (68) are established. We now claim that inequalities (66) are satisfied. By (70), 푥푖+푚−푝 ≦ 훽푖,, for 푖 + 푚 − 푝 ≤ 푚푖푛(푚, 푞). Thus the lower inequality in (66) is satisfied, provided 푖 ≦ 푚푖푛(푝, 푝 + 푞 − 푚). If 푖 > 푚푖푛(푝, 푝 + 푞 − 푚), then 푖 + 푚 − 푝 > 푚푖푛(푚, 푞)and hence 푥푖+푚−푝 = 0, so that automatically 훽푖, ≧ 푥푖+푚−푝. Thus the lower inequalities in (66) are satisfied. We show by induction on 푖 that 푥푖 ≧ 훽푖,. For 푖 = 1 this follows immediately from (69), since α1 ≧ βi,.

Suppose 푥푖−1 ≧ 훽푖−1. If we show that each of the three quantities entering into the minimum in (70) exceeds 훽푖, we may conclude that 푥푖 ≧ 훽푖. We may assume also that 푖 ≦ 푚푖푛(푝, 푞), since 훽푖 = 0 (≦ 푥1) for 푖 > 푚푖푛(푝, 푞).Thus (by (60)) 훼푖 ≧ 훽푖 if 푚 − 푝 < 푖, 훽푖−푚+푝, ≧ 훽푖,by (59). By induction 푥푖−1 ≧ 훽푖−1 ≧ 훽푖. Hence the inequality 푥푖 ≧ 훽푖 for all 푖 ≦ 푝 is established. 2 2 It is a known fact (see [231]), because 푥1 ≧ ⋯ ≧ 푥푞 satisfy (68), there exists an n-square unitary matrix V such that the eigenvalues of ∗ ∗ 푋 푋 = 푉[푗1, … , 푗푞|1, … , 푛]퐴 퐴푉[1, … , 푛|푗1, … , 푗푞] (71) are 2 2 2 푥1 , … , 푥푚푖푛 (푚,푞), … , 푥푞 (72) Here

푋 = 퐴푉[1, … , 푛|푗1, … , 푗푞] Thus 푋푋∗ has 2 2 2 푥1 , … , 푥푚푖푛 (푚,푞), … , 푥푚 (73) as eigenvalues. Because the inequalities (66) are satisfied, there exists an m-square unitary matrix U such that ∗ ∗ 푈푋푋 푈 [푖1, … , 푖푝|푖1, … , 푖푞] 2 2 2 has eigenvalues 훽1 ≧ 훽푚푖푛 (푝,푞) ≧ ⋯ = 훽푝 . It is now immediate that the submatrix 푈[푖1, … , 푖푝|1, … , 푚]퐴푉[1, … , 푛|푗1, … , 푗푞] of 푈퐴푉 has (59) as its singular values. The proof of Theorem (1.2.3) is now finished. We remark that the nonincreasing condition (59) is actually superfluous. We have Theorem (1.2.4). Theorem (1.2.4)[238]:Let arbitrary numbers β1 … . βmin (p,q)be given, such that (60) and (61) hold. Then the conclusions of Theorem (1.2.3) aye valid. Proof: The proof amounts to showing that, if (60) and (61) are valid for not necessarily decreasing numbers 훽1 … . 훽푚푖푛 (푝,푞)then (60) and (61) remain valid if 훽1 … . 훽푚푖푛 (푝,푞)are rearranged into decreasing order. More precisely, let 휎 be a permutation of 1, 2,…, 푠 = 푚푖푛(푝, 푞) such that 훽휎(1) ≧ 훽휎(2) ≧ ⋯ ≧ 훽휎(푠). If 휎(푖) ≧ 푖we then have 훽휎(푖) ≦ 훼휎(푖) ≦ 훼푖 if σ(푖) < 푖 then for some j < 푖 we have σ(푗) ≧ 푖 and hance 훽휎(푖) ≦ 훽휎(푗) ≦ 훼휎(푖). thus 훽휎(푖) ≦ 훼푖holds for all 푖. similarly, for 푖 ≦ 푚푖푛(푝 + 푞 − 푚, 푝 + 푞 − 푛),if 휎(푖) ≦ 푖then 훽휎(푖) ≧ 훽푖 ≧ 훼푖+푚−푝+푛−푞. if σ(푖) > 푖, then for some 푗 > 푖 we have σ(푗) ≦ 푖 But then 훽휎(푖) ≧ 훽휎(푗) ≧ 훼휎(푗)+푚−푝+푛−푞 ≧ 훼푖+푚−푝+푛−푞 thus 훽휎(푗) ≧ 훼푖+푚−푝+푛−푞 For all 푖 ≦ 푚푖푛(푝 + 푞 − 푚, 푝 + 푞 − 푛). 푚 For the next theorems we let 푄푚푝 denote the totality of ( 푝 )sequences 휔 = {푖1, … , 푖푝} of integers for which 1 ≦ 푖1 < ⋯ < 푖푝 ≦ 푚and we let Qna denote the totality of sequences푟 = {푗1, … , 푗푝} of integers for which 1 ≦ 푗1 < ⋯ < 푗푝 ≦ 푛. We let 퐴 [휔|휏] = 퐴[푖1, … , 푖푝|푗1, … , 푗푝] (74) be the 푝 × 푞 submatrix of 퐴 at the intersection of the rows w and the olumns 휏, and we let 훽휔휏.1 ≧ 훽휔휏.2 ≧ ⋯ ≧ 훽휔휏.푛푖푚(푝,푞) 17 be the singular values of (74). As before, we let 훽휔휏.푡 = 0for 푡 > 푚푖푛(푝, 푞). Theorem (1.2.5)[238]:Define rational numbers 휑0, … , 휑푚−푝, and 휓0, … , 휓푛−푞by the polynomial identities 푚−1 푚−푝 휆 + 푖 ∏ ( ) = ∑ 휑 휆푚−푝−푡 (75) 1 + 푖 푡 푖=푝 푡=0 푛−1 푛−푝 휆 + 푖 ∏ ( ) = ∑ 휓 휆푛−푝−푡 (76) 1 + 푖 푡 푖=푞 푡=0 For 푖 ≦ 푚푖푛(푝, 푞), define rational numbers 푑0, … . , 푑푚푖푛(푚+푛−푝−푞,푛−푖), and ′ ′ 푑0, … . , 푑푚푖푛(푚+푛−푝−푞,푛−푖) (depending on 푖) by the polynomial identities min(푚−푝,푞−푖) 푛−푞 푚푖푛(푚+푛−푝−푞,푛−푖) 푛−푝−푟 푛−푞−푠 푚+푛−푝−푞−휌 ( ∑ 휑푡휆 ) (∑ 휓푠휆 ) = ∑ 푑휌휆 (77) 푟=0 푠=0 휌=0 and 푚푖푛 (푚−푝,푞−푖) 푛−푞 푛−푝−푟 푛−푞−푠 ( ∑ 휑푚−푝−푟휆 ) (∑ 휓푛−푞−푠휆 ) 푟=0 푠=0 푚푖푛(푚+푛−푝−푞,푛−푖) ′ 푚+푛−푝−푞−휌 = ∑ 푑휌휆 (78) 휌=0 then 푚푖푛(푚+푛−푝−푞,푛−푖) 푚푖푛(푚+푛−푝−푞,푛−푖) 2 1 1 2 ′ 2 ∑ 푑휌훼푖+휌 ≦ 푚 푛 ∑ 훽휔휏,푖 ≦ ∑ 푑휌훼푖+휌 (79) ( ) ( ) 휌=0 푝 푞 휔∈푄푚푝휏∈푄푛푞 휌=0 Proof: Let X휏 = AV[1, … , n: j1, … , jq] and let 2 2 2 2 퓍τ,1 ≧ 퓍τ,2 ≧ ⋯ ≧ 퓍τ,min(p,q)+1 = ⋯ = 퓍τ,m(= 0) ∗ be the roots of 푋휏푋휏 ,. Then by (66) we have 2 2 2 퓍휏,1 ≧ 훽휔휏,푖 ≧ 퓍휏,푖+푚−푝 1 ≦ 푖 ≦ 푝. Using [231], we see that, for 푖 ≦ 푝 (and so for 푖 ≦ 푚푖푛(푝, 푞)), 푚−푝 푚−푝 2 1 2 2 ∑ 휑휏푥푟,푖+푟 ≦ 푚 ∑ 훽휔휏,푖 ≦ ∑ 휑푚−푝−푟 푥푟,푖+푟. 휏=0 ( 푝 ) 휔∈푄휏∈푄 푟=0 Since 푥휏,푖+푟 whenever 푖 + 푟 > 푚푖푛(푚, 푞), we get 푛−푝 푛−푞 2 1 2 2 ∑ 휓푠훼푖+푠 ≦ 푛 ∑ 푥휏,푖 ≦ ∑ 휓푛−푞−푠 푥휏,푖+푟. (80) ( ) 푠=0 푝 휔∈푄푛푞 푠=0 2 2 By (68), 훼푖 ≧ 퓍휏,푖 ≧ 훼푖+푛−푞 for 1 ≦ 푖 ≦ 푞, and hence, by [231,304],

푛−푞 푛−푞 2 1 2 2 ∑ 휓푠훼푖+푠 ≦ 푛 ∑ 푥푟,푖 ≦ ∑ 휓푛−푞−푠 푥푖+푠, (81) ( ) 푠=0 푞 휔∈푄푛푞 푠=0 For 푖 ≦ 푞. 푛 Summing (80) over -c and dividing by (푞), , upon using (81) we obtain 푚푖푛 (푚−푝,푞−푖) 푛−푞 2 1 1 2 ∑ 휑휏 ∑ 휓푠훼푖+푟+푠 ≦ 푚 푛 ∑ 훽휔휏,푖 ( ) ( ) 휏=0 푠=0 푝 푞 휔∈푄푚푝휏∈푄푛푞 min(푚−푝,푞−푖) 푛−푞 2 ≦ ∑ 휑푚−푝−휏 ∑ 휓푠푛−푝−푠훼푖+휏+푠 (82) 휏=0 푠=0 For 푖 ≦ 푚푖푛(푝, 푞). 2 On the left side of (82), the coefficient of 훼푖+푝, is 푚푖푛 (푚−푝,푞−푖) 푛−푞

∑ ∑ 휑푟휓푠 푟=0,푟+푠=휌 푠=0,푟+푠=휌 For 0 ≦ 휌 ≦ 푚푖푛(푚 + 푛 − 푝 − 푞, 푛 − 푖). However, 푚푖푛 (푚−푝,푞−푖) 푛−푞

푑휌 = ∑ ∑ 휑푟휓푠 푟=0,푟+푠=휌 푠=0,푟+푠=휌 for 0 ≦ 휌 ≦ 푚푖푛(푚 + 푛 − 푝 − 푞, 푛 − 푖). Thus the lower bound in (79) is established. 2 On the right side of (82), the coefficient of αi+p, is min (m−p,q−i) n−q

∑ ∑ φm−p−τψn−q−s τ=0,τ+s=ρ s=0,τ+s=ρ for0 ≦ 휌 ≦ 푚푖푛(푚 + 푛 − 푝 − 푞, 푛 − 푖). However, 푚푖푛 (푚−푝,푞−푖) 푛−푞 ′ 푑휌 = ∑ ∑ 휑푚−푝−휏휓푛−푞−푠 휏=0,휏+푠=휌 푠=0,푟+푠=휌 for0 ≦ 휌 ≦ 푚푖푛(푚 + 푛 − 푝 − 푞, 푛 − 푖). The result is now at hand. If 푝 and 푞 are large and i is small, so that min(푚 + 푛 − 푝 − 푞. 푛 − 푖) = 푚 + 푛 − 푝 − 푞, 2 2 formulas (79) provide convex combinations of 훼푖 , … , 훼푖+푚−푝−푛−푞. which serve as upper and 2 lower bounds for the mean of the βωτ,i Thus Theorem (1.2.5) provides a result sharper than can be established by applying Theorem (1.2.2), since Theorem (1.2.2) only asserts that the

2 2 2 훽휔휏,푖between αi and 훼푖+푚−푝−푛−푞.. To see that in fact we have convex combinations, notice that (for these values of 푝, 푞, 푖), 푚+푛−푝−푞 푚+푛−푝−푞 푚−푞 푛−푞 푚−푝 푛−푞

∑ 푑휌 = ∑ ∑ ∑ 휑휏휓푠 = ∑ ∑ 휓푠 = 1, 휌=0 휌=0 푠=0,휏+푠=휌 푠=0,푟+푠=휌 푠=0 푠=0 since 푚−푝 푛−푞

∑ 휑휏 = 1 = ∑ 휓푠. 휏=0 푠=0 Similarly 푚−푛−푝−푞 푚−푛−푝−푞 푚−푝 푛−푞 ′ ∑ 푑휌 = ∑ ∑ ∑ 휑푚−푝−푟휓푛−푞−푠 휌=0 휌=0 푟=0,푟+푠=휌 푠=0,푟+푠=휌 푚−푝 푛−푞

= ∑ ∑ 휑푚−푝−휏휓푛−푞−푠 = 1 휏=0,휏+푠=휌 푠=0,휏+푠=휌 When p and q are small and 푖 large, so that 푚푖푛(푚 − 푛 − 푝 − 푞, 푛 − 푖) = 푛 − 푖, formula (79) 2 2 may be regarded as providing subconvex combinations of αi , … , αn. (convex combinations of 2 2 2 훼푖 , … , 훼푛., 0) which serve as bounds for the mean of the 훽휔휏,푖. Theroem(1.2.6)[238]:. Let

푓휔,휏(휆) = (휆 − 훽휔,휏,푖) … (휆 − 훽휔,휏,푚푖푛 (푝,푞)), 2 2 푓휔,휏(휆) = (휆 − 훼1.) … (휆 − 훼푚푖푛 (푚,푛).) Then 푝−푚푖푛 (푝,푞) ∑ 휆 푓휔,휏 (휆)

휔∈푄푚푝휏∈푄푛푞 1 1 푑푚−푝 푑푛−푞 = 휆푚−푞 휆푛−푚푖푛 (푝,푞)푓(휆) (83) (푚 − 푞)! (푛 − 푞)! 푑휆푚−푞 푑휆푛−푞 ∗ ∗ Proof: Since the matrices 훽휔,휏훽휔,휏 are 푝 × 푝 principal submatrices of the 푚 × 푚 matrix 푋휏푋휏 , we find (see [ 231,33]) that ∗ 2 푚−푚푖푛(푚,푞). ∑ (휆 − 훽휔휏.1) … (휆 − 훽휔휏,푚푖푛 (푝,푞))휆

휔∈푄푚푝휏∈푄푛푞 1 푑푛−푝 = (휆 − 훼2) … (휆 − 훼2 )휆푚−푚푖푛(푚,푛). (푛 − 푞)! 푑휆푛−푞 1 푚푖푛(푚,푞) ∗ Since XτXτ, is a principal 푞 × 푞 submatrix of the 푛 푥 푛 matrix 퐴*퐴, we have 2 2 휌−푚푖푛(푚,푞). ∑ (휆 − 푥휏.1) … (휆 − 푥휏,푚푖푛 (푝,푞))휆

휏∈푄푛푞 1 푑푛−푝 = (휆 − 훼2) … (휆 − 훼2 )휆푛−푚푖푛(푚,푛). (푛 − 푞)! 푑휆푛−푞 1 푚푖푛(푚,푛) Thus

휌−푚푖푛(푝,푞). ∑ 푓휔,휏(휆)휆

휏∈푄푛푞 1 푑푚−푝 = 휆푚−휌 ∑ (휆 − 푥2 ) … (휆 − 푥2 )휆휌−푚푖푛(푚,푞). (푛 − 푞)! 푑휆푚−푞 휏.1 휏,푚푖푛(푚,푞) 휏∈푄푛푞 1 1 푑푚−푝 푑푛−푝 = 휆푚−휌 휆푚−푚푖푛(푚,푛)푓(휆). (푚 − 푞)! (푛 − 푞)! 푑휆푛−푞 푑휆푛−푞 The proof is complete. We now give the promised second proof of Theorem (1.2.2). For any 푚 × 푛 matrix 퐴 with singular values 훼1 ≧ ⋯ ≧ 훼푚푖푛 (푚,푛)the roots of the (푚 + 푛) −square Hermitian matrix 0 퐴 푀 = [ ] 퐴∗ 0 Are ±α1, … , ±αmin (m,n),0 (with multiplicity 푚 + 푛 − 2 min (푚, 푛)). to see this, observe that −1 1푚 휆 퐴 휆퐼푛 −퐴 d푒푡(휆1푚+푛 − 푀) = 푑푒푡 [ ] 푑푒 [ ∗ ] 0 퐼푛 −퐴 휆퐼푛 −1 ∗ 휆퐼푚 − 휆 퐴퐴 0 = 푑푒푡 [ ∗ ] −퐴 휆퐼푛 푛 −1 ∗ 푛−푚 2 ∗ = 휆 푑푒푡(휆1푚 − 휆 퐴퐴 ) = 휆 푑푒푡(휆 1푚 − 퐴퐴 ) The principle (푝 + 푞)square submatrix of 푀, obtained by deleting all rows and columns except rows and columns 푖1, . . . , 푖푝, 푚 + 푗1, … , 푚+푗푞, is 0 A[i1, … , i푝: j1, … , jq] [ ∗ ]. A[i1, … , i푝: j1, … , jq] 0 Using the inequalities connecting the eigenvalues of a (푝 + 푞) −principal submatrix of Hermitian matrix 푀 with the eigenvalues of 푀, we obtain the inequalities (60) and (61).

Chapter 2 Interlacing Inequalities for Singular Values of Submatrices We provide a complete noncommutative analog of the famous cycle of theorems characterizing the function theoretic generalizations of 퐻∞. A sample of our other results: we prove a Kaplansky density result for a large class of these algebras, and give a necessary condition for when every completely contractive homomorphism on a unital subalgebra of a C∗-algebra possesses a unique completely positive extension. As an application, we solve the longstanding open problem concerning the noncommutative generalization, to Arveson’s noncommutative 퐻푝 spaces, of the famous ‘outer factorization’ of functions f with log |f| integrable. Using the Fuglede-Kadison determinant, we also generalize many other classical results concerning outer functions. Section (2.1) Unique Extensions Function algebras are subalgebras of 퐶(퐾)-spaces, or equivalently, subalgebras of commutative 퐶∗ −algebras. Thus function algebras are examples of operator algebras (subalgebras of general 퐶∗-algebras). Much work has been done to transfer results or perspectives from function theory to operator algebraic settings. One such setting, is the theory of noncommutative 퐻푝 spaces associated with Arveson’s maximal subdiagonal subalgebras of finite von Neumann algebras. Many of the central results from abstract analytic function theory, and in particular much of the classical generalized Hp function theory, may be generalized almost verbatim to subdiagonal algebras. The proofs in the noncommutative case however, while often modeled loosely on the ‘commutative’ arguments of Helson and Lowdenslager [100,13,79,32] and others, usually require substantial input from the theory of von Neumann algebras and noncommutative 퐿푝-spaces, see [300,177, 153, 176, 279, 163, 58, 56]. In fact in many cases like Szegö’s theorem – completely new proofs have had to be invented. We tackle what appears to us to be the main ‘classical’ results which have resisted generalization to date, namely those referred to in the generalized function theory literature from the 1960’s as, respectively, the 퐹. and 푀. Riesz, Gleason and Whitney, Szegö Lp, and Kolmogorov, theorems. We the following statement: essentially all of the generalized 퐻푝 function theory as summarized in [282] for example, extends further to the setting of subdiagonal algebras. In Arveson’s setting, we have a weak*-closed unital subalgebra 퐴 of a von Neumann algebra 푀 possessing a faithful normal tracial state τ, such that if Φ is the unique conditional expectation from M onto 풟 = 퐴 ∩ 퐴∗ satisfying 휏 = 휏 ∘ 훷, then 훷 is a homomorphism on 퐴. Take note that here 퐴∗ denotes the set {푎: 푎∗ ∈ 퐴} and not the Banach dual of 퐴. For the sake of clarity we will write 푋∗ for the Banach dual of a normed space 푋. We say that a subalgebra A of the type described above is a tracial subalgebra of 푀. If in addition 퐴 + 퐴∗ is weak* dense in 푀 then we say that A is maximal subdiagonal (see [300, 240]). A large number of very interesting examples of these objects were given by Arveson [300], and others (see e.g. [161, 178]). If 풟 is one dimensional we say that 퐴 is antisymmetric; if further 푀 is commutative then 퐴 is called a weak* Dirichlet algebra [282]. For antisymmetric maximal subdiagonal algebras, many of the ‘commutative’ proofs from [282] require almost no change at all. It is worth saying that classical notions of ‘analyticity’ correspond in some very vague sense to the 22 case that 풟 is ‘small’. Indeed if 퐴 = 푀 then 풟 = 푀 and 훷 is the identity map, so that the theory essentially collapses to the theory of finite von Neumann algebras, which clearly is far removed from classical concepts of ‘analyticity. Indeed for our 퐹. and 푀. Riesz theorem to hold, we show that it is necessary and sufficient for 풟 to be finite dimensional. Because of this, in our several applications of this theorem we assume dim(풟) < ∞. A subsidiary theme is ‘unique extensions’ of maps on A. We begin with some results on this topic. Recall from [58] that a subalgebra A of M has the unique normal state extension property if there is a unique normal state on M extending τ|A. If, on the other hand, for every state ω of M with 휔 ∘ 훷 = 휔 on A, we always have that 휔 ∘ 훷 = 휔 on M, then we say that A has the 훷 -state property. The major unresolved question in [58] was whether a tracial subalgebra with the unique normal state extension property is maximal subdiagonal. We make what we feel is substantial progress on this question. In particular, we show that the question is equivalent to the question of whether every tracial subalgebra with the 훷-state property is maximal subdiagonal, and equivalent to whether every tracial subalgebra satisfying a certain variant of the well known ‘factorization’ property actually has ‘factorization’.We also give an interesting necessary condition for when completely contractive homomorphisms possess a unique completely positive extension. Our unique extension results play a role in the proof of our 퐹. and 푀. Riesz theorem, and are the primary thrust of the Gleason-Whitney theorem. We prove our Szegö퐿푝 formula, and generalized Kolmogorov theorem. The first noncommutative 퐹. and 푀. Riesz theorem for subdiagonal algebras was the pretty theorem of Exel in [241]. This result assumes norm density of 퐴 + 퐴∗, and antisymmetry. (We are aware of the 퐹. and 푀. Riesz theorem of Arveson [301,9,54] and Zsido’s extension there of [161,259,4,8,85,160,17,284], but this result is quite distinct from the ones discussed above.) Although some of the steps of our proof parallel those of [241,204,99,19,12], the arguments are for the most part quite different. Indeed generally the proofs will be modeled on the classical ones, but do however require some rather delicate additional machinery. We remark that there are other, more recent, noncommutative variants of 퐻∞ besides the subdiagonal algebras see e.g. [90,5]. Here too one finds noncommutative generalizations of classical 퐻푝-theoretic results, such as the Szegö infimum theorem, these variants are in general quite unrelated, with only a formal correspondence. For a functional 휔 ∈ 푀⋆, we will need to compare the property 휔 = 휔 ∘ 훷 on A, with the property 휔 = 휔 ∘ 훷 on M. On this topic we begin with the following remarks. It is easy to see, since 훷 is idempotent, that 휔 = 휔 ∘ 훷 on 퐴 iff 퐴0 ⊂ 퐾푒푟(휔). Here, 퐴0 = 퐴 ∩ 퐾푒푟(휔), a closed two-sided ideal in A. For normal functionals one can say more, although this will not play an important role for us. If 푓 ∈ 퐿1(푀) let 휔푓 = 휏(푓 ·). From the last paragraph, 휔푓 = 휔푓 ∘ 훷 on A ff 휏(푓퐴0) = (0). On the other hand, 휔푓 = 휔푓 ∘ 훷on 푀 iff 휏(푓푎) = 휏(푓훷(푎)) = 휏(훷(푓)푎) for all 푎 ∈ 푀 iff 푓 = 훷(푓) iff 푓 ∈ 퐿1(풟). Proposition (2.1.1)[61]: If 퐴 is a tracial subalgebra of 푀 then the unique normal state extension property is equivalent to the following property: whenever 휔 is a normal state of 푀 satisfying 휔 = 휔 ∘ 훷 on 퐴, then 휔 = 휔 ∘ 훷 on 푀.

Proof:Suppose that A has the unique normal state extension property, and suppose that 휔 is a 1 normal state of 푀 satisfying 휔 = 휔 ∘ 훷 on 퐴. If 휔 = 휏(푓 ·), where 푓 ∈ 퐿 (푀)+, then by the 1 remarks preceding Proposition (2.1.1) we have that 휏(푓퐴0) = (0). Hence 푓 ∈ 퐿 (퐷) by [58,307,11]. Hence 휔 = 휔 ∘ 훷 on 푀. 1 For the converse, note that if 푔 ∈ 퐿 (푀)+with휏 = 휏(푔 · ) on 퐴, then since 휏 = 휏 ∘ 훷, we have that 휏(푔. ) = 휏(푔 · ) ∘ 훷 on A, and hence that 휏(푔. ) = 휏(푔. ) ∘ 훷 on M. By the remarks 1 above, 푔 ∈ 퐿 (풟)+. But then the fact that 휏 = 휏(푔 ·) on D is enough to force 푔 = 핀. So A has the unique normal state extension property. We say that a subalgebra 퐴 of 푀 has factorization if given 푏 ∈ 푀+ ∩ 푀−1 we can find 푎 ∈ 퐴−1 with 푏 = 푎∗푎(or equivalently 푏 = 푎푎∗). It is shown in [300,310,18] that any maximal subdiagonal algebra has factorization. Thus it is logmodular, namely any such b is a limit of terms of the form 푎∗푎 with 푎 ∈ 퐴−1. In fact, in the category of tracial algebras factorization or logmodularity are equivalent to maximal subdiagonality [58]. By the next result such algebras satisfy a formally much stronger property than that of the last proposition: Theorem (2.1.2)[61]: Let 퐴 be a logmodular subalgebra of a C∗-algebra 푀, and let 휓 be a positive contractive projection from 푀 onto a subalgebra of 퐴 containing 핝푀, which is a homomorphism on 퐴. Then for any state 휔 of 푀, we have that ω = ω ∘ 휓on M, whenever 휔 = 휔 ∘ 휓 on 퐴. Proof: If 푎 ∈ 퐴−1 then by hypothesis we have 휔(휓(푎)푎−1) = 휔(휓(휓(푎)푎−1)) = 휔(휓(푎)휓(푎−1)) = 휔(핀) = 1 By the Cauchy-Schwarz and Kadison-Schwarz inequality we deduce: 1 ≤ 휔(휓(푎)휓(푎)∗) 휔((푎−1)∗푎−1) ≤ 휔(휓(푎푎∗)) 휔((푎−1)∗푎−1) = 휔((푎푎∗)) 휔((푎푎∗)−1). We can now follow the proof of [56] or [61,227]. Since A is logmodular, for any 푏 ∈ 푀−1 ∩ 푀+ we have that 1 ≤ 휔(휓(푏))휔(푏−1). This leads to the equation 1 ≤ 휔(휓(푒푡푢)) 휔(푒−푡푢) = ′ 푓(푡), for푢 ∈ 푀푠푎. Differentiating and noting that 푓 (0) = 0, yields 휔(푢) = 휔(휓(푢)) as required. When applied to tracial algebras and their associated canonical conditional expectations, the preceding result still holds under a formally weaker hypothesis. Specifically we say that a tracial subalgebra 퐴 of 푀 with canonical conditional expectation Φ has conditional factorization if given any 푏 ∈ 푀+ ∩ 푀−1, we have 푏 = |푎|for some element 푎 ∈ 퐴 ∩ 푀−1 with 훷(푎)훷(푎−1) = 1. Corollary (2.1.3)[61]:. A tracial subalgebra of 푀 with conditional factorization has the 훷 −state property. We say that 퐴 has the unique state extension property if if there is a unique state on 푀 extending 휏|퐴. This is a formally weaker property than the 훷-state property: Proposition (2.1.4)[61]: Let A be a weak* closed unital subalgebra of 푀. If A has the 훷- state property then it has the unique state extension property. The converse is true if A is antisymmetric. Proof: Suppose that 휔 is a state of 푀 extending τ|A. Then ω ∘ 훷 = 휏 ∘ 훷 = 휏 = 휔 on A. By the 훷-state property, on M we have 휔 = 휔 ∘ 훷 = 휏 ∘ 훷 = 휏. For the converse we need only note that if A is antisymmetric, then 휔 ∘ 훷 = 휔 on A forces 휏 = 휔 on 퐴.

Corollary (2.1.5)[61]: Suppose that A is a tracial subalgebra of 푀 with the unique normal state extension property. Then 퐴∞ = 푀 ∩ [퐴]2is a tracial subalgebra with the 훷-state property. Proof: First note that by [58], 퐴∞is a tracial subalgebra of 푀 with respect to the same 훷 and 휏. By [58], 퐴∞has conditional factorization. Corollary (2.1.3) now gives the conclusion. Corollary (2.1.6)[61]: The open question from [58] as to whether every tracial subalgebra with the unique normal state extension property is maximal subdiagonal, is equivalent to the question of whether every tracial subalgebra with the 훷-state property is maximal subdiagonal. It is also equivalent to whether every tracial subalgebra with the unique state extension property is maximal subdiagonal. It is also equivalent to whether every tracial subalgebra with conditional factorization has factorization. Proof: Suppose that every tracial subalgebra with the 훷-state property is maximal subdiagonal, and suppose that A has the unique normal state extension property. By Corollary (2.1.5), 퐴∞ has the 훷 -state property.Henceit is maximal subdiagonal, and therefore satisfies 퐿2-density. Consequently 퐴 satisfies 퐿2-density, and so 퐴 is maximal subdiagonal by [58]. Similarly, suppose that every tracial subalgebra with conditional factorization has factorization, and suppose that A has the 훷 -state property. By results above, 퐴 has the unique normal state extension property, and so by [58], 퐴∞ has conditional factorization. By hypothesis, 퐴∞ has factorization. Thus it is maximal subdiagonal by[4], and thus as in the last paragraph A is maximal subdiagonal. In [86], Lumer considered the property of ‘uniqueness of representing measure’, namely the property that every multiplicative functional on 퐴 ⊂ 퐶(퐾) has a unique extension to a state on 퐶(퐾), He showed how this condition could be used as another possible axiom from which all the generalized 퐻푝 theory may be derived. The natural noncommutative generalization of Lumer’s property, is that every completely contractive representation of 퐴 has a unique completely positive extension to푀. It is known that maximal subdiagonal algebras have this property [60, 55]. Although we have not settled the converse yet, we can say that every unital subalgebra of 푀 which has this property must in some sense be a large subalgebra of M. The following result represents some sort of converse to many of the preceding results which established various unique extension properties as a consequence of maximal subdiagonality. ∗ ∗ In the following result we use the 퐶 -envelope 퐶푒 (A) of an operator algebra 퐴. See e.g. [56,16,85,131,160] for the definition of this, and for its universal property. ∗ Theorem (2.1.7)[61]: Suppose that 퐴 is a subalgebra of a unital 퐶 -algebra 퐵 such that 핀퐵 ∈ 퐴, and suppose that 퐴 has the property that for every Hilbert space 퐻, every completely contractive unital homomorphism π: A → B(H) has a unique completely contractive (or equiv. ∗ completely positive) extension B →B(H). Then 퐵 = 퐶푒 (퐴), the C∗-envelope of 퐴. Proof: (i) (The case that 퐴 is a C∗-subalgebra of 퐵.) In this case, since contractive homomorphisms on C∗-algebras are ∗-homomorphisms (see e.g. [56]), we must prove that if every unital ∗-homomorphism 휋: 퐴 → 퐵(퐻) has a unique completely contractive extension 퐵 → 퐵(퐻), then 퐴 = 퐵. To see this, let 휌: 퐵 → 퐵(퐻) be the universal representation of 퐵. Then휌 is unital, and hence so is 휋 = 휌|퐴. Let 푈 be a unitary in (퐴)′. Then since 푈∗휌(. )푈 = 휌 on A, we have by hypothesis that 푈∗휌(·)푈 = 휌 on 퐵, and thus 푈 ∈ 휌(퐵)′. Thus 휋(퐴)′ = 휌(퐵)′, and it follows that 휋(퐴)′′ = 휌(퐵)′′. If 휌̃ is the unique normal extension of 휌

25 to 퐵∗∗, then 휌̃ is faithful on 퐵∗∗ and it has range 휌(퐵)′′. The restriction of 휌̃ to the copy ∗ 퐴⊥⊥of 퐴∗∗inside 퐵∗∗ has range 휋(퐴)′′ = 휋̅̅(̅̅퐴̅̅)푤 , and is therefore surjective. This forces the copy of 퐴∗∗ inside 퐵∗∗ to be all of 퐵∗∗. Thus 퐴 = 퐵 ∩ 퐴⊥⊥ = 퐵. (ii) (The general case.) Let 퐶 = 퐶∗(퐴), the C∗-algebra generated by 퐴 in 퐵. Since 퐴 ⊂ 퐶, it follows from the hypothesis that every unital ∗-homomorphism 휋: 퐶 → 퐵(퐻) has a unique completely contractive extension 퐵 → 퐵(퐻). By (i),퐶 = 퐵. ∗ ∗ By virtue of this fact, we need only prove that 퐶 (퐴) = 퐶푒 (퐴) under the assumptions ∗ ∗ of the theorem. By the universal property of 퐶푒 (퐴), there is a ∗-epimorphism 휃: 퐵 = 퐶 (퐴) → ∗ 퐶푒 (퐴) restricting to the ‘identity map’ on 퐴. If 퐵 ⊂ 퐵(퐻) then the canonical map from the ∗ ∗ copy of 퐴 in Ce(A), to 퐴 ⊂ 퐵(퐻), has a completely positive extension 훷: 퐶푒 (퐴) → 퐵(퐻). On 퐴, the map Φ ∘ θ is the identity map, so that by hypothesis 훷 ∘ 휃 = 푖퐵. Thus 휃 is one-to-one, and hence 퐶∗(퐴)is a 퐶∗-envelope of 퐴. Corollary (2.1.8)[61]: Suppose that 퐴 is a tracial subalgebra of 푀with the property that for every Hilbert space 퐻, every completely contractive unital homomorphism 휋: 퐴 → 퐵(퐻) has a unique completely contractive (or equiv. completely positive) extension 퐵 → 퐵(퐻). Then 퐴 generates 푀 as a 퐶∗-algebra. Indeed, 푀 is a 퐶∗-envelope of 퐴. The classical form of the 퐹. and 푀. Riesz theorem (see e.g. [147]) is known to fail for weak* Dirichlet algebras; and hence it will fail for subdiagonal algebras too. However there is an equivalent version of the theorem which is true for weak* Dirichlet algebras[146, 282], and we will focus on this variant here. We shall say that a tracial subalgebra 퐴 of 푀 has the 퐹 &푀 Riesz property if for every bounded function 휌 on 푀 which annihilates 퐴0, the normal and singular parts휌푛 and 휌푠 annihilate 퐴0 and 퐴 respectively. During our investigation we shall have occasion to make use of the polar decomposition of normal functionals on a von Neumann algebra. We take the opportunity to point out that for our purposes we shall assume such a polar decomposition to be of the form 휔(푎) = |휔|(푢푎) for some partial isometry, rather than휔(푎) = |휔|(푎푢) which seems to be more common among the proponents of noncommutative 퐿푝- spaces. The following result shows that to study the 퐹 & 푀 Riesz property, we may restrict our attention to algebras for which the diagonal 풟 is finite dimensional: Proposition (2.1.9)[61]: If a tracial subalgebra 퐴 of 푀 satisfies the 퐹 & 푀 Riesz property then the diagonal 퐷is finite dimensional. ⋆ ⋆ Proof: Let 휓 ∈ 퐷 . Then 휓 ∈ 푀 annihilates A0. By the 퐹&푀 Riesz property, 휓 ∘ 훷 agrees with (휓 ∘ 훷) on A, and so 휓 = 휓 ∘ 훷|퐷 is weak* continuous on 풟. Thus 풟 is reflexive, and therefore finite dimensional. Lemma 2.1.10)[61]: Let 퐴 be a maximal subdiagonal subalgebra of 푀. Let 휔 be a state of 푀, and let (휋휔, 픥휔, 훺휔) be the 퐺푁푆 representation of 푀. Further, let 휔 be the orthogonal ̅̅̅̅̅̅̅̅̅̅̅̅̅̅ projection of ω onto the closed subspace 휋휔(퐴0)훺휔. (a) The following holds: (i) There exists a central projection 푝0 in 휋휔(푀)′′such that for any 휉, 휓 ∈ 픥휔the functionals 푎 → 휋휔(푎)푝0휉, 휓 and 푎 → 휋휔(핝 − 푝0)휉, 휓 on 푀 are respectively the normal and singular parts of the functional 푎 → 휋휔(푎)푝0휉, 휓. In particular, the triples

(푝0휋휔, 푝0픥휔, 푝0훺휔) and ((핝 − 푝0) 휋휔, (핝 − 푝0)ℎ휔, (핝 − 푝0)훺휔) are copies of the GNS representations of ωn and ωs respectively. (ii) 휔0: 푎 → 〈휋휔(푎)(훺휔 − 훺0), 훺휔 − 훺0〉 defines a positive functional of M satisfying ω0 = ω0 ∘ Φ. (b) Suppose that in addition 푑푖푚(풟) < ∞. 1/2 1⁄ (i) Then ω0 is a normal functional of the form 휔0 = 휏(푔 · 푔 2) for some 푔 ∈ 퐷+. Moreover 푝0(훺휔 − 훺0) = 훺휔 − 훺0, and p0Ωω is the orthogonal projection of 푝0훺휔 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ onto 푝0(휋휔(퐴0)훺휔). ̅̅̅̅̅̅̅̅̅̅̅̅̅̅ (ii) If 휔 is singular, then for any 푓 ∈ 퐷 we have that 휋휔(푓)훺휔 ∈ 휋휔(퐴0)훺휔. (c) Suppose that dim(퐷) < ∞ and 훺휔 ∉ 휋휔(퐴0)훺휔. If ω0 is faithful on 퐷, then there exists a sequence {푎푛} ⊂ 퐴 such that 휋휔(푎푛)(훺휔 − 훺0) → 푝0훺휔. Proof: (a)(i): This is essentially the content of [180]. (a)(ii): Let (휋휔, 픥휔, 훺휔) and 훺0 be as in the hypothesis, and define a positive functional ω0 on 푀 by 휔0: 푎 → 〈휋휔(푎)(훺휔 − 훺0), 훺휔 − 훺0〉. Let f ∈ 퐴0 be given. By construction 휋휔(푓)훺휔 ⊥ (훺휔 − 훺0). Since 퐴0 is an ideal, 휋휔(푓푎)훺휔 ∈ 휋휔(퐴0)훺휔 for each 푎 ∈ 퐴0. Since A0 belongs to 휋휔(퐴0)훺휔, we may of course select a sequence {푏푛} ⊂ 퐴0 for which 휋휔(푏푛)훺휔convergeto 훺0. Hence 휋휔(푓푏0)훺휔 converges to 휋휔(푓)훺0. Thus 휋휔(푓푏0)훺휔 ∈ 휋휔(푓)훺0, which forces 휋휔(푓)훺0 ⊥ (훺휔 − 훺0). From the previous two centered equations it is now clear that 퐴0 ⊂ 퐾푒푟(휔0). Thus 휔0 = 휔0 ∘ 훷 on A by the remarks preceding Proposition (2.1.1). Hence 휔0 = 휔0 ∘ 훷on 푀 by Corollary (2.1.3). (b) (i): Since 퐷 is finite dimensional, we can find 푔 ∈ 퐷+ so that 휔0(푎) = 휏(푔푎) for all푎 ∈ 풟. Since 휔0 ∘ 훷 = 휔0, we conclude that for any 푎 ∈ 푀, 휔0(푎) = 휔0(훷(푎)) = 휏(푔훷(푎)) = 휏(훷(푔푎)) = 휏(푔푎), There by establishing the first part of the claim. For the second part, note that since 휔0 is clearly normal, we have by part (a)(i) that 0 = 휋휔(푎)(핝 − 푝0)(훺휔 − 훺0), 훺휔 − 훺0 for all 푎 ∈ 푀. For 푎 = 핝 this yields 0 = ‖(핝 − p0)(Ωω − Ω0)‖, or equivalently 푝0(훺휔 − 훺0) = 훺휔 − 훺0. From this fact, we may now conclude that 〈푝0휋휔(푎)훺휔, 푝0(훺휔 − 훺0)〉 = 휋휔(푎)훺휔, 훺휔 − 훺0 = 0 for all 푎 ∈ 퐴0. Thus 푝0(훺휔 − 훺0) ⊥ 푝0휋휔(퐴0)훺휔. Now select a sequence {푏푛} ⊂ 퐴0 so that 휋휔(푏푛)훺휔 →0. By continuity, 푝0훺0 = 푙푖푚푛 푝0휋휔(푏푛)훺휔 ∈ 푝0휋휔(퐴0)훺휔. From these considerations it is clear that 푝0훺0 is the orthogonal projection of 푝0훺휔 onto 푝0휋휔(퐴0)훺휔. (b) (ii): If ω is singular, then ∗ 0 = 휔푛(푎푏) = 〈휋휔(푎푏)푝0훺휔, 훺휔〉 = 〈 푝0휋휔(푏푛)훺휔, 휋휔 (푎 ) 〉 27 for all푎, 푏 ∈ 푀 Since 훺휔 is cyclic, this is sufficient to force 푝0 = 0. But then 훺휔 = 훺0 = 푝0(훺휔 − 훺0) = 0by part (b)(i). As before select {푏푛} ⊂ 퐴0 so that 휋휔(푏푛)훺휔 → 훺0 = 훺휔. For any 푓 ∈ 퐷 the ideal property of A0 then ensures that 휋휔(푓)훺휔 = 푙푖푚푛휋휔(푓푏푛)훺휔 ∈ 휋휔(퐴0)훺휔. 1 (c): Suppose that 휔푛, the normal part of ω, is of the form 휔푛 = 휏(ℎ · ) for some ℎ ∈ 퐿 (푀)+. As noted earlier, (푝0휋휔, 푝0픥휔, 푝0훺휔) is a copy of the 퐺푁푆 representation engendered by 휔푛. If now we compute the representation of 휔푛 from first principles, it is clear that 2 푝0픥휔corresponds to the weighted Hilbert space 퐿 (푀, ℎ) obtained by equipping M with the inner product 〈푎, 푏〉 = 휏(ℎ1/2휎푏∗푎ℎ1/2), 푎, 푏 ∈ 푀, and taking the completion. Note that 퐿2(푀, ℎ) can be identified unitarily, and as 푀 modules, with the closure of 푀ℎ1/2 in 퐿2(푀). For any a ∈ 푀 considered as an element of 퐿2(푀, ℎ) we will write a instead of 푎. The canonical ∗-homomorphism representing 푀 as an algebra of bounded operators on 퐿2(푀, ℎ) is of course given by defining 휋푛(푏)휓푎 = 휓푎푏 , 푎, 푏 ∈ 푀, 2 and then extending this action to all of 퐿 (푀, ℎ). Since 휔푛 is normal, 휋푛 (corresponding to 푝0휋휔) is σ-weakly continuous and satisfies 휋푛(푀) = 휋푛(푀)′′. Thus 퐾푒푟(휋푛) is 휎 -weakly closed two-sided ideal, and hence we can find a central projection 푒 ∈ 푀 so that (핝 − 푒)푀 = 푘푒푟(휋푛). Restrict 휋푛to 푎∗-isomorphism from eM onto 휋푛(푀). Then for any 푎, 푏, 푐 ∈ 푀 we have 1/2 ∗ 1/2 〈휋푛(푐)휓푎, 휓푏〉ℎ = 휏(ℎ 푏 (푒푐푒)푎ℎ ) (0) Let 휓 denote the orthogonal projection of 휓핀 onto the closure of {휓푎: 푎 ∈ 퐴0}. (0) (Note that 휓핝 and 휓 of course correspond to 푝0훺휔 and 푝0훺0 in parts (a) and (b) of the proof.) Since 퐿2(푀, ℎ) may be viewed as a subspace of 퐿2(푀), let 퐹 ∈ 퐿2(푀)be the element corresponding to 휓(0). It is easy to see that 푒퐹 = 퐹. From parts (a) and (b) we now have that 1 (0) (0) ∗ 1/2 휔0 = 〈휋푛(·)(휓핝 − 휓 ), 휓핝 − 휓 〉ℎ = 휏((ℎ2푒 − 퐹 ) . (ℎ 푒 − 퐹)). This in turn ensures that 1 |ℎ2푒 − 퐹∗|2 = 푔 1/2 where g is as in part (b). Thus ℎ 푒 − 퐹 ∈ 푀. Since by assumption ω0 is faithful on 풟, it follows that 푆푢푝푝(푔) = 핝. Since 풟 is finite dimensional, g must be invertible. But then ℎ1/2푒 − 퐹 must also be invertible, by the previous centered equation. (Recall that if ab is invertible in a finite von Neumann algebra then both a and b are invertible.) The polar decomposition of ℎ1/2푒 − 퐹∗ is of the form ℎ1/2푒 − 퐹∗ = 푢푔1/2 for some unitary 푢 ∈ 푀. From this it is clear that 1 (ℎ2푒 − 퐹)−1 = 푢푔−1/2. 1/2 −1/2 2 2 Clearly ℎ 푢푔 ∈ 퐿 (푀). Hence we may select {푎푛} ⊂ 푀 converging in 퐿 (푀) to ℎ1/2푢푔−1/2 = ℎ1/2(ℎ1/2푒 − 퐹)−1.By the previously established correspondences we then have 1 ‖휓 − 휋 (푎 )(휓 − 휓(0))‖ = 휏(|ℎ2푒 − (푎 푒)(ℎ1/2푒 − 퐹)|2)1/2 핝 푛 푛 핝 ℎ 푛 28

→ 휏(|ℎ1/2푒 − ℎ1/2푒|2)1/2 = 0. This implies, in the notation of parts (a) and (b), that 휋휔(푎푛)(훺휔 − 훺0) → 푝0훺휔. It remains 1/2 −1/2 to show that we may select {푎푛} ⊂ 퐴, or equivalently, that ℎ 푢푔 ∈ [퐴]2. For this, it ∗ 1/2 −1/2 ∗ suffices by the L2 density of A + A to show that ℎ 푢푔 ⊥ [퐴0]2.So let 푎 ∈ 퐴0 be given, and observe that 1 1 1 1 1 1 1 1 − − − 휏 (푎ℎ2푢푔 2) = 휏 (푔−1푎ℎ2푢푔 2푔) = 휏 (푔−1푎ℎ2푢푔 2) = 휏 (푔−1푎ℎ2(ℎ2푒 − 퐹∗) 1/2 ∗ −1 1/2 〈 (0)〉 = (ℎ 푒 − 퐹 )( 푔 푎ℎ ) = 휓푔−1푎, 휓핝 − 휓 ℎ = 0 (0) (The last equality follows from the ideal property of A0 and the fact that 휓핝 − 휓 is orthogonal to {휓푎: 푎 ∈ 퐴0}. ) The claim therefore follows. Corollary (2.1.11)[61]: Let A be a maximal subdiagonal algebra with 푑푖푚(풟) < ∞. The following are equivalent: (i) A satisfies the 퐹&푀 Riesz property. (ii) Whenever ω annihilates 퐴0, the normal and singular parts ωn and 휔푠, will separately annihilate 퐴0. (iii) Whenever ω annihilates 퐴, the normal and singular parts, ωn and 휔푠, will separately annihilate퐴0. (iv) Whenever ω annihilates 퐴, the normal and singular parts, ωn and 휔푠 will separately annihilate 퐴. Proof: The implications (i) ⇒ (ii) ⇒ (iii) are clear. If (iii) holds, let 휔 be a bounded linear functional which annihilates 퐴0. Since 훷 is a normal map onto 풟, and 풟 is finite dimensional, the functional defined by 휔풟 = 휔|풟 ∘ 훷 is normal. Then 휌 = 휔 − 휔풟 defines a functional which annihilates A. From (iii) we then have that 휌푛 and 휌푠 separately annihilate A0. The normality of ωD ensures that 휌푛 = 휔푛 − 휔풟 , 휌푠 = 휔푠. Since by construction 휌 = 휔 − 휔풟 annihilates A0, we conclude that 휔푛and 휔푠separately annihilate 퐴0. This proves (ii). To prove the validity of (i), it remains to show that any singular functional 휔 which annihilates 퐴0, also annihilates 풟. For such , the ‘modulus’ |휔| is still singular (see e.g. [174, 241], or the argument in the first part of the proof of the next theorem). 2 Let (πω, 픥ω, Ωω) be the GNS representation of |ω|. For each 푎 ∈ 푀 we have |휔(푎)| ≤ ∗ ‖휔‖|휔|(푎 푎). By a standard argument this implies that there exists a vector 휂 ∈ 픶휔 such that 휔(·) = 〈휋휔(·)훺휔, 휂〉. Let 푑 ∈ 풟 be given. By part (b)(ii) of Lemma (2.1.10) we may select a sequence {푓푛} ⊂ 퐴0 so that πω(d)Ωω = lim πω(fn)Ωω. But then n 휔(푑) = 〈휋휔(푑)훺휔, 휂〉 = 푙푖푚〈휋휔(푑)훺휔, 휂〉 = 푙푖푚 휔(푓푛) = 0 푛 푛 as required. The equivalence with (iv) is now obvious.

Theorem (2.1.12)[61]:Let 퐴 be a maximal subdiagonal algebra. Then 퐴 satisfies the 퐹&푀 Riesz property if and only if 푑푖푚(풟) < ∞.

Proof: We proved the one direction in Proposition (2.1.9). For the other, let 휔 be a bounded linear functional on 푀 which annihilates 퐴0, and let ωnand 휔푠 be the normal and singular parts 1 of 휔. Write 휔푛 = 휏(ℎ ·), for some ℎ ∈ 퐿 (푀). We extend 휔, 휔푛, and 휔푠, uniquely to normal functionals on the enveloping von Neumann algebra (the double commutant in the universal representation) and define |휔|, |휔푛|and |ωs|, to be the absolute values of these extensions restricted to 푀. Then from for example ([173], cf. [240]) applied to 휔 and 휏, we have that as functionals on 푀, |휔푛| and |ωs| are respectively the normal and singular parts of |휔|, and that |휔| = |휔푛| + |휔푠|. We note from [134] that there is no danger of confusion as regards the absolute value of ωnsince the absolute value of 휔푛as a functional on 푀 and as a functional on the enveloping von Neumann algebra coincide on 푀. Now consider the positive functional 휌 given by 휌 = 휏 + |휔|. Let (휋휌, 픶휌, 훺휌) be the 퐺푁푆 representation constructed from ρ, and define ρ0 by 휌0(푎) = 〈휋휌(푎)(훺휌 − 훺0), 훺휌 − 훺0〉, where 휌0 is the orthogonal projection of ρ onto the closure of + {휋휌(푎)훺휌: 푎 ∈ 퐴0}. For any 푓 ∈ 퐴0 and any 푑 ∈ 풟 , we have by construction.That 1 2 1 2 1/2 2 ‖휋휌(푑 )(훺휌 − 휋휌(푓)훺휌)‖ = 휌(|푑2(핝 − 푓)| ) ≥ 휏(|푑2(핝 − 푓)| ) = 휏(푑 − 푑푓 − 푓∗푑 + |푑1/2푓|2) = 휏(푑 + |푑1/2푓|2) ≥ 휏(푑). On selecting a sequence {푓푛} ⊂ 퐴0 so that 휋휌(푓)훺휌 → 0, it follows that ρ0(푑) = 1/2 2 ‖휋휌(푑 )(훺휌 − 훺0)‖ ≥ 휏(푑). Hence 휌0 is faithful on 풟, and 훺휌 ≠ 훺0. Thus we may apply all of Lemm(2.1.10) to (휋0, 픥0, 훺0) . Next notice that for each a in the enveloping von Neumann algebra we have |휔(푎)|2 ≤ ‖휔‖|휔|(푎∗푎) ≤ ‖휔‖휌(푎∗푎). Thus on restricting to elements of 푀, and employing a standard argument, this implies that there exists a vector 휂 ∈ 휋휌 such that 휔(·) = 〈휋휌(·)훺휌, 휂〉. Now consider the related functional 휔̃(·) = 〈휋휌(·)(훺휌 − 훺0), 휂〉. Select a sequence {푓푛 ⊂ 퐴0 so that 휋휌(푓푛)훺0 → 훺0. Let 푎 ∈ 퐴0 be given. Since 퐴0 is an ideal, and since 휔 annihilates 퐴0, we conclude that ω̃(푎) = 〈휋휌(푎)(훺휌 − 훺0), 휂〉 = 푙푖푚〈휋휌(푎(핝 − 푓푛))(훺휌 , 휂〉 = 푙푖푚 휔(푎(핝 − 푓푛)) = 0. 푛 푛 Thus ω̃ also annihilates 퐴0. By part (c) of the Lemma we can find a sequence {푎푛} ⊂ 퐴 such that 휋휌(푎푛)((훺휌 − (훺0) → 푝0(훺휌. Let 푎 ∈ 퐴0 be given. Since A0 is an ideal, and since ω̃ annihilates A0, we may now conclude that 휔푛(푎) = 〈휋휌(푎)훺휌, 휂〉 = 푙푖푚〈휋0(푎(핝 − 푓푛))(훺휌 , 휂〉 = 푙푖푚 휔̃(푎(핝 − 푓푛)] = 0 푛 푛 Thus 휔푛 annihilates A0. But then so does 휔푠 = 휔 − 휔푛. It now follows from Corollary (2.1.11) that 퐴 satisfies the 퐹 & 푀 Riesz property.

Corollary (2.1.13)[61]: If 퐴 is a maximal subdiagonal algebra with 풟 finite dimensional, and if 휔 ∈ 푀∗annihilates 퐴 + 퐴∗, then 휔 is singular. ∗ Proof: Since 퐴 satisfies the 퐹&푀 Riesz property, 휔푛. annihilates 퐴. Similarly, since 퐴 . ∗ satisfies the 퐹&푀 Riesz property, 휔푛annihilates A . Since 퐴 is subdiagonal, 휔푛 = 0. Corollary (2.1.14)[61]: If 퐴 has the 퐹 & 푀 Riesz property, then any positive functional on 푀 which annihilates 퐴0 is normal. Proof:If 휔 is a state on 푀 which annihilates 퐴0, and if 퐴 has the 퐹 & 푀 Riesz property, then the (positive) singular part of 휔 is 0 since it must annihilate 핝. We say that an extension in 푀⋆ of a functional in 퐴⋆ is a Hahn-Banach extension if it has the same norm. If 퐴 is a weak* closed subalgebra of 푀 then we say that 퐴 has property (퐺푊1) if every Hahn-Banach extension to 푀 of any normal functional on 퐴, is normal on 푀. We say that A has property (퐺푊2) if there is at most one normal Hahn-Banach extension to 푀 of any normal functional on A. We say that A has the Gleason-Whitney property (퐺푊) if it possesses (퐺푊1) and (퐺푊2). This is simply saying that there is a unique Hahn-Banach extension to 푀 of any normal functional on A, and this extension is normal. Of course normal functionals on 퐴 or on 푀 have to be of the form 휏(푔 ·) for some 푔 ∈ 퐿1(푀). Theorem (2.1.15)[61]: If 퐴 is a tracial subalgebra of 푀 then 퐴 is maximal subdiagonal if and only if it possesses property (퐺푊2). If 풟 is finite dimensional, then A is maximal subdiagonal if and only if it possesses property (퐺푊). Proof: Suppose that 퐴 possesses property (퐺푊2). To show that 퐴 is maximal subdiagonal, it suffices to show that if 푔 ∈ 퐿1(푀),with 휏(푔(퐴 + 퐴∗)) = 0, then 푔 = 0. By considering real and imaginary parts we may assume that 푔 = 푔∗. Then 휏(|푔| ·) and 휏((|푔| + 푔) ·) are positive normal functionals on 푀 which agree on 퐴. They are also Hahn- Banach extensions, since the norm of a positive functional is achieved at 1. Thus by (GW2), these functionals agree on 푀, and so |푔| + 푔 = |푔|. That is, 푔 = 0. In the remainder of the proof suppose that 퐴 is maximal subdiagonal. Suppose that 푓, 푔 ∈ 퐿1(푀) correspond to two normal Hahn-Banach extensions to 푀 of a given functional on 퐴. Then ‖푓‖1 = ‖푔‖1, and this quantity equals the norm of the restriction to 퐴. We have 휏((푓 − 푔)퐴) = 0; since 퐴 is subdiagonal it follows from [153] that h = 푔 − 푓 ∈ [퐴0]1. In order to establish (퐺푊2), we need to show that ℎ = 0. Since Ball (퐴) is weak* compact, and since ‖푓‖1 equals the norm of the above-mentioned restriction to A, there exists a ∈ A of norm 1 with 휏(푓푎) = ‖푓‖1. It is evident that |푎푓|2 = 푓∗푎∗푎푓 ≤ 푓∗푓 = |푓|2 . 1 1 푝 Now 0 ≤ 푇 ≤ 푆 in 퐿 (푀) implies that 푇2 ≤ 푆2(see e.g. [163], and we thank David Sherman for this reference). It follows that |푎푓| ≤ |푓|. On the other hand, 휏(|푓|) = 휏(푓푎) = 휏(푎푓) ≤ 휏(|푎푓|).Thus‖ |푓| − |푎푓|‖1 = 휏(|푓| − |푎푓|) = 0, and so |푓| = |푎푓|.The functional 휓 = 휏(푎푓. ) on 푀 must be positive since 휓(핝) = 휏(푎푓) = 휏(|푓|) = 휏(|푎푓|) = ‖휓‖. Thus 푎푓 ≥ 0, and 푎푓 = |푎푓| = |푓|. Since ℎ ∈ [퐴0]1 we have

휏((푓 + ℎ)푎) = 휏(푓푎)‖푓‖1 = ‖푔‖1 = ‖푓 + ℎ‖1 An argument similar to that of the last paragraph shows that 푎(푓 + ℎ) = |푓 + ℎ| ≥ 0.Thus 푎ℎ is self-adjoint. Since ℎ ∈ [퐴0]1 it is easy to see that 휏(푎ℎ퐴) = 0. Therefore from the self- 31 adjointness of ah one may deduce that 휏(푎ℎ(퐴 + 퐴∗)) = 0. Because 퐴 is subdiagonal, it follows that 푎ℎ = 0. Thus |푓| = 푎푓 = 푎(푓 + ℎ) = |푓 + ℎ| Let e be the left support projection of a. Then 푒⊥ is the projection onto 퐾푒푟(푎∗). We have |푓|푒⊥ = 푓∗푎∗푒⊥ = 0. It follows that 푓푒⊥ = 0. Thus 0 = 푒⊥푓∗푓푒⊥ = 푒⊥|푓 + ℎ|2푒⊥ = 푒⊥(푓 + ℎ)∗(푓 + ℎ)푒⊥ = 푒⊥ℎ∗ℎ푒⊥. Hence he⊥ = 0. To show that ℎ푒 = 0, we reproduce the ideas in the argument in the second paragraph of the proof. Namely, note that |(푓푎)∗|2 ≤ |푓∗|2, so that |(푓푎)∗| ≤ |푓∗|. But ∗ ∗ ∗ ∗ (|푓 |) = ‖푓‖1 = 휏(푓푎) ≤ 휏(|(푓푎) |), and as before this shows that |(푓푎) | = |푓 |. Then also τ(푓푎) = 휏(|(푓푎)∗|), and as before this shows that 푓푎 ≥ 0. Similarly, (푓 + ℎ)푎 ≥ 0. So ha is again selfadjoint, and this implies as before that ℎ푎 = 0. Thus ℎ푒⊥ = 0, and so ℎ = ℎ푒 + ℎ푒⊥ = 0 as required. Now suppose that, in addition, 풟 is finite dimensional, and that 휌 is a Hahn-Banach extension of a normal functional ω on 퐴. By basic functional analysis, 휔is the restriction of a normal functional ω on M. We may write 휌 = 휌푛 + 휌푠, where ωn and ωs are respectively the normal and singular parts, and ‖휌‖ = ‖휌푛‖ + ‖휌푠‖.Then 휌 − 휔̃ annihilates 퐴, and hence by our 퐹. and 푀. Riesz theorem both the normal and singular parts, 휌푛 − 휔̃ and 휌s respectively, annihilate 퐴0. Hence they annihilate 퐴, and in particular 휌푛 = 휔 on 퐴. But this implies that ‖ρn‖ + ‖ρs‖ = ‖ρ‖ = ‖ω‖ ≤ ‖ρn‖ We conclude that 휌푠 = 0. Thus A also satisfies (퐺푊1), and hence (퐺푊).There is another (simpler) variant of the Gleason-Whitney theorem [149], which transfers more easily to our setting: Theorem (2.1.16)[61]:Let 퐴 be a maximal subdiagonal subalgebra of 푀 with 풟finite dimensional. If 휔is a normal functional on 푀 then ω is the unique Hahn-Banach extension of ∗ its restriction to 퐴 + 퐴 . In particular, ‖휔‖ = ‖휔퐴+퐴∗‖for any휔 ∈ 푀∗. Proof: Let ρ be a Hahn-Banach extension of the restriction of ω to A + A∗. We may write 휌 = 휌푛 + 휌푠, where 휌푛 and 휌푠, are respectively the normal and singular parts, and ‖휌‖ = ‖휌푛‖ + ∗ ‖휌푠‖. Then 휌 − 휔 annihilates A + 퐴 . By Corollary (2.1.14), 휌푛 − 휔 = (휌 − 휔)푛 = 0. As in the last part of the previous proof, this implies that 휌푠 = 0. So 휌푠 = 휌푛 = 휔. Corollary (2.1.17)[61]:(Kaplan sky density theorem for subdiagonal algebras) Let 퐴 be a maximal subdiagonal subalgebra of M with D finite dimensional. Then the unit ball of 퐴 + 퐴∗is weak* dense in Ball(푀). Proof: If C is the unit ball of 퐴 + 퐴∗, it follows from the last remark that thepre-polar of C is Ball(M⋆). By the bipolar theorem, C is weak* dense in Ball (푀). Arveson formulated the Szegö theorem for 퐿2(푀) in terms of the Kadison-Fuglede determinant ∆(·). The long-outstanding open question of whether general maximal subdiagonal algebras satisfy the Szego theorem for 퐿2(푀), was eventually settled in the affirmative in [163]. We will now extend this result to 퐿푝(푀). We refer the reader to [300, 58] for the properties of the Kadison-Fuglede determinant which we shall need.

푝 푝 Lemma (2.1.18)[61]:∆(푏 ) = ∆(푏) for 푝 ≥ 1 and 푏 ∈ 푀+.

Proof: By the multiplicativity property of ∆, the relation clearly holds for dyadic rationals. We may assume that 0 ≤ 푏 ≤ 1. In this case, by the functional calculus it is clear that 푏푞 ≤ 푏푝 if 0 < 푝 ≤ 푞. If q is any dyadic rational bigger than 푝 then ∆(푏)푝 = ∆(푏푝) ≤ ∆(푏푝) 1 1 It follows that ∆(푏푝) ≤ ∆(푏푝). Replacing 푝 by 1/푝, we have ∆(bp)p ≤ ∆ ((bp)p) = ∆(b), which gives the other direction. Theorem (2.1.19)[61]:.(Szegö theorem for 퐿푝(푀)) Suppose that 퐴 is maximal subdiagonal, 1 and 1 ≤ 푝 < ∞. If ℎ ∈ 퐿 (푀)+ then 푝 ∆(ℎ) = 푖푛푓 {휏(ℎ|푎 + 푑| ) ∶ 푎 ∈ 퐴0, 푑 ∈ 풟, ∆(푑) ≥ 1}. Proof: We set 푝 푆푝 = {|푎| : 푎 ∈ 퐴, (휙(푎)) ≥ 1}, 푆 = {푎∗푎: 푎 ∈ 퐴−1, ∆(푎) ≥ 1}. By the modification in [58] of a trick of Aversion’s from [300], it suffices to show that the closure of 푆푝 equals the closure of 푆. First we show that 푆 ⊂ 푆푝. Indeed, if 푏 ∈ 푆 then b is 1 1 invertible, and therefore so is . Since 퐴 has factorization, there is an 푎 ∈ 퐴−1 with |a| = bp . 푝 By Lemma (2.1.19) and Jensen’s formula [300, 163] we have 1 1 ∆(훷(푎)) = ∆(푎) = ∆(|푎|) = ∆ (푏푝) = ∆(푏)푝 ≥ 1. 푝 Hence 푏 = |푎| ∈ 푆푝. 푝 Suppose that 푏 ∈ 푆푝 .If 푏 = |푎| where 훥(훷(푎)) ≥ 1 then by Jensen’s inequality [300, 241] we have 훥(푎) = 훥(|푎|) ≥ 1. Hence by Lemma (2.1.18) we have 훥(푏) ≥ 1. If 푛 ∈ ℕ then 1 since 퐴 has factorization, there exists a c ∈ A−1with 푏 + 1 = 푐∗푐. Thus 푛 1 Δ(푐)2 = 훥(푏 + 1) ≥ 훥(푏) ≥ 1. 푛 1 Thus b + 1 = c∗c ∈ S, and we deduce that 푏 ∈ 푆̅. Hence S̅̅̅ ⊂ S̅. n p Note that the following generalized Kolmogorov theorem is not true for all maximal subdiagonal algebras. For example, take 퐴 = 푀 = 퐿∞[0, 1]. Theorem (2.1.20)[61]:.Suppose that 퐴 is an antisymmetric maximal subdiagonal algebra. 1 1 2 ∗ −1 − −1 Ifℎ ∈ 퐿 (푀)+ then inf{휏(ℎ|핝 + 푓| ): 푓 ∈ 퐴0 + 퐴0 } is either 휏(ℎ ) 2 , if ℎ exists in the sense of unbounded operators and is in 퐿1(푀); or the infimum is 0 if h−1 ∉ L1(M). More 1 푝 푝 ∗ −1 generally, if 1 ≤ 푝 < ∞ then 푖푛푓{휏(|(핝 + 푓)ℎ | ): 푓 ∈ 퐴0 + 퐴0} is either 0 if ℎ ∉ 1 퐿1/(푝−1)(푀), o푟 휏(ℎ− 1/푝−1 ) − 1 ifℎ−1 ∈ 퐿1/(푝−1)(푀). 푝 1 Proof: We formally follow the proof of Forelli as adapted in [282]. Let ℎ ∈ 퐿1(푀) , and + + 푝 1 1 = 1. Define 퐿푝(푀, ℎ) to be the completion in 퐿푝(푀) of 푀ℎ푝. Note that if 푒 is the support 푞 projection of 푎 positive 푥 ∈ 퐿푝(푀) then it is well known (see e.g. [175]) that 퐿푝(푀)푒 equals 푝 푝 푝 the closure in 퐿 (푀) of 푀푥. Hence 퐿 (푀, ℎ) = 퐿 (푀)푒, where e is the support projection of

ℎ. Now for any projection 푒 ∈ 푀 it is an easy exercise to prove that the dual of 퐿푝(푀) is eLq(M) (see e.g. [175]). It follows that the dual of 퐿푝(푀, ℎ) is 퐿푞(푀, ℎ). 1 푝 푞 푝 푞 1 ∗ 푝 If k ∈ 퐿 (푀, ℎ) then 푘ℎ ∈ 퐿 (푀)퐿 (푀) ⊂ 퐿 (푀). We view 퐴0 + 퐴0 in 퐿 (푀, ℎ) as its 1 ∗ 푝 q image (A0 + A0)h , and let 푁 be the annihilator of this in L (M, h). That is, 푔 ∈ 푁 iff 푔 ∈ 퐿푞(푀, ℎ) and 1 1 푝 ∗ ∗ 푝 0 = 휏(ℎ (퐴0 + 퐴0)푔) = 휏((퐴0 + 퐴0 )푔ℎ ). 1 1 Since 푔ℎp ∈ 퐿1(푀) the last equation holds iff 푔ℎ푝 = 푐핝, where 푐 is a constant. Since h is 1 − selfadjoint, if c ≠ 0 then it follows that ℎ 푝 exists in the sense of unbounded operators, and its closure is the constant multiple 푑푔 ∈ 퐿푞(푀), where 푑 = 푐−1. (Since we are in the finite case, 1 there is no difficulty with τ-measurability here, this is automatic [183]). If 푐 = 0 then 푔ℎ 푝 = 1 0 which implies that 푔 = 0. To see the last statement note that if ℎ 푝 is viewed as a selfadjoint unbounded operator on a Hilbert space 퐻, and if 푒 is its support projection, which equals the 1 1 1 1 1 ̅̅̅̅̅̅1 support projection of hq , then 푒ℎ푝 = ℎ푝 , and so 푒ℎ푝푒 = 푒ℎ푝. Since 푔 ∈ 푀ℎ푞 , we have 1 1 푔푒 = 푔. However 푔푒 = 0 since 푔ℎ푝 = 0. Thus if g has norm 1 then c ≠ 0, ℎ푝 ∈ 퐿푞(푀) 1 푞 1 − and |푑| = ‖ℎ푝‖ = 휏(ℎ 푝 )푞 . 퐿푞(푀) The infimum in the theorem is the pth power of the norm of 핝 in the quotient space of 퐿푝(푀, ℎ) ∗ ∗ ⊥ modulo the closure of 퐴0 + 퐴0. Since the dual of this quotient is (퐴0 + 퐴0 ) = 푁, this 1 푝 infimum equals the pth power of sup{|휏 (푔ℎ )| : 푔 ∈ 푁, ‖푔‖퐿푞(푀) ≤ 1}. This equals 0 if no 푞 1 1 1 − − − − 푔 ∈ 푁 has norm 1; otherwise it equals τ(ℎ 푝 ) 푞 = 휏(ℎ 푝−1 ) 푞 by the above. Indeed, the 1 1 1 infimum is 0 iff 휏(푔ℎ푝 ) = 0 for all 푔 ∈ 푁. Since 푔ℎ푝 is constant, this occurs iff 푔ℎ푝 = 0, 1 ∗ 푝 which as we saw above happens iff 푔 = 0. Thus the infimum is 0 iff 푁 = (0) iff (퐴0 + 퐴0)ℎ 1 is dense in 퐿푝(푀, ℎ). Since ℎ푝 ∈ 퐿푝(푀, ℎ), the latter condition implies that there is a sequence 1 1 ∗ 푝 푝 −1/푝 푞 (푔푛) in 퐴0 + 퐴0 with 푔푛ℎ → ℎ in 푝 −norm. If ℎ ∈ 퐿 (푀) then by Hölder’s inequality we have 휏(|푔푛 − 핝 |) → 0, which is impossible since 1 = |휏(푔푛 − 핝)| ≤ 휏(|푔푛 − 핝|). Section (2.2) Szeg퐨̈ ’s Theorem and Outers for Noncommutative 퐇퐩 It has long been of great importance to operator theorists and operator algebraists to find noncommutative analogues of the classical ‘inner-outer factorization’ of analytic functions. We recall some classical results: If 푓 ∈ 퐿1 with푓 ≥ 0, then ∫ 푙표푔 |푓| > −∞iff 푓 = |ℎ| for an outer ℎ ∈ 퐻1 (iff 푓 = |ℎ|푝 for an outer ℎ ∈ 퐻푝). We will call this the Riesz-Szegö theorem. If 푓 ∈ 퐿1 with ∫ 푙표푔 |푓| > −∞, then 푓 = 푢ℎ, where u is unimodular and h is outer. Outer functions may be defined in terms of a simple equation involving ∫ 푙표푔 |푓|Such results are usually treated as consequences of the classical Szegö theorem, which is really a distance formula in terms of the entropy 푒푥푝(∫ 푙표푔 |푓|), and which in turn is intimately related to the 34

Jensen inequality (see e.g. [147]). In the noncommutative situation one wishes, for example, to find conditions on a positive operator 푇 which imply that 푇 = |푆| for an operator S which is in a ‘noncommutative Hardy class’, or, even better, which is ‘outer’ in some sense. There are too many such results to attempt a listing of them (see e.g. [88]). Central parts of this topic still seem to be poorly understood. As an example of this, we cite the main and now classical result of [6], concerning a Riesz-Szegö like factorization of a class of 퐵(퐻)-valued functions on the unit interval, which has resisted generalization in some important directions. We generalize the classical results above to the noncommutative 퐻푝 spaces associated with Arveson’s remarkable subdiagonal algebras [300]. Our generalization solves an old open problem (see the discussion in [88], and [178]). The approach which we take has been unavailable until now (since it relies ultimately on the recent solution in [163] of a 40 year old open problem from [300]). Moreover, the approach is very faithful to the original classical function theoretic route (see e.g. [147]), proceeding via noncommutative Szegö theorems. We have attempted to demonstrate that all the results in ([282]) the ‘generalized 퐻푝- theory’ for abstract function algebras from the 1960s, extend in an extremely complete and literal fashion, to the noncommutative setting of Arveson’s subdiagonal subalgebras of von Neumann algebras [300]. This may be viewed as a very natural merging of generalized Hardy space, von Neumann algebra, and noncommutative 퐿푝 space, techniques. See [62]. It completes the noncommutative extension of the basic Hardy space theory. As posited by Arveson, one should use the Fuglede-Kadison determinant ∆(푎) = 푒푥푝(휏(푙표푔 |푎|)) where τ is a trace, as a natural replacement in the noncommutative case for the quantity ∫ 푙표푔 푓 above. We use properties of the Fuglede-Kadison determinant to give several useful variants of the noncommutative Szegö theorem for 퐿푝(푀), including the one usually attributed to Kolmogorov and Krein. As applications, we generalize the noncommutative Jensen inequality, and generalize many of the classical resuts concerning outer functions, to the noncommutative 퐻푝 context. For a set S, we write S+for the set {푥 ∈ 푆 ∶ 푥 ≥ 0} see [62]. . We assume throughout that 푀 is a von Neumann algebra possessing a faithful normal tracial state τ. The existence of such τ implies that 푀 is a so-called finite von Neumann algebra, and that if 푥∗푥 = 1 in 푀, then 푥푥∗ = 1 too. Indeed, for any 푎, 푏 ∈ 푀, ab will be invertible precisely when a and b are separately invertible. We will also need to use a well known fact about inverses of an unbounded operator , and in our case 푇 will be positive, selfadjoint, closed, and densely defined. We recall that 푇 is bounded below if for some 휆 > 0 one has 푘푇 (휂)푘 ≥ 휆‖휂‖for all 휂 ∈ 푑표푚(푇 ). This is equivalent to demanding that |푇 | ≥ 휀1 for some 휀 > 0, and of course in this case, |푇 | has a bounded positive inverse. 퐴 (finite maximal) subdiagonal subalgebra of 푀 is a weak* closed unital subalgebra 퐴 of 푀 such that if 훷 is the unique conditional expectation guaranteed by [181] from 푀 onto 퐴 ∩ 퐴∗ ≝ 퐷 which is trace preserving (that is, 휏 ∘ 훷 = 휏), then: 훷(푎1푎2) = 훷(푎1)훷(푎2), 푎1, 푎2 ∈ 퐴. (1) One also must impose one further condition on A. There is a choice of at least eight equivalent, but quite different looking, conditions [62]; Arveson’s original one (see also [240]) is that 퐴 + 퐴∗ is weak* dense in M. In the classical function algebra setting [288], one

35 assumes that 풟 = 퐴 ∩ 퐴∗ is one dimensional, which forces 훷 = 휏(·)1. If in our setting this is the case, then we say that 퐴 is antisymmetric. The simplest example of a maximal subdiagonal algebra is the upper triangular matrices 퐴 in 푀푛. Here 훷 is the expectation onto the main diagonal. There are much more interesting examples from free group von Neumann algebras, Tomita-Takesaki theory, etc (see e.g. [300,161,178]). On the other end of the spectrum, 푀 itself is a maximal subdiagonal algebra (take 훷 = 퐼푑). It is therefore remarkable that so much of the classical 퐻푝 theory does extend to all maximal subdiagonal algebras. 푝 푝 By analogy with the classical case, we set 퐴0 = 퐴 ∩ 퐾푒푟(훷) and set 퐻 or 퐻 (퐴) to 푝 푝 be [퐴]푝, the closure of 퐴 in the noncommutative 퐿 space 퐿 (푀), for 푝 ≥ 1. More generally 푝 we write [푆]푝 for this closure of any subset 푆. We will often view 퐿 (푀) inside 푀̃, the set of unbounded, but closed and densely defined, operators on H which are affiliated to M. This is a ∗-algebra with respect to the ‘strong’ sum and product (see [183]). We order 푀̃ by its cone of positive (selfadjoint) elements. The trace τ extends naturally to the positive operators in 푀̃,. If 푝 1 ≤ 푝 < ∞, then 퐿 (푀, 휏) = {푎 ∈ 푀̃ ∶ 휏(|푎|푝) < ∞}, equipped with the norm ‖ · ‖푝 = 휏(| · |푝)1/푝 (see e.g.[77]). We abbreviate 퐿푝(푀, 휏) to 퐿푝(푀). Arveson’s Szegö formula is: 2 훥(ℎ) = 푖푛푓{휏(ℎ|푎 + 푑| ) ∶ 푎 ∈ 퐴0, 푑 ∈ 퐷, 훥(푑) ≥ 1} 1 for all ℎ ∈ 퐿 (푀)+. Here Δ is the Fuglede-Kadison determinant, originally defined on 푀 by 훥(푎) = 푒푥푝 휏(푙표푔 |푎|) 푖푓 |푎| > 0, and otherwise, 훥(푎) = 푖푛푓 훥(|푎| + ℰ1), the infimum taken over all scalars ℰ > 0 (see [26, 300]). We will discuss this determinant in more detail. Unfortunately, the just-stated noncommutative Szegö formula, and the (no doubt more important) associated Jensen’s inequality 훥(훷(푎)) ≤ 훥(푎), 푎 ∈ 퐴, resisted proof for nearly 40 years, although Arveson did prove them in his extraordinary original [300] for the examples that he was most interested in. The second proved in [163] that all maximal subdiagonal algebras satisfy these formulae. Settling this old open problem opened up the theory. 푝 An element ℎ ∈ 퐻 is said to be outer if 1 ∈ [ℎ퐴]푝. This definition is in line with e.g. Helson’s definition of outers in the matrix valued case he considers in [102]. We now state a sample of our results about outers. For example, we are able to improve on the factorization theorems from e.g. [58] in several ways: namely we show that if 푓 ∈ 퐿푝(푀) with 훥(푓) > 0 then 푓 may be essentially uniquely factored 푓 = 푢ℎ with 푢 unitary and ℎ outer. There is a much more obvious converse to this, too. We now have an explicit formula for the 푢 and ℎ. We refer to a factorization 푓 = 푢ℎ of this form as a Beurling-Nevanlinna factorization. It follows that in this case if 푓 ≥ 0 then 푓 = |ℎ| with h outer. This gives a solution to the problem posed in [88], and in [178]. If ℎ ∈ 퐻푝, and h is outer then 훥(ℎ) = 훥(훷(ℎ)). A converse is true: if 훥(ℎ) = 훥(훷(ℎ)) > 0 then h is outer. It follows that under some restrictions on 풟 = 퐴 ∩ 퐴∗, h is outer iff 훥(ℎ) = 훥(훷(ℎ)) > 0. There are many factorization theorems for subdiagonal algebras (see e.g. [300, 178, 149, 176, 88]), but as far as we know there are no noncommutative factorization results involving outers or the Fuglede-Kadison determinant. We mention for example Arveson’s original factorization result from [300], or Marsalli and West’s Riesz factorization of any 푓 ∈ 퐻1 as a 36

1 1 product푓 = 푔ℎ with 푔 ∈ 퐻푝, ℎ ∈ 퐻푞, + = 1. Some also require rather stronger 푝 푞 hypotheses, such as 푓−1 ∈ 퐿2(푀) (see e.g. [178]) The commutative case of most of the topics was settled in [281]. While certainly gave us motivation to persevere in our endeavor, we follow completely different lines, and indeed the results work out rather differently too. In particular, the quantity 휏(푒푥푝(훷(푙표푔 |푓|))), which plays a central role in most of the results in [281], seems to us to be unrelated to outers or factorization in the noncommutative setting. We remark that numerical experiments do seem to confirm the existence of a Jensen inequality 휏(푒푥푝(훷(푙표푔|푎|))) ≥ 휏(|훷(푎)|) for subdiagonal algebras. We remark that there are many other, more recent, generalizations of H∞, based around multivariable analogues of the Sz-Nagy-Foias model theory for contractions. The unilateral shift is replaced by left creation operators on some variant of Fock space. Many are currently intensively pursuing these topics, they are very important and are evolving in many directions. Although these theories also contain variants of Hardy space theory, they are quite far removed from subdiagonal algebras. For example, if one compares Popescu’s theorem of Szegö type from [90] with the Szegö theorem for subdiagonal algebras discussed here, one sees that they are only related in a very formal sense. The Fuglede-Kadison determinant Δ, and its amazing properties, is perhaps the main tool in the noncommutative 퐻푝 theory. In [277], Fuglede and Kadison study the determinant as a function on M. We will define it for elements of 퐿푞(푀) for any 푞 > 0. In fact, as was pointed out to us by Quanhua Xu, L. G. Brown investigated the determinant and its properties in the early 1980s, on a much larger class than 퐿푞(푀) (see [163]); indeed recently Haagerup and Schultz have thoroughly explicated the basic theory of this determinant for a very general class of τ −measurable operators (see [286]) as part of Haagerup’s amazing attack onthe invariant subspace problem relative to a finite von Neumann algebra. We will define the Fuglede-Kadison determinant for an elementh ℎ ∈ 퐿푞(푀), for any 푞 > 0, as follows. We set 훥(ℎ) = 푒푥푝 휏(푙표푔 |ℎ|) if |ℎ| > 휀1 for some 휀 > 0, and otherwise, 훥(ℎ) = 푖푛푓 훥(|ℎ| + 휖1), the infimum taken over all scalars ϵ > 0. To see that this is well-defined, we adapt the argument in [57], making use of the Borel functional calculus for unbounded operators applied to the inequality 1 0 ≤ 푙표푔 푡 ≤ 푡푞, 푡 ∈ [1, ∞). 푞 Notice that for any 0 < 휀 < 1, the function 푙표푔 푡 is bounded on [ε, 1]. So given ℎ ∈ 1 퐿 (푀)+ with ℎ ≥ 휀핝, it follows that (푙표푔 ℎ)푒[0,1] is similarly bounded. The previous centered 1 1 equation ensures that 0 ≤ (푙표푔 ℎ) ≤ ℎ푞푒[1, ∞) ≤ ℎ푞. Here푒 denotes the spectral 푒[1,∞) 푞 푞 [0,휆] resolution of ℎ. Thus if ℎ ∈ 퐿푞(푀) and ℎ ≥ 휖 then 푙표푔 ℎ ∈ 퐿1(푀). The following are the basic properties of this extended determinant which we shall need. Full proofs may be found in [286], which are valid for a very general class of unbounded operators (see also [62] for another (later) proof for the Lp(M) class).

Theorem (2.2.1)[66]: If 푝 > 0 and ℎ ∈ 퐿푝(푀) then 37

(i) 훥(ℎ) = 훥(ℎ∗) = 훥(|ℎ|). 푝 (ii) If ℎ ≥ 푔 푖푛 퐿 (푀)+ then 훥(ℎ) ≥ 훥(푔). (iii) If ℎ ≥ 0 then 훥(ℎ푞) = 훥(ℎ)푞 for any q > 0. (iv) 훥(ℎ푏) = 훥(ℎ)훥(푏) = 훥(푏ℎ) for any 푏 ∈ 퐿푞(푀) and any 푞 > 0. Throughout, 퐴 is a maximal subdiagonal algebra in 푀. We consider versions of Szegö’s formula valid in 퐿푝(푀) rather than 퐿2(푀). We will also prove a generalized Jensen inequality, and show that the classical Verblunsky-Kolmogorov- Krein strengthening of Szegö’s formula extends even to the noncommutative context. 1 It is proved in [61] that for ℎ ∈ 퐿 (푀)+ and 1 ≤ 푝 < ∞, we have 푝 훥(ℎ) = 푖푛푓{휏(ℎ|푎 + 푑| ): 푎 ∈ 퐴0, 푑 ∈ 풟, 훥(푑) ≥ 1}. We now prove some perhaps more useful variants of this formula 푞 1 푞 푝 푝 푞 Lemma (2.2.2)[66]:Ifℎ ∈ 퐿 (푀)+and 0 < 푝, 푞 < ∞, we have Δ(h) = 푖푛푓{휏(|ℎ 푏| ) : 푏 ∈ 푞 1 푝 푝 푞 푀+, 훥(푏) ≥ 1} = inf{휏(|푏ℎ | ) : 푏 ∈ 푀+, 훥(푏) ≥ }.The infimums are realized on the commutative von Neumann subalgebra 푀0 generated by ℎ, and are unchanged if in addition we also require 푏 to be invertible in 퐵. Proof: That the two infimums in the displayed equation are equal follows from the fact that ∗ 푝 ‖푥‖푝 = ‖푥 ‖푝for 푥 ∈ 퐿 (푀) (see [275]). Thus we just prove the first equality in that line. For 푏 ∈ 푀+, 훥(푏) ≥ 1, we have by Theorem (2.2.1) (iii) that 훥(|ℎ푞/푝푏|푝) = 훥(|ℎ푞/푝푏|푝) = 훥(|ℎ푞/푝푏|푝). Consequently, using facts from Theorem (2.2.1) again, we have 휏(|ℎ푞/푝푏|푝) ≥ 훥(|ℎ푞/푝푏|푝) = [훥(ℎ푞/푝)훥(푏)]푝 ≥ 훥(ℎ푞/푝)푝 = 훥(ℎ)푞. To complete the proof, it suffices to find, given ε > 0, an invertible b in (M0)+, the von 푞 1 Neumann algebra generated by ℎ, with 훥(푏) ≥ 1 and 휏(|ℎ푝 푏|푝)푞 < 훥(ℎ) + 휀. But for any 푏 ∈ 푞 푝 푝 푞 푝 (푀0)+ we have |ℎ 푏| = ℎ 푏 by commutativity, and then the result follows from an analysis of Arveson’s original definition of Δ(h) (see [59]). In particular since훥(ℎ푞) = 푞 푝 푖푛푓{휏(ℎ 푏 ): 푏 ∈ (푀0)+, 훥(푏) ≥ 1} by [59], an application of Theorem (2.2.1) (3) 1 푞 푝 푞 ensuresthat 훥(ℎ) = 푖푛푓{휏(ℎ 푏 ) : 푏 ∈ (푀0)+, 훥(푏) ≥ 1}. q Corollary (2.2.3)[66]:If h ∈ L (M)+ and 0 푝, 푞 < ∞, we have 푞 1 훥(ℎ) = 푖푛푓{휏(|ℎ푝 푎|푝)푞: 푎 ∈ 퐴, 훥(훷(푎)) ≥ 1} 푞 1 = 푖푛푓{휏(|ℎ푝|푝)푞: 푎 ∈ 퐴, 훥(훷(푎)) ≥ 1}. The infimums are unchanged if we also require a to be invertible in 퐴, or if we require 훷(푎)to be invertible in 풟. Proof:That the two infimums in the displayed equation are equal follows from the fact that ∗ 푝 ∗ ‖푥‖푝 = ‖푥 ‖ for 푥 ∈ 퐿 (푀) (see [277]), and by replacing 퐴 with 퐴 , which is also subdiagonal. Thus we just prove the first equality in that line. For 푎 ∈ 퐴, 훥(훷(푎)) ≥ 1 we have 푞 1 푞 1 푞 1 휏(|ℎ푝 푎|푝)푞 = 휏(|푎∗ℎ푝|푝)푞 = 휏(||푎∗|ℎ푝|푝)푞 ≥ 훥(ℎ),

38 by Lemma (2.2.2), since ∆(|푎∗|) = ∆(푎∗) = ∆(푎) ≥ ∆(훷(푎)) ≥ 1 (using Jensen’s inequality). Thus Δ(h) is dominated by the first infimum. On the other hand, by the previous 푞 1 푝 푝 푞 result there is an invertible b ∈ M+ with 훥(푏) ≥ 1 and 휏(|ℎ 푎| ) < 훥(ℎ) + 휀. By factorization, we can write b = |a∗| for an invertible a in A, and by Jensen’s formula [300, 163] we have 훥(훷(푎)) = 훥(푎) = 훥(푎∗) = 훥(푏 ≥ 1. Then 푞 1 푞 1 1 휏(|ℎ푝 푎|푝)푞 = 휏(|푎∗ℎ푝|푝)푞 = 휏(|푏 푎|푝)푞 < 훷(ℎ) + 휀. Corollary (2.2.4)[66]: (Generalized Jensen inequality) Let A be a maximal subdiagonal algebra. For any ℎ ∈ 퐻1 we have 훥(ℎ) ≥ 훥(훷(ℎ)). Proof: Using the퐿1-contractivity of 훷 we get 휏(||ℎ|푎|) = 휏(|ℎ푎|) = 휏(||훷(ℎ)|훷(푎)|), 푎 ∈ 퐴. Taking the infimum over such a with 훥(훷(푎)) ≥ 1, we obtain from Corollary (2.2.3), and Theorem (2.2.1) applied to 풟, that 훥(ℎ) = 훥(|ℎ|) ≥ 훥(|훷(ℎ)|) = 훥(훷(ℎ)). We recall that although 퐿푝(푀) is not a normed space if 1 > 푝 > 0, it is a socalled 푝 vlinear metric space with metric given by ‖푥 − 푦‖푝 for any 푥, 푦 ∈ 퐿푝(see [277]). Thus although the unit ball may not be convex, continuity still respects all elementary linear operations. 푞 푞 푞 푝 푝 Corollary (2.2.5)[66]:.Let ℎ ∈ 퐿 (푀)+ and 0 < 푝, 푞 < ∞. If ℎ ∈ [ℎ 퐴0]푝, then훥(ℎ) = 0. 푞 푞 푝 푝 Conversely, if 퐴 is antisymmetric and Δ(ℎ) = 0, then ℎ ∈ [ℎ 퐴0]푝. Indeed if 퐴 is antisymmetric, then 푞 1 푝 푝 푞 훥(ℎ) = 푖푛푓{휏(|ℎ (1 − 푎0)| ) : 푎0 ∈ 퐴0}. Proof:The first assertion follows by taking a in the infimum in Corollary (2.2.3) to be of the form 1 − 푎0 for 푎0 ∈ 퐴0. 푞 1 푞 푝 푝 푞 푝 If A is antisymmetric, and if 푡 ≥ 1 with 휏(|ℎ (푡1 − 푎0)| ) < 훥(ℎ) + 휀, then 휏(|ℎ (1 − 1 푝 푞 푎0)| ) < 훥(ℎ) + 휀. From this the last assertion follows that the infimum’s in Corollary (2.2.3) can be taken over terms of the form 1 + 푎0 where 푎0 ∈ 퐴0. If this infimum was 0 we 푞 푞 푝 푝 could then find a sequence 푎푛 ∈ 퐴0 with ℎ (1 + 푎푛) → 0 with respect to ‖·‖푝. Thus ℎ ∈ 푞 푝 [ℎ 퐴0]푝. We close with the following version of the Szegö formula valid for general positive linear functionals. Although the classical version of this theorem is usually attributed to Kolmogorov and Krein, we have been informed by Barry Simon that Verblunsky proved it first, in the mid 1930’s (see e.g. [267]): Theorem (2.2.6)[66]:Noncommutative Szegö -Verblunsky- Kolmogorov- Krein theorem) Let ωbe a positive linear functional on 푀, and let 휔n and 휔푠 be its normal and singular parts 1 respectively, with 휔푛 = 휏(ℎ) forℎ ∈ 퐿 (푀)+. If 푑푖푚(퐷) < ∞, then 훥(ℎ) = 푖푛푓{휔(|푎|2): 푎 ∈ 퐴, 훥(훷(푎)) ≥ 1}. The infimum remains unchanged if we also require 훷(푎) to be invertible in 풟.

Proof: Suppose that 푑푖푚(풟) < ∞. All terminology and notation will be as [61], the preamble to the proof of the noncommutative 퐹&푀. Riesz [61]. For the sake of clarity we pause to highlight the most important of these. If (휋휔, 퐻휔, 훺휔) is the 퐺푁푆 representation engendered by 휔, there exists a central projection 푝0 in 휋휔(푀)′′ such that for any 휉, 휓 ∈ 퐻휔 the functionals 푎 → 〈휋휔(푎)푝0휉, 휓〉 and 푎 → 〈휋휔(푎)(1 − 푝0)휉, 휓〉 on 푀 are respectively the normal and singular parts of the functional 푎 → 〈휋휔(푎)휉, 휓〉 [180]. In this representation 훺0 will denote the orthogonal projection of Ωω onto the closed subspace 휋휔(퐴0)훺휔. Let 푑 ∈ 풟 be given. We may of course select a sequence (푓푛) ⊂ 퐴0 so that 푙푖푚푛→∞휋휔(푓푛)훺휔 = 0. By the ideal property of 퐴0 and continuity, it then follows that ̅̅̅̅̅̅̅̅̅̅̅̅̅̅ 휋휔(푑)훺0 = 푙푖푚 휋휔 (푑푓푛)훺휔 ∈ 휋휔(퐴0)훺휔. 푛→∞ Once again using the ideal property of A0, the fact that 훺휔 − 훺0 ⊥ 휋휔(퐴0)훺휔 now ∗ forces 휋휔(푑)훺0, 휋휔(푎)훺휔 = 〈훺휔 − 훺0, 휋휔(푑 푎)훺휔〉 = 0 for every 푎 ∈ 퐴0. Therefore ̅̅̅̅̅̅̅̅̅̅̅̅̅̅ 휋휔(푑)훺0(훺휔 − 훺0) ⊥ 휋휔(퐴0)훺휔. From the facts in the previous two centered equations, it now follows that 휋휔(푑)훺0is the ̅̅̅̅̅̅̅̅̅̅̅̅̅̅ orthogonal projection of 휋휔(푑)훺휔 onto 휋휔(퐴0)훺휔. Using this fact we may now repeat the arguments of [61] for the functional 휔푑(·) = 〈휋휔(·)휋휔푑)(훺휔 − 훺0), 휋휔(푑)(훺휔 − 훺0)〉 to conclude that ωd is normal, with 푝0(휋휔(푑)(훺휔 − 훺0)) = 휋휔(푑)(훺휔 − 훺0), where 푝0 is the central projection in 휋휔(푀)′′ mentioned above, and also: 푝0휋휔(푑)(훺휔 − 훺0) ⊥ 푝0휋휔(퐴0)훺휔 and 푝0(휋휔(푑)훺0) is the orthogonal projection of p0(πω(d)Ωω) onto 푝0(휋휔(퐴0)훺휔). Thus we arrive at the fact that 2 푖푛푓 휔(|푑 + 푎| ) = 푖푛푓 〈휋휔(푑)훺휔 + 휋휔(푎)훺휔, 휋휔(푑)훺휔 + 휋휔(푎)〉 푎∈퐴0 푎∈퐴0 2 = 푖푛푓 ‖휋휔(푑)훺휔 − 휋휔(푎)‖ = 〈휋휔(푑)(훺휔 − 훺0), 휋휔(훺휔 − 훺0)〉 푎∈퐴0 = 〈푝0휋휔(푑)(훺휔 − 훺0), 푝0휋휔(푑)(훺휔 − 훺0)〉 = 푖푛푓 〈푝0휋휔(푑)훺휔 + 푝0휋휔(푎)훺휔, 푝0휋휔(푑)훺휔 + 푝0휋휔(푎)〉 푎∈퐴0 2 2 = 푖푛푓 휔푛(|푑 + 푎| ) = 푖푛푓 휏(ℎ|푑 + 푎| . 푎∈퐴0 푎∈퐴0 On taking the infimum over all 푑 ∈ 풟 with 훥(푑) ≥ 1, the result follows from Corollary (2.2.3). Throughout A is a maximal subdiagonal algebra. We recall that if ℎ ∈ 퐻1then h is outer if 1 [ℎ퐴]1 = 퐻 . An inner element is a unitary which happens to be in 퐴. 푝 1 푝 Lemma (2.2.7)[66]:Let 1 ≤ 푝 ≤ ∞. Then ℎ ∈ 퐿 (푀) and ℎ is outer in 퐻 , iff [ℎ퐴]푝 = 퐻 . (Note that [·]∞ is the weak* closure.) If these hold, then ℎ ∉ [ℎ퐴0]푝. 푝 1 1 Proof: It is obvious that if [ℎ퐴]푝 = 퐻 . then [ℎ퐴]1 = 퐻 . Conversely, if [ℎ퐴]1 = 퐻 and ℎ ∈ 푝 퐿 (푀), then the first part of the proof of [58] applied to [ℎ퐴]푝actually shows that [ℎ퐴]푝 = 푝 [ℎ퐴]1 ∩ 퐿 (푀)for all 1 ≤ 푝 ≤ ∞. Hence by [148], we have 푝 1 푝 푝 [ℎ퐴]푝 = [ℎ퐴]1 ∩ 퐿 (푀) = 퐻 ∩ 퐿 (푀) = 퐻 .

If ℎ ∈ [ℎ퐴0]푝then1 ∈ [ℎ퐴]푝 ⊂ [[ℎ퐴0]푝퐴]푝 ⊂ [ℎ퐴0]푝. Now continuously extends to a map p p 푝 which contractively maps L (M) onto L (풟) (see e.g. [176]). If ℎ푎푛 → 1 in 퐿 , with 푎푛 ∈ 퐴0, then 훷(ℎ푎푛) = 훷(ℎ)훷(푎푛) = 0 → 훷(1) = 1, This forces 훷(핝) = 1, a contradiction. Lemma (2.2.8)[66]:If ℎ ∈ 퐻1 is outer then as an unbounded operator h has dense range and trivial kernel. Thus ℎ = 푢|ℎ| for a unitary 푢 ∈ 푀. Also, 훷(ℎ) has dense rangeand trivial kernel. Proof: If ℎ is considered as an unbounded operator, and if 푝 is the range projection of h, then 1 ⊥ since there exists a sequence (푎푛)in A with ℎ푎푛 → 1in 퐿 -norm, we have that 푝 = 0. Thus the partial isometry 푢 in the polar decomposition of ℎ is an isometry, and hence is a unitary, in 푀. It follows that |h| has dense range, and hence it, and ℎ also, have trivial kernel. For the last part note that 1 1 퐿 (풟) = 훷(퐻 ) = 훷([ℎ퐴]1) = [훷(ℎ)풟]1. Thus we can apply the above arguments to 훷(ℎ)too. There is a natural equivalence relation on outers: Proposition (2.2.9)[66]:If ℎ ∈ 퐻푝 is outer, and if 푢 is a unitary in 풟, then ℎ′ = 푢ℎ is outer in Hp too. If ℎ, 푘 ∈ 퐻푝 are outer, then |ℎ| = |푘| iff there is unitary 푢 ∈ 풟 with ℎ = 푢푘. Such u is unique. Proof: The first part is just as in the classical case. If ℎ, 푘 ∈ 퐻1 and |ℎ| = |푘|, then it follows, as in [281], that ℎ = 푢푘 for a unitary 푢 ∈ 푀 with 푢 , 푢∗ ∈ 퐻1. Thus 푢 ∈ 퐻1 ∩ 푀 = 퐴 (see [148]), and similarly 푢∗ ∈ 퐴, and so 푢 ∈ 풟. The uniqueness of u follows since the left support projection of an outer is 1 (see proof of Lemma (2.2.8)). 2 1 As in the classical case, if ℎ ∈ 퐻 is outer, then h 2is outer in 퐻 . Indeed one may follow the proof on [281], and the same proof shows that a product of any two outers is outer (see also the last lines of the proof of Theorem (2.2.13) below). We do not know whether every outer in H1 is the square of an outer in 퐻2. The first theorem is a generalization of the classical characterization of outers in 퐻푝: Theorem (2.2.10)[66]: Let 퐴 be a subdiagonal algebra, let 1 ≤ 푝 ≤ ∞ and ℎ ∈ 퐻푝. If h is outer then 훥(ℎ) = 훥(훷(ℎ)). If훥(ℎ) > 0, this condition is also sufficient for h to be outer. Note that if 푑푖푚(퐷) < ∞, then 훷(ℎ) will be invertible for any outer h by Lemma (2.2.8). Thus in this case it is automatic that 훥(훷(ℎ)) > 0. Proof: The case for general p follows from the p = 1 case and Lemma (2.2.7). Hence we may suppose that 푝 = 1. 1 First suppose that ℎis outer. Given any 푑 ∈ 퐿 (퐷) and any a0 ∈ [A0]1, we clearly have ∗ that 휏(|푑 − 푎0|) ≥ 휏(|푑| − 푢 푎0) = 휏(|푑|), where 푢is the partial isometry in the polar decomposition of 푑. In other words, for any 푎 ∈ [퐴]1 we have ‖훷(푎)‖1 = 푖푛푓 ‖푎 − 푎0‖1 = 푖푛푓 ‖푎 − 푎0‖1 . 푎0∈퐴0 푎0∈|퐴0|1 Therefore ̃ ̃ ̃ 휏(|훷(ℎ) 푑|) = 푖푛푓 휏(|ℎ 푑 − 푎0|), 푑 ∈ 풟. 푎0∈퐴0 Notice that [ℎ퐴0]1 = [[ℎ퐴]1퐴0]1 = [[퐴]1퐴0]1 = [퐴0]1. Thus the above equality may alternatively be written as 41

̃ ̃ ̃ 휏(|훷(ℎ) 푑|) = 푖푛푓 휏(|ℎ 푑 − 푎0|), 푑 ∈ 풟. 푎0∈퐴0 ̃ ̃ Finally notice that |훷(ℎ)푑 ̃| = ||훷(ℎ)|푑 ̃| and |ℎ( 푑 − 푎0)| = ||ℎ|( 푑 − 푎0)|.Therefore if now we take the infimum over all 푑̃ ∈ 풟with훥( 푑̃) ≥ 1, Szegö’s theorem will force 훥(훷(ℎ)) = 훥(|훷(ℎ)|) = 훥(|ℎ|) = 훥(ℎ). Next suppose that 훥(ℎ) = 훥(훷(ℎ)) > 0. We will first consider the case ℎ ∈ [퐴]2.By Lemma ̃ ∗ (2.2.7) we then need only show that h is outer with respect to[퐴]2. Replace h by ℎ = 푢 ℎ where u is the partial isometry in the polar decomposition of 훷(ℎ). If we can show that eh is outer, it (and hence also 푢∗) will have dense range, which would force 푢∗ to be a unitary. Thus ℎ = 푢ℎ̃ would then also be outer. Now notice that by construction |ℎ| ≥ |ℎ̃| and 훷(ℎ̃) = |훷(ℎ)|. From this and the generalized Jensen inequality we have 훥(ℎ) = 훥(|ℎ|) ≥ 훥(|ℎ̃|) = 훥(ℎ̃) ≥ 훥(훷(ℎ̃)) = 훥(훸(ℎ)) = 훥(ℎ). Thus 훥(ℎ̃) = 훥(훷(ℎ̃)) > 0. We may therefore safely pass to the case where 훷(ℎ) ≥ 0. By multiplying with a scaling constant, we may also clearly assume that 훥(ℎ) = 1. For any 푑 ∈ 풟 and 푎 ∈ 퐴0 we have 휏(|1 − ℎ(푑 + 푎)|2) = 휏(1 − 훷(ℎ)푑 − 푑∗훷(ℎ)) + 휏(|ℎ(푑 + 푎)|2). (2) To see this, simply combine the fact that 휏 ∘ 훷 = 휏 with the observation that 훷(ℎ(푑 + 푎)) = 훷(ℎ)훷(푑 + 푎) = 훷(ℎ)푑. With 푑, a as above, notice that 휏(|ℎ(푑 + 푎)|2) = 휏(||ℎ|(푑 + 푎)|2). By Szegö’s theorem in the form of Corollary (2.2.3), we may select sequences (푑푛) ⊂ {푑 ∈ −1 풟 ∶ 훥(푑) ≥ 1}, (푎푛) ⊂ 퐴0, such that 2 2 2 푙푖푚 (|ℎ(푑푛 + 푎푛)| ) = 훥(|ℎ| ) = 훥(ℎ) = 1. 푛→∞휏 Claim: we may assume the 푑푛’s to be positive. To see this, notice that the invertibility of the ∗ 푑푛’s means that for each 푛 we can find a unitary 푢푛 ∈ 풟 so that 푑푛푢푛 = |푑 푛|. Since for each 푛 we have 2 2 ∗ 2 휏(|ℎ(푑푛 + 푎푛)| ) = 휏(|ℎ(푑푛 + 푎푛)푢푛| ) = 휏(|ℎ(|푑 푛| + 푎푛푢푛)| ), 1/2 1/2 the claim follows. Notice that then 휏(훷(ℎ)푑푛) = 휏(푑푛 훷(ℎ)푑푛 ) ≥ 0. Using in turn the 2 퐿 -contractivity of 훷, the fact that 훷(ℎ(푑푛 + 푎푛)) = 훷(ℎ)푑푛, and Hölder’s inequality, we conclude that 2 2 2 휏(|ℎ(푑푛 + 푎푛)| ) ≥ 휏(|훷(ℎ)푑푛| ) ≥ 휏(|훷(ℎ)푑푛|) 2 2 ≥ 휏(훷(ℎ)푑푛) ≥ 훥(훷(ℎ)) = 1. 2 Since 푙푖푚 휏(|ℎ(푑푛 + 푎푛)| ) = 1, we must therefore also have 푙푖푚 휏(훷(ℎ)푑푛) = 1.But if this 푛→∞ 푛→∞ 2 is the case then equation (2) assures us that ℎ(푑푛 + 푎푛) → 1 in 퐿 -norm as푛 → ∞. That is, 1 ∈ [ℎ퐴]2. Clearly ℎ must then be outer.Now let ℎ ∈ [퐴]1. By noncommutative Riesz factorization (see [181]) we may select ℎ1, ℎ2 ∈ [퐴]2 so that ℎ = ℎ1ℎ2. Since훥(ℎ1)훥(ℎ2) = 훥(ℎ) = 훥(훷(ℎ)) = 훥(훷(ℎ1))훥(훷(ℎ2)) > 0 and 훥(ℎ푖) ≥ 훥(훷(ℎ푖)) for each 푖 = 1, 2 (by the generalized Jensen inequality), we must have 훥(ℎ푖) = 훥(훷(ℎ푖)) for each 푖 = 1, 2. Thus both ℎ1 and ℎ2 must be outer elements of [A]2. Consequently [ℎ퐴]1 = [ℎ1ℎ2퐴]1 = [ℎ1[ℎ2퐴]2]1 = [ℎ1[퐴]2]1 = [[ℎ1[퐴]2]2]1 = [[퐴]2]1 = [퐴]1, so that h is outer as required. Note that: In the general non-antisymmetric case, one can have outers with

훥(ℎ) = 0. Indeed in the case that 퐴 = 푀 = 퐿∞[0, 1], then outer functions in 퐿2 are exactly the ones which are a.e. nonzero. One can easily find an increasing function ℎ: [0, 1] → (0, 1] 1 satisfying 훥(ℎ) = 0, or equivalently 푙표푔 ℎ = −∞. See also [280]. This prompts the ∫0 following: Definition (2.2.11)[66]: We say that h is strongly outer if it is outer and 훥(ℎ) > 0.Note that if 푑푖푚(퐷) < ∞, then every outer h is strongly outer. 1 1 1 Corollary (2.2.12)[66]: Let 1 ≤ 푝, 푞, 푟 ≤ ∞ with = + and let ℎ = ℎ ℎ with ℎ ∈ 푝 푞 푟 1 2 1 푞 푟 푟 퐻 and ℎ2 ∈ 퐻 . If 훥(ℎ) > 0 then ℎ is outer in 퐻 iff both ℎ1and ℎ2 are outer. 푝 Corollary (2.2.13)[66]: If푓 ∈ 퐿 (퐷) with 훥(푓) > 0 then f is outer. Indeed there exist 푑푛 ∈ 풟 푝 with 훥(푓푑푛) ≥ 1, and 푓푑푛 → 1 in 2-norm. Also, any 푓 ∈ 퐿 (푀) with 훥(푓) > 0 has left and right support projections equal to 1. That is, as an unbounded operator it is one-to-one and has dense range. Proof: For the first assertion note that 훷(푓) = 푓 and so 훥(푓) = 훥(훷(푓)) > 0. An inspection of the proof of the theorem gives the dn with the asserted properties. Thus f clearly has left support projection 1, and by symmetry the right projection is 1 too. Finally note that for the last assertion we may assume that 푀 = 풟. 푝 푝 Corollary (2.2.14)[66]: If 1 ≤ 푝 ≤ ∞and 훥(ℎ) > 0 then ℎ is outer in 퐻 iff [퐴ℎ]푝 = 퐻 . Proof: Replacing 퐴 by 퐴∗, it is trivial to see that 훥(ℎ) = 훥(훷(ℎ)) > 0, is equivalent to 훥(ℎ∗) = 훥(훷(ℎ∗)) > 0. The latter is equivalent to h∗ being outer in 퐻2(퐴∗) = (퐻2)∗; or 2 ∗ ∗ ∗ equivalently, to (퐻 ) = [ℎ 퐴 ]2. Taking adjoints again gives the result. 2 Proposition (2.2.15)[66]: If ℎ ∈ 퐻 , then ℎ is outer iff the wandering subspace of [ℎ퐴]2(see [280, 62]) has a separating cyclic vector for the 풟 action, and 2 2 ‖훷(ℎ)‖ = 푖푛푓{휏(|ℎ(1 − 푎0)| ): 푎0 ∈ 퐴0}. 1 Proof: (Following [281].) For 푥 ∈ 퐿1(푀) set 훿(푥) = 푖푛푓{휏(||푥| (1 − 푎 )|2): 푎 ∈ 퐴 }. 2 0 0 0 2 2 2 First suppose that ℎ ∈ 퐻 is outer. Then [ℎ퐴]2 ⊖ [ℎ퐴0]2 = 퐻 ⊖ [퐴0]2 = 퐿 (풟), which has a separating cyclic vector. We next prove that if ℎ ∈ 퐻2 is outer, then ‖훷(ℎ)‖2 = 훿(|ℎ|2). To do this we view 훷 as the orthogonal projection from 퐿2(푀) onto 퐿2(풟), which 2 restricts to the orthogonal projection 푃 from [A]2 onto L (풟). For any orthogonal projection 푃 from a Hilbert space onto a subspace 퐾, we have ‖푃(휁)‖ = 푖푛푓{‖휁 − 휂‖: 휂 ∈ ⊥ 2 2 퐾 }. Thus‖훷(ℎ)‖ = 푖푛푓{휏(|ℎ − 푎0| ): 푎0 ∈ [퐴0]2}.Since h is outer, we have 2 [[ℎ퐴]2퐴0]2 = [퐻 퐴0]2, 표푟 [ℎ퐴0]2 = [퐴0]2. Thus 2 2 2 ‖훷(ℎ)‖ = 푖푛푓{휏(|ℎ − ℎ푎0| : 푎0 ∈ 퐴0} = 훿(|ℎ| ). Conversely, suppose that the wandering subspace of [ℎ퐴]2 has a separating cyclic vector. By 2 2 2 [57], we have [ℎ퐴]2 = 푢퐻 for a unitary 푢 ∈ [ℎ퐴]2 ⊂ 퐻 . We have ℎ = 푢푘, with 푘 ∈ 퐻 , and ∗ 2 2 [퐴]2 = 푢 [ℎ퐴]2 = 퐻 . So 푘 is outer. If ‖훷(ℎ)‖ = 훿(ℎ), then using the notation in the last paragraph, ‖훷(푢)훷(푘)‖2 = 훿(|푢푘|2) = 훿(|푘|2) = ‖훷(푘)‖2. That is, 휏(훷(푘)∗(1 − 훷(푢)∗훷(푢))훷(푘)) = 0. Since by Lemma (2.2.8) the left support ∗ projection of 훷(푘) is 1, the functional 푎 → 휏(훷(푘) 푎훷(푘)) is faithful on 푀+ (indeed, ∗ ∗ 휏(훷(푘) 푎훷(푘) ≠ 0 for any non-zero 푎 ∈ 푀+, which forces 훷(푢) 훷(푢) = 1. A simple

43 computation shows that 훷(|푢 − 훷(푢)|2) = 1 − 훷(푢)∗훷(푢) = 0, so that 푢 = 훷(푢) ∈ 풟.Thus h = uk is outer.A classical theorem of Riesz-Szegö states that if 푓 ∈ 퐿1 with 푓 ≥ 0, then R log 푓 > −∞ iff 푓 = |ℎ| for an outer ℎ ∈ 퐻1 iff 푓 = |ℎ|2 for an outer ℎ ∈ 퐻2. We now turn to this issue in the noncommutative case. We are adapting ideas of Helson-Lowdenslager and Hoffman: Lemma (2.2.16)[66]:Suppose that A is a maximal subdiagonal algebra, and that 푘 ∈ 2 2 퐿 (푀)with 푘 ∉ [푘퐴0]2. Let 푣 be the orthogonal projection of 푘 onto [푘퐴0]2. Then |푘 − 푣| = 훷(|푘 − 푣|2) ∈ 퐿1(풟). Also, 훥(|푘 − 푣|) ≥ 훥(푘). Proof:Suppose that 푘푎푛 → 푣,with 푎푛 ∈ 퐴0.Clearly 푘 − 푣 ⊥ 푘(1 − 푎푛)푎0 ∈ 푘퐴0for all 푎0 ∈ 2 퐴0. In the limit, 푘 − 푣 ⊥ (푘 − 푣)푎0.That is,τ(|푘 − 푣| 푎0)=0, which by [59] implies that |푘 − 푣|2 = 훷(|푘 − 푣|2) ∈ 퐿1(풟).For the last assertion, note that by Lemma (2.2.2) we have 2 2 훥(|푘 − 푣| ) = 푖푛푓{휏(|(푘 − 푣)푑| ): 푑 ∈ 풟+with 훥(푑) ≥ 1}.But since 푣푑 ∈ [푘퐴0]2 for every 푑 ∈ 풟, we may apply Szeg표̈’s theorem to conclude that this infimum majorises in푓{휏(|푘푑 − 2 2 2 푘푎0| ): 푑 ∈ 퐷+with 훥(푑) ≥ 1, 푎0 ∈ 퐴0} = 훥(|푘| ) = 훥(푘) ,using the fact that |푘푑 − 푘푎0| = ||푘|(푑 − 푎0)|. Theorem (2.2.17)[66]:Suppose that 퐴 is a maximal subdiagonal algebra, and that 푘 ∈ 퐿2(푀). Let 푣be the orthogonal projection of 푘 onto [푘퐴0]2. If 훥(푘) > 0, then 푘 has an (essentially unique) Beurling-Nevanlinna factorization 푘 = 푢ℎ, where 푢 is a unitary in 푀, and equals the partial isometry in the polar decomposition of 푘 − 푣, and h is strongly outer and equals 푢∗푘. We also have 훥(푘) = 훥(푘 − 푣).If|푘 − 푣|is bounded below then (푘 − 푣)푑 = 푢 for some 푑 ∈ 풟. 1 Proof: By Corollary (2.2.5), 푘 ∉ [푘퐴0]2. By the Lemma, |푘 − 푣|2 ∈ 퐿 (풟). Let 푢 be the partial isometry in the polar decomposition of 푘 − 푣. Since 훥(푘 − 푣) ≥ 훥(푘) > 0 by the Lemma, we deduce from Corollary (2.2.13) that 푢 is surjective, and hence is a unitary. In the case that |푘 − 푣| is bounded below let 푑 = |푘 − 푣| − 1 ∈ 풟+, and then 푢 = (푘 − 푣)푑. Let ∗ 2 ∗ ∗ 2 ℎ = 푢 푘 ∈ 퐿 (푀). We claim that 휏(푢 푘푎0) = 0 for all 푎 0 ∈ 퐴0, so that ℎ = 푢 푘 ∈ 퐿 (푀) ⊖ ∗ 2 [퐴0 ]2 = 퐻 . To see this, let 푒푛 be the spectral projection of |푘 − 푣| corresponding to the interval [0, 1/푛]. Then by elementary spectral theory, and since 푘 − 푣 = 푢|푘 − 푣|, we have 1 1 1 − 푒 = |푘 − 푣|푟 for some 푟 ∈ 풟. (Take 푟 = 푔(|푘 − 푣|) where 푔 is 휒( , ∞)). Thus 푛 푡 푛 ∗ ∗ ∗ ∗ 휏(푎0푘 푢(1 − 푒푛)) = 휏(푎0푘 (푘 − 푣)푟) = 0, ∗ since 푘푎0푟 ∈ [푘퐴0]2 and 푘 − 푣 ⊥ [푘퐴0]2. On the other hand, by the Borel functional calculus it is clear that 푒푛 → 푒 strongly, where e is the spectral projection of |푘 − 푣| corresponding to {0}. Since 훥(|푘 − 푣|) ≥ 훥(푘) > 0 by the Lemma, it is easy to see by spectral theory that 푒 = 0 (this is essentially corresponds to the fact that a positive function f ∗ ∗ which is 0 on a nonnull set has ∫ 푙표푔 푓 = −∞). We conclude that (푎0푘 푢푒푛) → 0, and it follows that ∗ ∗ ∗ ∗ ∗ ∗ 휏(푎0푘 푢) = 휏(푎0푘 푢푒푛) + 휏(푎0푘 푢(1 − 푒푛)) = 0. To see that 푢∗푘 is outer, we will use the criterion in Theorem (2.2.10). We claim that 훷(푢∗푘) = ∗ |푘 − 푣|. To see this, note that by the last paragraph we have τ(푢 푥) = 0 for any 푥 ∈ [푘퐴0]2 and in particular for 푥 = 푣푐 for any 푐 ∈ 풟. We have 휏(훷(푢∗푘)푐) = 휏(푢∗푘푐) = 휏(푢∗(푘 − 푣)푐) = 휏(|푘 − 푣|푐).

Since this holds for any 푐 ∈ 풟 we have 훷(푢∗푘) = |푘 − 푣|. Thus we have by the generalized Jensen inequality (2.2.4) that 훥(푘) = 훥(푢∗푘) ≥ 훥(훷(푢∗푘)) = 훥(|푘 − 푣|) ≥ 훥(푘). Hence ℎ = 푢∗푘 is outer by Theorem (2.2.10). The uniqueness now follows. Corollary (2.2.18)[66]:Suppose that 퐴 is a maximal subdiagonal algebra with 풟finite dimensional, and that 푘 ∈ 퐿2(푀) with 훥(푘) > 0. Let 푣 be the orthogonal projection of k onto [푘퐴0]2. Then |푘 − 푣| is invertible, and all the conclusions of the previous theorem hold. Proof: By the above, |푘 − 푣| ∈ 퐿1(풟) = 풟, and 훥(|푘 − 푣|) ≥ 훥(푘) > 0. Thus |푘 − 푣|is invertible since 풟 is finite dimensional. The rest follows from the previous theorem. We next give a refinement of the ‘Riesz factorization’ into a product of two 퐻2 functions: Corollary (2.2.19)[66]:. If 퐴 is a maximal subdiagonal algebra with 풟finite dimensional, and if 푓 ∈ 퐿1(푀) with 훥(푓) > 0, then there exists an outer ℎ2 ∈ 퐻2, an invertible 푑 ∈ 풟 with 1 훥(푑) = , and an ℎ ∈ [푓퐴 ] such that 푓 − ℎ ∈ 퐿2(푀), and 푓 = (푓 − ℎ )푑ℎ . If also √훥(푓) 1 0 1 1 1 2 1 1 2 푓 ∈ 퐻 , then this can be arranged with ℎ1 ∈ 퐻 , 훷(ℎ1) = 0, and 푓 − ℎ1 ∈ 퐻 . 1 Proof: Let 푘 = |푓|2 . By Corollary (2.2.5) we have 푘 ∉ [푘퐴0]2. If 푢, 푣 are as in Theorem (2.2.16), and if 푓 = 푤|푓| = 푤푘2 is the polar decomposition of 푓, then ∗ 푓 = (푤푘푢)(푢 푘) = (푤푘(푘 − 푣))푑ℎ2 = (푓 − ℎ1)푑ℎ2 ∗ where ℎ2 = 푢 푘 and ℎ1 = 푤푘푣. 2 2 1 If 푘푎푛 → 푣 in L norm, with 푎푛 ∈ 퐴0, then 푓푎푛 = 푤푘 푎푛 → 푤푘푣 in 퐿 norm. Thus ℎ1 ∈ −1 2 [푓퐴0]1. Also, 푓 − ℎ1 = 푤푘푢푑 ∈ 퐿 (푀) (recall that since 풟 is finite dimensional, 푑 − 1 = 1 1 1 2 |푘 − 푣| ∈ 풟). If 푓 ∈ 퐻 , then ℎ1 ∈ [푓퐴0]1 ⊂ 퐻 , and 훷(ℎ1) = 0.So 푓 − ℎ1 ∈ 퐻 ∩ 퐿 (푀) ⊂ 2 ∗ 2 퐿 (푀) ⊖ [퐴 ]2 = 퐻 . Corollary (2.2.20)[66]:.If 퐴 is a maximal subdiagonal algebra, and if 푓 ∈ 퐿1(푀) with 훥(푓) > 0, then there exists a strongly outer ℎ ∈ 퐻1, and a unitary 푢 ∈ 푀 with 푓 = 푢ℎ. Proof: By the proof of Corollary (2.2.19), and in that notation, we have 푓 = 푤푘푢ℎ2 for an outer ℎ2. Note that 푤 is a unitary, since 푓 has dense range (Corollary (2.2.13)). Since 훥(푤푘푢) = 훥(푘) > 0, we have by the last theorem that 푤푘푢 = 푈ℎ1 for a unitary 푈 and 2 strongly outer ℎ1 ∈ 퐻 . Let ℎ = ℎ1ℎ2. Corollary (2.2.21)[66]:If 퐴 is a maximal subdiagonal algebra, and 푓 ∈ 퐿푝(푀) then 훥(푓) > 0 iff 푓 = 푢ℎ for a unitary 푢 and a strongly outer ℎ ∈ 퐻푝. Moreover, this factorization is unique up to a unitary in 풟. Proof: (⇒) By Corollary (2.2.20) we obtain the factorization with outer ℎ ∈ 퐻1. Since |푓| = |ℎ| we have ℎ ∈ 퐿푝(푀) ∩ 퐻1 = 퐻푝 (using [153]), and 훥(ℎ) > 0. (⇐) We have Δ(푓) = 훥(푢)훥(ℎ) > 0. The uniqueness of the factorization was discussed after Proposition (2.2.9). Note that. The 푢 in the last result is necessarily in [푓퐴]푝indeed if ℎ푎푛 → 1 with 푎푛 ∈ 퐴, then 푓푎푛 = 푢ℎ푎푛 → 푢. Corollary (2.2.22)[66]:If 퐴 is a maximal subdiagonal algebra, then 푓 ∈ 퐻푝 with 훥(푓) > 0 iff 푓 = 푢ℎ for an inner 푢 and a strongly outer ℎ ∈ 퐻푝. Moreover, this factorization is unique up to a unitary in 풟. 45

1 1 1 Proof: Clearly f is also in H . Then 푢 is necessarily in [푓퐴]푝 ⊂ 퐻 . So 푢 ∈ 퐻 ∩ 푀 = 퐴(see [149]). Thus 푢is ‘inner’ (i.e.is a unitary in 퐻∞ = 퐴). An obvious question is whether there are larger classes of subalgebras of 푀 besides subdiagonal algebras for which such classical factorization theorems hold. The following shows that, with a qualification, the answer to this is in the negative: Proposition (2.2.23)[66]: Suppose that 퐴is a tracial subalgebra of 푀 in the sense, such that 2 every 푓 ∈ 퐿 (푀) with 훥(푓) > 0 is a product 푓 = 푢ℎ for a unitary 푢 and an outer h ∈ [A]2. Then 퐴 is a finite maximal subdiagonal algebra. Proof: Suppose that 퐴 is a tracial subalgebra of 푀 with this factorization property. We will show that 퐴 satisfies the ‘퐿2-density’ and the ‘unique normal state extension’ properties which together were shown in [59] to imply that 퐴is subdiagonal. As in [59, 153,149], 퐴∞ is the tracial algebra 퐴∞ = 푀 ∩ [퐴]2 extending 퐴. If 푥 ∈ 푀 is strictly positive, then 훥(푥) > 0 by e.g. Theorem (2.2.1) (ii). So 푥 = 푢ℎ for a unitary 푢 and ℎ ∈ 퐻2. Clearly ℎ is bounded, so that ℎ ∈ 1 ∗ −1 −1 퐴∞, and 푥 = (푥 푥)2 = |ℎ|. Also, ℎ ∈ 퐴∞, since if ℎ푎푛 → 1 then 푎푛 → ℎ . Thus 퐴∞, has ∗ the ‘factorization’ property and so is maximal subdiagonal [59]. Hence 퐴∞ + 퐴∞, and therefore ∗ 2 1 also 퐴 + 퐴 , is dense in 퐿 (푀). Next, suppose that 푔 ∈ 퐿 (푀)+ satisfies 휏(푔퐴0) = 0. We need 1 to show that 푔 ∈ 퐿 (풟)+. Since τ((g + 1)A0) = 0, we can replace g with g + 1 if necessary, 1 2 to ensure that 훥(푔) > 0. Let 푓 = 푔2 ∈ 퐿 (푀). Then 훥(푓) > 0, 푓 ⊥ [푓퐴0]2, and by hypothesis ∗ ∗ 푓 = 푢ℎfor an outer ℎ ∈ [퐴]2 and some unitary 푢 in 푀. Since ℎ = 푢 푓 ⊥ 푢 ∈ [푓퐴]2 = 1 [ℎ퐴]2 = [퐴0]2, and ℎ ∈ [퐴]2, it follows that ℎ ∈ [퐷]2. Thus 푔 ∈ [풟]1 = 퐿 (풟). This verifies the ‘unique normal state extension’ property of [59]. The following generalizes [145]: 1 Corollary (2.2.24)[66]:If 푓 ∈ 퐿 (푀)+, then the following are equivalent: (i) 훥(푓) > 0, (ii) 푓 = |ℎ|푝for a strongly outer ℎ ∈ 퐻푝, (iii) 푓 = |푘|푝 for 푘 ∈ 퐻푝 with 훥(훷(푘)) > 0. 1 1 Proof: (i) ⇒(ii) By a previous result, 훥(푓푝 ) > 0, and so by the last result we have푓푝 = 푢ℎ, 1 1 푝 푝 where ℎ is outer in 퐻푝, and u is unitary. Thus 푓 = (푓푝푓푝 )2 = (ℎ∗ℎ)2 = |ℎ|푝. (ii) ⇒(iii) This follows from Theorem (2.2.10) . (iii) ⇒(i) If 푓 = |푘|푝 for 푘 ∈ 퐻푝 with 훥(훷(푘)) > 0, then 훥(푓) = 훥(푘)푝 ≥ 훥(훷(푘))푝 > 0 by Theorem (2.2.1) and the generalized Jensen inequality. Of course in the case that 풟 is finite dimensional one can drop the word ‘strongly’ in the last several results. In particular, in the case that the algebra 퐴 is antisymmetric, these results and their proofs are much simpler and are spelled out in our survey [62].

Chapter 3 The Algebraic Structure and Quasi-Radial Quasi-Homogegeneous Symbols The 푘-dimensional representations form a generalized maximal ideal space with a canonical surjection onto the ball of 푘 × 푘푛 matrices which is a homeomorphism over the open ball analogous to the fibration of the maximal ideal space of 퐻∞ over the unit disk. The algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C∗-algebra, and for 푛 = 1 all of them collapse to the algebra generated by Toeplitz operators with radial symbols. Section (3.1)The Non-Commutative Analytic Toeplitz Algebras In [68, 208, 211, 212], a good case is made that the appropriate analogue for the analytic Toeplitz algebra in 푛 non-commuting variables is the WOT-closed algebra generated by the left regular representation of the free semigroupon n generators. It obtain a compelling analogue of Beurling's theorem and inner-outer factorization. We add further evidence. The result is a short exact sequence determined bya canonical homomorphism of the automorphism group onto this algebra onto the group of conformal automorphisms of the unit ball of ℂ푁. The kernelis the subgroup of quasi-inner automorphisms, which are trivial modulothe WOT-closed commutator ideal. Additional evidence of analytic properties comes from the structure of k- dimensional (completely contractive) representations, which have a structure very similar to the fibration of the maximal ideal space of 퐻∞ over the unit disk. An important tool in our analysis is a detailed structure theory for WOT-closed right ideals. Curiously,left ideals remain more obscure. The non-commutative analytic Toeplitz algebra 픏푛 is determined by the left regular representation of the free semigroup ℱ푛 on 푛 generators 푧1, … , 푧푛which acts on ℓ2(ℱn) by ⋋ (휔) 흃풗 = 흃휔풗 for 푣, 휔 in ℱ푛. In particular, the algebra 픏푛 is the unital, WOT-closed algebra generated by the isometries 퐿퐼 =⋋ (푧푖) for 1 ≤ 푖 ≤ 푛. This algebra and its norm-closed version (the non-commutativedisk algebra) were introduced by Popescu [214] in an abstract sense in connection with a non-commutative von Neumann inequality and further studied in [208, 214, 209, 212, 213]. For 푛 = 1, weobtain the algebra generated by the unilateral shift, the analytic Toeplitz algebra. The corresponding algebra for the right regular representation is denoted ℜn. This algebra is unitarily equivalent to 픏n and is also equal to the commutantof 픏푛. (see [68,98]). It contains the classification of the WOT-closed right and two-sided ideals of 픏n. These ideals are determined by their range, which is always a subspace in 퐿푎푡ℜ푛; and this pairing is a complete lattice isomorphism.The ideal is two-sided when the range is also in 퐿푎푡 픏푛. This is the keytool needed to establish classify the weak-* continuous multiplicative linear functionals on 픏n. We obtain some factorization results for right ideals that allow us to show that a WOT-closed right ideal is finitely generated algebraically precisely when the wandering subspace of the range space is finite dimensional; and otherwise, they require a countably infinite set of generators even as a WOT-closed right ideal. We examine the representation space of 픏n. The multiplicative linear functionals have a structure that parallels the maximal ideal space of 퐻∞. This provides a natural homomorphism ∞ of 픏n into the space 퐻 (𝔹푛) of bounded analytic functions on the ball. Strikingly, the dilation theory for non-commuting 푛-tuples allows us to obtain an analogous structurefor 푘-

47 dimensional representations for every 푘 < ∞. In particular, the open ball 𝔹푛,푘 of strict contractions in ℳ푘,푛푘 sits homeomorphically in a canonical way in this space. Automorphisms of 픏n are shown to be automatically norm and WOT continuous. We show that there is a natural homomorphism from 퐴푢푡픏푛 onto 퐴푢푡(𝔹푛), the group of conformal automorphisms of the ball of ℂ푛, determined by their action on the WOT −continuous linear functional 휑휆 forλ ∈ 𝔹n. The kernel of this map is the ideal of automorphisms which are trivial modulo the WOT-closed commutator ideal. In order to show that this homomorphismis surjective, we determine all automorphisms of 픏n of the form 퐴푑푊 for unitary 푊. Using certain automorphisms of the Cuntz-Toeplitz algebra found by Voiculescu [221,299], we are able to obtain an isomorphism of this subgroup 퐴푢푡푢(픏푛) with 퐴푢푡(𝔹푛). Thus the automorphism group of 픏n is a semidirect product. We will write 퐿 = [퐿1 … 퐿2] both for then-tuple of isometries and the isometric operator from (푛) ℋ푛 into ℋ푛. By 퐿푣 or 푣(퐿)we will denote the corresponding word 휆(푣) in the n tuple. We allow 푛 = ∞. In this case, ℂ푛 is replaced by a separable Hilbert space ℋ, and the unit ball 𝔹nbecomes the unit ball of ℋ endowed with the weak topology. This occasionally causes additional difficulties which will be pointed out as necessary. The full Fock space of a Hilbert space ℋ is the Hilbert space

퐹(ℋ) = ∑ ⨁ℋ⨁푘 푘≥0 where ℋ⨁0 = ℂ and ℋ⨁푘 is the tensor product of 푘 copies of ℋ. When ℋ = ℂ푛 with orthonormal basis ζifor 1 < 푖 < 푛, the Fock space has anorthonormal basis 휁휔 = 푘 휁푖1 ⨂ … ⨂ 휁푖푘 for all choices of 휔 = (푖1 … 푖푘) in{1, … , 푛} and 푘 ≥ 0 (with the convention that ⨁0 휁∅ spans ℋ . For each vector 휁in ℋ, there is a left creation operator ℓ(휁)휉 = 휁⨂휉. Clearly, there is a natural isomorphism of Fock space onto ℋn, where 푛 = 푑푖푚 ℋgiven by sending 휁휔 to 휉휔. This unitary equivalence sends ℓ(휁푖 ) to 퐿푖. The following heuristic is useful when working with operators in 픏푛. If 퐴 = ∑휔 푎휔퐿휔is a finite linear combination of the set {퐿휔: 휔 ∈ ℱ푛}, then 퐴휉1 = ∑휔 푎휔휉휔; conversely, given a finite linear combination of basis vectors 휁 = ∑휔 푎휔휉휔, the operator 퐴 = ∑휔 푎휔퐿휔 belongs to 픏n and satisfies 퐴휉1 = 휁. Sometimes this operator will be denoted by 퐿휁 . This correspondence of course cannot be extended to infinite combinations. However, notice that for an arbitrary element 퐴 of 픏n, 퐴 is completely determined by its action on ξ1: indeed,퐴휉푣 = 퐴푅푣휉1 = 푅푣퐴휉1. So if 퐴휉1 = ∑휔 푎휔휉휔, we have

퐴휉푣 = ∑ 푎휔휉휔푣 = ∑ 푎휔(퐿휔휉푣). 휔 It is useful to view the formal sum ∑푤 푎푤퐿푤 as the Fourier expansion of 퐴. In particular [68], the Cesaro sums |휔| ∑(퐴) = ∑ (1 − ) 푎 퐿 푛 휔 휔 푛 |휔|<푛 converge in the strong-* topology to 퐴. 48

The algebra 픏푛 contains no non-scalar normal elements. Every non-scalar element of 픏푛 is injective and has connected spectrum containing more than one point. So 픏n contains no nonzero compact operators,quasinilpotent elements or non-scalar idempotents. In particular, 픏n is semisimple.(See [114,68].) 푛 If ℳ is an invariant subspace for 픏푛, the wandering subspace is 풲 = ℳ ⊖ ∑푖=1 퐿푖ℳ. ∑ By the analogue of the 풲old decomposition [210], it follows that ℳ = ω∈ℱn ⨁Lω풲 . The invariant subspaces of the analytic Toeplitz algebra are determined by Beurling's [40] as the subspaces 휔퐻2 where 휔 is an inner function in 퐻∞. These subspaces are always cyclic with 2 2 2 wandering subspace 휔퐻 ⊖ 푧휔퐻 = ℂ휔. The subspace 휔퐻 is the range of 푇휔, which is an ∞ isometry in 퐻 = 픏1 = ℜ1. The analogue of Beurling'stheorem is: Theorem (3.1.1)[156]:([208, 68]). Every invariant subspace of 픏푛 is generated by a wandering subspace. Thus it is the direct sum of cyclic subspaces. The cyclic invariant subspaces of 픏n are precisely the ranges of isometries in ℜ푛; and the choice of isometry is unique up to a scalar. If ℳ is a cyclic invariant subspace for 픏n, then its wandering subspace is 1-dimensional. If 휉 is a wandering vector for ℳ, then we denote the corresponding isometry in ℜ푛 by ℜ휉. Explicitly, we have the formula, ℜ휉휉휔 = 퐿휔휉. Conversely, any isometry in ℜ푛is an ℜ휉 for some 픏푛-wandering vector휉: Similarly, we see that any isometry in 픏푛 has the form 퐿휉 for someℜ푛-wandering vector ξ. By analogy, the isometries of 픏n are called inner; and the elements with dense range are called outer. An element A in 픏n is inner if and only if ‖퐴‖ = ‖퐴휉1‖ = 1. As a corollary, one obtains the following analogue of inner-outer factorization: Corollary (3.1.2)[156]: Every 퐴 in 픏n factors as 퐴 = 퐿휉퐵 where 퐿휉 is an isometry in 픏푛 and 퐵 belongs to 픏푛 and has dense range. This factorization is uniqueup to a scalar. The operator 퐵is invertible if and only if 퐴 has closed range. We also need to understand the structure of the eigenvectors for the adjoint analogous to the point evaluations in the unit disk associated to eigen-values of the backward shift. ∗ Theorem (3.1.3)[156]: (cf. [21] and[68,69,196,197]). The eigenvectors for 픏n are the vectors 푛 2 1⁄2 2 1⁄2 −1 푣휆 = (1 − ‖휆‖ ) ∑ 휔(휆)휉휔 = (1 − ‖휆‖ ) (퐼 − ∑ 휆푖 퐿푖) 휉1

휔∈ℱ푛 푖=1 For ⋋ in the unit ball 𝔹푛. They satisfy ∗ 퐿푖푣휆 = 휆푖푣휆 And (푝(퐿)푣휆, 푣휆) = 푝(휆) for every polynomial 푝 = ∑휔 푎휔휔 in the semigroup algebra ℂℱ푛. This extends to the map 휑휆(퐴) = (퐴푣휆, 푣휆), which is a WOT-continuous multiplicative linear ⊥ functional on 픏n. The vector 푣휆 is cyclic for픏n. The subspace ℳ휆 = {푣휆} is 픏푛invariant, and its wandering subspace풲⋋is n-dimensional, spanned by 2 1⁄2 휉휆,푖 = 휆푖휉1 − (1 − ‖휆‖ ) 퐿푖푣휆for1 ≤ 푖 ≤ 푛 These results are used in [68] to show that 픏푛 is hyper-reexive. Moreover,for every weak-* continuous linear functional 푓 on 픏nwith ‖푓‖ < 1, there are vectors 휉 and 휁 such that 푓(퐴) = (퐴휉, 휁)for all A in 픏n and‖휉‖‖휁‖ < 1. This yields the immediate consequence which will be important on several occasions.

Corollary (3.1.4)[156]: ([68,199]). The weak-* and WOT topologies on 픏n coincide. We identify the WOT-closed right and two-sided ideals of 픏푛. Let 퐼푑푟(픏푛),퐼푑ℓ(픏푛) and 퐼푑(픏푛) denote the sets of all WOT-closed right, left and two-sided ideals respectively. The important observation is that these ideals can be identified by their ranges. If 픍 belongs to

퐼푑푟(픏푛), then the subspace 픍휉1belongs to 퐿푎푡ℜ푛. To see this, note that ℜ푛픍휉1 = 픍ℋ푛휉1 = 픍ℋ푛 = 픍픏푛휉1 = 픍휉1 Thus 픍휉1 = 픍ℋ푛 is the range of 픍 and is ℜ푛 invariant. When 픍 belongs to Idℓ(픏n); we have 픏n픍ξ1 = 픍ξ1; so 픍ξ1 is 픏n invariant. Hence when 픍 is a two-sided ideal, 픍휉1 belongs to 퐿푎푡(픏푛)⋂퐿푎푡(ℜ푛): Conversely, if ℳ belongs to 퐿푎푡(ℜ푛), we shall see during the proof of Theorem (3.1.5) that the set {퐴 ∈ 픏푛: 퐴휉1 ∈ ℳ} belongs to 퐼푑푟(픏푛). It will follow that when 픍 is a right ideal, the subspace 픍휉1determines 픍 and moreover 픍 is two-sided precisely when 픍휉1is also 픏푛 invariant. We do not make the same claims for left ideals. One should note that when 픍 is a left ideal,

픍휉1is not equal to 픍ℋ푛. The full range of the ideal is not a complete invariant. There are technical difficulties for left ideals that we were not able to resolve; but analogous results are plausible. We remark that 퐼푑푟(픏푛) and 퐼푑(픏푛) form complete lattices with the operations of intersection and WOT-closed sum.

Theorem (3.1.5)[156]:Let 휇: 퐼푑푟(픏푛) → Lat(ℜ푛) be given by 휇(픍) = 픍ξ1. Then μ a complete lattice isomorphism. The restriction of μ to the set Id(픏n) is a complete lattice isomorphism onto 퐿푎푡 픏푛 ∩ 퐿푎푡ℜ푛. The inverse map ι sends a subspace ℳ to 휄(ℳ) = {퐽 ∈ 픏푛 ∶ 퐽휉1 ∈ 휉} Proof: We have seen above that ℳ = 휇(픍) is a subspace of the appropriate type for right and two-sided (and even left) ideals. Conversely, we now check that ι sends invariant subspaces to ideals of the appropriate type. So fix a subspace ℳ in 퐿푎푡(ℜ푛) and consider 휄(ℳ). It is clear that 휄(ℳ) is a WOT −closed subspace of 픏n. Suppose that 퐽 is in ι(ℳ)and A belongs to 픏n. Then 퐽퐴휉1 ∈ 퐽ℋ푛 = 퐽ℜ푛휉1 = ℜ푛 퐽 휉1 ⊂ ℳ Whence 휄(ℳ) is a right ideal. And if ℳ is in Lat 픏푛, then for 퐽 in ι(ℳ) and 퐴 in 픏n, : 퐴 퐽 휉1 ∈ 퐴 ℳ ⊂ ℳ So ι(ℳ) is a left ideal. Thus 휄(퐿푎푡픏푛 ∩ 퐿푎푡ℜ푛) is contained in Id(픏푛). Next we show that μι is the identity map. Fix ℳ in Lat(ℜ푛). By the definitions of the maps involved, we have μι(ℳ) is contained in ℳ. To see the opposite inclusion, let {휁푗} be an 푛 orthonormal basis for the ℜn wandering subspace 풲 = ℳ ⊖ ∑푖=1⊕ 푅푖ℳ . Then

ℳ = ∑⊕ ℜ푛[휁푗] = ∑⊕ 푅푎푛퐿휁푗 푗 푗

( ) Since 퐿휁푗휉1 = 휁푗 belongs to 퓜, it follows that 퐿휁푗 lies in ι ℳ . So ( ) ( ) ( ) ∑⊕ 푅푎푛퐿휁푗 ⊂ 휄 ℳ ℋ푛 = 휄 ℳ 휉1 = 휇(휄 ℳ ) 푗 Therefore 휇 휄(ℳ) = ℳ. Now fix 픍 in 퐼푑푟(픏푛). As before, the defnitions involved show that 픍 is contained in 휄 휇 (픍). To see that this is an equality, we first show that for every 픍 in ℋn, 픍휉 = 휄 휇 (픍)휉 (1) Since 픏푛|휉| is a cyclic invariant subspace for 픏n, it may be written as 픏푛[휉] = 푅푎푛푅휂 where η is a wandering vector for 픏푛[휉]. Thus 픍휉 = 픍픏푛휉 = 픍 푅휂ℋ푛 = 푅휂픍 ℋ푛 = 푅휂ℳ evidently, the same computation for ι μ (픍) yields the same result; hence (1) holds. Suppose that 푓 is a WOT-continuous linear functional on 픏n which annihilates the ideal 픍. By [68,152], there are vectors 휉 and 휂 such that 푓(퐴) = (퐴휉, 휂) for all A in 픏n. Since 푓(픍) = 0, it follows that η isorthogonal to 픍휉. Then by the previous paragraph, 휂 is also orthogonal to휄 휇 (픍)휉 and thus 푓 also annihilates 휄 휇 (픍). By the Hahn-Banach Theorem,we therefore have 픍 = 휄 휇 (픍). Thus we have established that μ is a bijective pairing between 퐼푑푟(픏푛)and 퐿푎푡ℜ푛 which carries −1 퐼푑(픏푛) onto Lat픏푛 ∩ Latℜ푛and 휄 = 휇 . If 픍1and 픍2 are WOT-closed right ideals, then

휇(픍1 + 픍2) = (픍1 + 픍2)ℋ푛 = 픍1ℋ푛 + 픍2ℋ푛 = 휇(픍1) ⋁ 휇(픍2) and hence sums are sent to spans. Similarly, ifℳ1 and ℳ2 are subspaces in 퐿푎푡(픏푛) ∩ 퐿푎푡ℜ푛, then 휄(ℳ1 ∩ ℳ2) = {퐽 ∈ 픏푛: 퐽휉1 ∈ ℳ1 ∩ ℳ2} = {퐽 ∈ 픏푛: 퐽휉1 ∈ ℳ1} ∩ {퐽 ∈ 픏푛: 퐽휉1 ∈ ℳ2} = ι(ℳ1) ∩ ι(ℳ2) It follows that 휇 preserves intersections. Finally, to see that μ is complete, note that if 픍푘 is an increasing union (or decreasing intersection) of ideals,we have

μ (⋃ 픍푘) = ⋃ 픍푘 ℋ푛 = ⋁ 휇(픍푘) 푘 푘 푘 and similarly for intersections. Therefore μ is a complete lattice isomorphism. Corollary (3.1.6)[156]:If J belongs to 픏푛, then the WOT-closed (two-sided) ideal〈퐽〉generated by 퐽 consists of all elements 퐴 in 픏n such that 퐴휉1 lies in픏푛퐽ℋ푛.

Proof: The ideal 〈퐽〉 is determined by its range, and this must be the least element ℳ of 퐿푎푡(픏n) ∩ 퐿푎푡ℜncontaining Jξ1. Thus ℳ = 픏푛ℜ푛퐽휉1 = 픏푛 퐽 ℜ푛휉1 = 픏푛 퐽ℋ푛 By Theorem (3.1.5), it follows tha〈퐽〉 = 휄(ℳ)t. Theorem (3.1.5) enables us to characterize the WOT-continuous multiplicative linear functionals on 픏n. Theorem (3.1.7)[156]: Suppose 휑 is a (non-zero) WOT-continuous multiplicative linear functional on 픏n. Then there exists λ in 𝔹n such that φ = φ휆 51

Proof: Let 픍 = 푘푒푟 휑. Then 픍is a WOT-closed two sided maximal ideal of codimension one. Set ℳ = 휇(픍) and note that 휉1 ∉ ℳ. (If not, Theorem (3.1.5) implies I belongs to 휄(ℳ) = 픍, which is impossible.) In fact,ℳ has codimension one. To see this, let 풩 = ℳ + ℂ휉1. If 풩 ≠ ⊥ ℋn, let 휁 be a unit vector in 풩 . Choose 퐴 in 픏nso that ‖휁 − 퐴휉1‖ < 1 and set 훼 = 휑(퐴). Since 퐴 − 훼퐼 is in 픍, 퐴휉1 − 훼휉1belongs to ℳ. Hence 1 = |(휁 − 훼휉1, 휁)| = |(휁 − 퐴휉1, 휁) + (퐴휉1 − 훼휉1, 휁)| = |(휁 − 퐴휉1, 휁)| < 1, which is absurd. So 풩 = ℋn and hence ℳ belongs to Lat(픏푛) ∩ Lat ℜ푛and has codimension ⊥ ∗ one. Thus ℳ is a 1-dimensional invariant subspace for픏n. By Theorem (3.1.3), there is a ⊥ point 휆 in 𝔹n such that ℳ = {푣휆} . ByTheorem (3.1,5), 푘푒푟 휑 = 휄(ℳ) = 푘푒푟 휑휆. Therefore 휑 = 휑휆. We present, as an example, an ideal which will be important later. Let 푒 denote the WOT- s closure of the commutator ideal of 픏n. The space ℋn is the symmetric Focz space spanned by 1 the vectors ∑ 휉 , where ω is inℱ , 푘 = |휔|, 푆 is the symmetric group on 푘 elements, 푘! 휎∈푆푘 휎(휔) 푛 푘 and 휎(휔) is the word with the terms in ω permuted by σ. Also recall that for λ in 𝔹푛, φλis the multiplicative linear functional on 픏푛 given by 휑휆(퐴) = (퐴푣휆, 푣휆) as in Theorem (3.1.3). Proposition (3.1.8)[156]:The WOT-closure of the commutator ideal is

푒 = 〈퐿푖퐿푗 − 퐿푗퐿푖 ∶ 푖 ≠ 푗〉 = ⋂ 푘푒푟 휑휆

휆∈𝔹푛 The corresponding subspace in Lat(픏푛) ∩ Lat ℜ푛is ( ) 휇 푒 = 푠푝푎푛 {휉푢푧푖푧푗푣 − 휉푢푧푗푧푖푣 ∶ 푖 ≠ 푗, 푢, 푣 ∈ ℱ푛} 푠⊥ ⊥ = ℋ푛 = 푠푝푎푛 {푣휆: 휆 ∈ 𝔹푛} Proof: Let 픍 be the WOT-closed ideal generated by the set of commutators {퐿푖퐿푗 − 퐿푗퐿푖: 푖 ≠ 푗}. Clearly e̅ ⊃ 픍. On the other hand, consider the set of operators of the form 퐴(퐵퐶 − 퐶퐵)퐷for 퐴, 퐵, 퐶, 퐷 in ℒ푛. These elements span a WOT-dense subset of e̅. Moreover, since the polynomials in the Liare WOT-dense in ℒn, we may further suppose that each of 퐴, 퐵, 퐶, 퐷 is such a polynomial. Thus by expanding, it suffices to show that operators of the form퐿푢(퐿푣퐿휔 − 퐿휔퐿푣)퐿푥 belong to 픍 for all words 푢, 푣, 휔, 푥 in ℱn. Now, every permutation of 풦 objects is the product of interchanges (푖, 푖 + 1) for some1 ≤ 푖 < 풦. Using this, it follows that 퐿휔 − 퐿휎(휔)belongs to 픍 for everyω in ℱ푛 and every σin푆|휔|. Therefore it follows that 퐿푢(퐿푣퐿휔 − 퐿휔퐿푣)퐿푥belongs to 픍. Thus 픍 = 푒̅.

The subspace 휇(푒) = 휇(픍) is the smallest ℒ푛ℛ푛invariant subspace containing {휉푧푖푧푗 −

휉푧푗푧푖: 푖 ≠ 푗} which is the subspace spanned by the vectors of the form 휉푢푧푖푧푗푣 − 휉푢푧푗푧푖푣. 푠 It is now clear that ℋ푛 is orthogonal to 휇(푒).On the other hand, a vector 휁 = ∑휔 푎휔 휉휔is orthogonal to 휇(푒) if and only if it is orthogonal to every 휉휔 − 휉휎(휔) for 휔 ∈ ℱ푛and 휎 ∈ 푆|휔|. 푠 Hence 푎휎(휔) = 푎휔; whence it follows that 휁 belongs to ℋ푛. 푠 푠 Next we show span{푣휆: 휆 ∈ 𝔹푛} = ℋ푛. Evidently, each 푣휆 belongs to ℋ푛. Let 푄푘denote the projection onto span{휉휔: |휔| = 푘}. For each λ in𝔹푛and 푧in 핋, 푚 푣푧̅휆 = ∑ 푧 푄푚푣휆 푚≥0 52

Thus by considering the -valued integrals 푘 . ℋn ∫휋 푧̅ 푣푧̅휆 푑푧 For 푘 ≥ 0, it follows that 푄푘푣휆 lies in span{푣휆: 휆 ∈ 𝔹푛}. Now it is an easy exercise to show 1 that the set of all 푄 푣 ′s contains each ∑ 휉 for |휔| = 푘. Hence span{푣 : 휆 ∈ 𝔹 } = 푘 휆 푘! 휎∈푆푘 휎(휔) 휆 푛 푠 ℋ푛. It is clear that the multiplicative linear functionals 휑휆vanish on the commutator,and hence on the γ closed ideal that it generates. Conversely, suppose that 퐴 in 픏nis not in é. Then 퐴 in 휉1 is not contained in휇(푒́). Therefore, since 휇(푒́) is the orthogonal complement of the set{푣휆: 휆 ∈ 𝔹푛}, thereis a 휆 in 𝔹푛 such that ∗ 2 1/2 0 ≠ (퐴휉1, 푣휆) = (휉1, 퐴 푣휆) = 휑휆(퐴)(휉1, 푣휆) = (1 − ‖휆‖ ) 휑휆(퐴). Thus 휑휆(퐴) ≠ 0; whence 퐴 is not in 푘푒푟 휑휆. Next we develop some useful lemmas about factorization in right ideals. In particular, they will allow us to determine when a right ideal is finitely generated. Recall from Theorem (3.1.1) that each isometry in 픏nhas the form 픏ζfor some ℛn-wandering vector ζ.

Lemma (3.1.9)[156]:Let 퐿휁푗, for1 ≤ 푗 ≤ 푘, be a finite set of isometries in 픏nwith pairwise 푘 orthogonal ranges ℳ푗. Let ℳ = ∑푗 ℳ푗 and 픍 = 휄(ℳ). Then 픍 equals {퐴 ∈ 픏푛: 푅푎푛(퐴) ⊂ ℳ}and every element of 픍 factors uniquely as 푘

퐴 = ∑ 퐿휉푗퐴푗 for 퐴푗 ∈ 픏푛 푗=1

Thus the (algebraic) right ideal generated by {퐿휁푗: 1 ≤ 푗 ≤ 푘} equals 풥. Proof: Clearly each ℳ푗 is ℛ푛invariant, and thus so is ℳ. Hence if 퐴 in 픏nsatisfies 퐴휉1 ∈ ℳ, then 퐴ℋ푛is contained in ℳ. Thus 픍 = {퐴 ∈ ℒ푛: 푅푎푛(퐴) ⊂ ℳ}

So 픍 is a WOT-closed right ideal containing 퐿휁푗 for 1 ≤ 푗 ≤ 푘. Conversely, suppose that A belongs to픍. Then since L L∗ is the orthogonal ζj ζj projection onto ℳj , we obtain the factorization. 푘 푘 퐴 = 퐿 퐿∗ 퐴 = 퐿 퐴 , (∑ 휁푗 휁푗) ∑ 휁푗 푗 푗=1 푗=1 Where 퐴 = 퐿∗ 퐴. This decomposition is unique because the퐿 , s are isometries with 푗 휁푗 휁푗 orthogonal ranges. We will show that each 퐴푗belongs toℒn.As 휁푗is a ℛ푛-wandering vector 푛 forℳ, it is orthogonal to ∑푖=1 ℛ푖ℳ. 푛 Now푁 = 푅푎푛(퐴)is contained in ℳ, whence ζj is also orthogonal to ∑푖=1 ℛ푖풩. Therefore, for any word win ℱ푛, ∗ ∗ (푅푖 퐴 휁푗휉휔) = (휁푗, 퐴푅푖휉휔) = (휁푗, 푅푖퐴휉휔) = 0, ∗ ∗ and so 푅푖 퐴 휁푗 = 0 . Now compute using [68] 퐴 푅 − 푅 퐴 = 퐿∗ 퐴푅 − 푅 퐿∗ 퐴 = 퐿∗ 푅 − 푅 퐿∗ 퐴 푗 푖 푖 푗 휁푗 푖 푖 휁푗 ( 휁푗 푖 푖 휁푗) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 휉1(푅푖 퐿휁푗휉1) 퐴 = 휉1(퐴 푅푖 휁푗) = 휉1(푅푖 퐴 휁푗) = 0 53

′ Therefore 퐴푗belongs to ℛ푛 = 픏푛. It is now evident that 퐴 belongs to the Algebraic right ideal generated by {퐿휁푗: 1 ≤ 푗 ≤ 푘}. 0,k An important special case concerns the (two-sided) ideals 픏n generated by {퐿휔: |휔| = 푘}, 0 which yields a useful decomposition of an arbitrary element of 픏n. In particular, the ideal 픏푛 ≔ 0,1 푛 ℒ푛 leads us to a unique decomposition of ℒn as 픏푛 = ℂ퐼 + ∑푖=1 퐿푖픏푛. This provides a handle on the algebraic rigidity of 픏nthat will prove useful for analyzing the automorphism group. Corollary (3.1.10)[156]:For 1 ≤ 푛 < ∞ and 푘 ≥ 1, every 퐴 in 픏n can be written uniquely as a sum

퐴 = ∑ 푎휔퐿휔 + ∑ 퐿휔퐴휔 |휔|<푘 |휔|=푘 Where 푎휔 ∈ ℂ and 퐴휔 ∈ 픏푛. Proof: The isometries {퐿휔: |휔| = 푘} have pairwise orthogonal ranges summing to ℳ = 푠푝푎푛{휉푣: 푣 ≥ 푘}. This subspace is픏푛ℛ푛 invariant, and thus by Theorem (3.1.5), the right ideal 0,k i(ℳ)is in fact two-sided. Lemma (3.1.9) shows that ι(ℳ) coincides with 픏n . Given 퐴in 픏n, write퐴휉1 = ∑ 푎휔휉휔. The coefficients 푎휔 for |ω| < 푘arethe unique constants such that (퐴 − ∑|휔|<푘 푎휔퐿휔)휉1lies in ℳ. Therefore by Lemma (3.1.9), this difference can be written uniquely as ∑|휔|=푘 퐿휔퐴휔. Example (3.1.11)[156]:Lemma (3.1.9) is not valid for countably many generators even with 푘 norm closure. Indeed, consider the isometries 퐿1퐿2in 픏2for 푘 ≥ 0. Their ranges are orthogonal, summing to the 픏푛ℛ푛-invariant subspace generated by휉푧2. So the WOT-closed right ideal 픍 that they generate is the two-sided WOT-closed ideal generated by 퐿2. Consider a sum of the form 푘 퐴 = ∑ 퐿1 퐿2ℎ푘(퐿1) 푘≥0 ∞ Where ℎ푘 will be functions in 퐻 . This will lie in 픍 provided that 퐴 is a bounded operator. However, it is a norm limit of finite sums of this type only if the series converges in norm. An easy computation shows that ∗ ∗ 퐴 퐴 = ∑ ℎ푘 (퐿1) ℎ푘(퐿1). 푘≥0 As 퐿1is a unilateral shift of infinite multiplicity, this sums to an operator unitarily equivalent 2 to the infinite implication of a Toeplitz operator with symbol ∑푘≥0|ℎ푘| . Thus 퐴 is bounded precisely when this sum is bounded. The sum of operators is norm convergent exactly when this sum of functionsis norm convergent. Constructing a sequence which is bounded but not norm convergent is easy. The algebra 퐻∞ is logmodular [118,251,200], and so if f is a non-negative real function in 퐿∞such that 푙표푔푓 is integrable, then there is a function ℎ in 퐻∞ such that |ℎ| = 푓. Choose a sequence of disjoint closed intervals Jk of the unit circle, each of positive length. Let 푓푘 = −푘 | |2 2 + 휒푗푘for 푘 ≥ 0, and let ℎ푘 be analytic functions with ℎ푘 = 푓푘.

Then 2 ∑|ℎ푘| = 2 + 휒퐽 where 퐽 = ⋃ 퐽푘 푘≥0 푘≥0 This sum is bounded. However, ‖ℎ푘‖ > 1 for all k, and thus this sum is not norm convergent. Moreover, this ideal is not finitely generated as a right ideal because ℳ has an infinite dimensional ℛn wandering space . Any element 퐽 in 픍 has 푅푎푛(퐽) contained in ℳ, and its projection onto . is a subspace of at most one dimension. The ranges of a set of generators must necessarily 푠푝푎푛푀; and thus countably many are required. We need a variant of Lemma (3.1.9) which is valid for countably generated ideals. Let k(픏n) denote the order k column space of 픏n, which is the set of all 푘-tuples of the form 퐴1 퐴 퐴 = [ 2] , 퐴 ∈ 픏 , 1 ≤ 푖 ≤ 푘 ⋮ 푖 푛 퐴푘 such that A is bounded with respect to the norm obtained by considering 퐴 as an element of (푘) ℬ(ℋ푛, ℋ푛 ). Similarly, let ℛk(픏n) denote the order 푘 row space of 픏nconsisting of operators 퐴 = [퐴1, 퐴2, … 퐴퐾], 퐴푖 ∈ ℒ푛, 1 ≤ 푖 ≤ 푘 such that A is bounded with respect to the norm obtained by considering 퐴 as an element of (푘) ℬ(ℋ푛 , ℋ푛).. For 푘 < ∞, this is all 푘-tuples, but the bounded-ness condition is non-trivial for 푘 = ∞. The following lemma shows, in particular, that the infinite row matrix, 퐿 = [퐿1, 퐿2, 퐿3, … ], 0 maps 풞∞(픏∞) bijectively onto 픏∞This result also applies to the two-sided WOT-closed ideal 풥 generated by the set {퐿2, ⋯ , 퐿푛}. The range of this ideal is the sum of the pairwise orthogonal ranges of 푘 {퐿1퐿푗: 푘 ≥ 0, 2 ≤ 푗 ≤ 푛}.

Lemma (3.1.12)[156]:Let, for퐿휁푗, for 푗 ≥ 1, be a countably infinite set of isometries in 픏nwith ∞ pairwise orthogonal ranges ℳj. Let ℳ = ∑j=1 ℳ j , and let 퐽 be the WOT-closed right ideal ι(ℳ). Then every element of J factors uniquely as A = ZX, where Z is the fixed isometry in ( ) ( ) ℛ∞ 픏n given by푍 = [퐿휁2, 퐿휁1, … ] and X is a bounded operator in 풞∞ 픏n . Hence A can be written uniquely as the WOT limit 푘

퐴 = WOT − 퐿푖푚 ∑ 퐿휁 퐴푗 푘−∞ 푗 푗=1 Proof:The proof begins as in Lemma (3.1.9). There is a unique decompositionof A as a WOT- convergent sum, 퐴 = WOT − ∑ 퐿 푋 ,where푋 = 퐿∗ 퐴. 푗≥1 휁푗 푗 푗 휁푗 The푋푗are elements of 픏n by the same computation. Thus defining 푋 to be the column operator with entries 푋푗, we obtain a formal factorization 퐴 = 푍푋. To see that 푋 is bounded, it suffices to compute that X*X = A*A.

Corollary (3.1.13)[156]: Every element 퐴 in 픏∞ decomposes uniquely as

A = ∑ 푎휔퐿휔 + 휔푘푋푘 |휔|<푘 ( ) 2 Where 푎휔 |휔|<푘belongs to ℓ , 푊푘 = [퐿휔푘,2, 퐿휔푘,1, … ], and {ωk,i} is a listing of all words of length k, and Xk belongs to C∞(픏∞). ∑ Proof: The identity 퐴휉1 = 휔∈ℱ푛 푎휔휉휔determines the coefficients 푎휔uniquely, and shows 2 that they belong to ℓ . For each 푗, the isometries 퐿휔푗,푖have pairwise orthogonal ranges, and hence the sun ∑|휔|=푗 푎휔퐿휔is norm convergent. Summing this over 푗 < 푘 yields the unique 0,푘 operator of this formin the same coset of 퐴 + 픏∞ The remainder is factored by Lemma (3.1.12).These lemmas allow us to determine when a right ideal is finitely generated. Theorem (3.1.14)[156]:Let J be a WOT-closed right ideal. If ℳ = 휇(픍)in 퐿푎푡ℛ푛has a finite dimensional wandering space of dimension 푘, then 픍 is generated by 푘 isometries as an algebraic right ideal. When this wandering subspace is infinite dimensional, 픍 is not finitely generated even as a WOT-closed rightideal. However, it is generated by countably many isometries as a WOT-closed right ideal. Proof: When the wandering space  is finite dimensional, choose an orthonormal basis ∑푘 {휁푗: 1 ≤ 푗 ≤ 푘}. Then ℳ = 푗=1 ⨁푅푎푛 퐿휁푗. Thus by Lemma (3.1.9), the isometries

{퐿휁푗: 1 ≤ 푗 ≤ 푘} generate 픍 as an algebraic right ideal. Similarly, when  is infinite dimensional, Lemma (3.1.12) yields a countable set of isometries which generate 픍 as a WOT- closed right ideal. Finally, suppose that 픍 is finitely generated as a WOT-closed right ideal, say by 푘 {퐴푗: 1 ≤ 푗 ≤ 푘}. Then the operators of the form ∑푗=1 퐴푗 퐵푗 for Bj in 픏n are WOT-dense in 픍. Therefore ̅̅푘̅̅̅̅̅̅̅̅̅̅

휇(픍) = ∑ 퐴𝑗ℋ푛 = ℛ푛[{퐴푗휉1: 1 ≤ 푗 ≤ 푘}] 𝑗=1 This subspace is finitely generated, and therefore has finite dimensional wandering space. In the category of unital operator algebras, we take the view point that the natural representations are the completely contractive unital representations. Given an operator algebra 프, for each 1 ≤ 푘 ≤ 풩0 we let 푅푒푝푘(풴) denote the set of completely contractive representations of 프 into ℬ(ℋ), whereℋ is a fixed Hilbert space of dimension 푘. Put the topology of pointwise weak-∗ convergence on this space. When 푘 < ∞, this is the topology of point wise (norm) convergence. Since the unit ball of ℬ(ℋ) is weak-* compact(and norm compact when 푘 < ∞), Tychonoff's Theorem shows that the set of contractive maps from 프 into ℬ(ℋ) is pointwise weak-* compact. When 푘 < 1, the collection of representations is closed in this topology, and thus is also compact. Unfortunately, the collection of representations is not closed when 푘 = ∞. For an example, consider the direct sum 푖푑(푛) of n copies of the identity representation of ℬ(ℋ)for 푛 ≥ 1. Since the direct sumof n copies of the 푛 unilateral shift S is unitarily equivalent to 푆 , we may find representations σn of ℬ(ℋ) on 푛 ℋsuch that 휎푛(S) = 푆 for every 푛. Note that no point wise weak-* limit point of this sequence 56 of representations is multiplicative, and hence the space of representations is not closed when 푘 = ∞. The natural equivalence relation on representations is unitary equivalence. When k < ∞, the unitary group 풰푘 is compact and acts on 푅푒푝푘(풴). Thus the quotient space is also compact and Hausdorff. This need not be the case for 푘 = 풩0 since unitary orbits of representations need not be closed in general. For these reasons, our standing assumption is that all representations of ℒn are on finite dimensional spaces. The familiar case of 푘 = 1 yields the set of multiplicative linear functionals. It is well known that multiplicative linear functionals are automatically completely contractive. In this case, unitary equivalence is the identity relation. Moreover, there is an objective pairing between the multiplicative linear functional and its kernel, a maximal ideal of co dimension 1. So 푅푒푝1(풴) is the direct analogue of the maximal space of a commutative Banach algebra. In a non-abelian algebra, there may be many maximal ideals of other co-dimensions. For 푘 > 1, it is clear that two similar representations will have the same kernel. In the case of 픏n, similar representations which are both completely contractive need not be unitarily equivalent. (Indeed, when 푛 = 1, simply consider two similar, but non-unitarily equivalent, contractions.) When 푘 < ∞ and a representation 훷 in 푅푒푝푘(풴) is irreducible (no invariant subspaces),the range must be all of ℳk = ℬ(ℋ). This is shows that every proper subalgebra of ℳk has a proper invariant subspace. Thus the kernel will be a maximal ideal of codimension k2.Conversely, if 훭 is a maximal ideal of 풴 of finite codimension, then there is a finite dimensional representation of 풴 on 풴/훭 with kernel 훭. This quotient is simple, and thus by Wedderburn's Theorem, 풴/훭 is isomorphicto ℳ푘 for some positive integer 푘. In particular, Μ has codimension k2.Restrict this representation to a minimal invariant subspace ℳ to obtaina representation 휋 and note that ℳ must have dimension k. Now π doesnot act on a Hilbert space. However, it is clearly a completely contractive representation. Any Hilbert space norm on 훭 is equivalent to the quotient norm, and thus will yield a completely bounded Hilbert space representation. Then by Paulsen's [206], this is similar to a completely contractive representation. This shows that the map from irreducible representations in 푅푒푝푘(풴) to the set 2 of maximal ideals of codimension 푘 is surjective.The algebra 픏n has many representations of every dimension. This will follow from Popescu's work on dilation theory for non-commuting n-tuples of operators. The case of 푘 = 1 is special and has some extra structure. So we will handle these special features separately. ∞ Recall the situation for 푛 = 1 in which 픏1 is isomorphic to 퐻 . There is a natural continuous ∞ projection 휋1 of the maximal ideal space 훭퐻∞ of 퐻 onto the closed disk 𝔻̅ given by evaluation at the coordinate function 푧. For each point 휆in 𝔻, there is a unique multiplicative linear functional 휑휆(ℎ) = ℎ(휆) extending evaluation of 푧 at λ. But for|λ| = 1, there is a very large space 훭휆 of multiplicative linear functionals taking the value λ at 푧. (See Hoffman [118] or Garnett [95].) The famous corona theorem of Carles on [47] shows that the point evaluations in the open unit disk are dense in 훭퐻∞. Even though 픏n is not commutative, the space 푅푒푝1(픏푛) of multiplicative linear functionals is very large. For representations of dimension greater than one, there are interesting parallels

57 with the case of multiplicative linear functional. The analysis is based on the extensive knowledge of dilation theory for non-commuting n-tuples. We reprise the results that we will need. Recall that if 훷 is a linear map of an operator algebra 풴 into ℬ(ℋ), then 훷(푘,ℓ) is the map from ℳk,ℓ(풴) into ℳk,ℓ(ℬ(ℋ)), each endowed withthe usual operator norms, given by (푘,ℓ) (푘) 훷 ([퐴푖푗]) = [훷(퐴푖푗)]. When ℓ = 푘, we write 훷 instead. The complete bound norm of (푘,ℓ) 훷 is defined to be ‖훷‖푐푏 = 푠푢푝푘,ℓ‖훷 ‖. The map 훷 is completely contractive if ‖훷‖푐푏 ≤ 1.See Paulsen's book [206] for details. Let 𝔹̅̅̅푛̅̅,푘̅denote the collection of all contractions in ℛ푛(ℬ(ℋ)) where 푑푖푚 ℋ = 푘; namely all (n) 푛 ∗ 1/2 푛-tuples푇 = [푇1 … 푇푛] in ℬ(ℋ , ℋ), such that 푑푖푚 ℋ = 푘 and ‖푇‖ = ‖∑푖=1 푇푖푇푖 ‖ ≤ 1. This is the higher dimension alanalogue of the 푛-ball. It is endowed with the product norm topology when 푘 < ∞ and the product weak-* topology when 푘 = 풩0. If 훷 is a (completely contractive) representation of ℒn on a Hilbert space ℋ, then the 푛- (1,푛) tuple푇 = 훷 (퐿) = (훷(퐿1), … , 훷(퐿푛)) is a contraction.Bunce [43], generalizing Frahzo [82], showed that every contraction 푇 has a dilation to an 푛-tuple of isometries 푆 = (푆1, … , 푆푛) with orthogonal ranges. Popescu [210] extended this to 푛 = ∞ and showed that there is a unique minimal isometric dilation of 푇. This yields a representation of the norm-closed algebra generated by 퐿 because the map taking each 퐿푖 to 푆푖 is a completely isometric isomorphism. Following this with the compression to the original space yields a homomorphism taking 퐿푖to 푇푖. However, this map usually does not extend naturally to a continuous map from 픏n into 퐴푙푔(푆). Popescu [209] determines when this has a wot-continuous extension to a representation of 픏n. Nevertheless, when 푘 = 푑푖푚 ℋ < ∞, we shall see that norm-continuous extensions always exist.The following is a technical lemma used in the proof of Theorem (3.1.16) below. 0,푗 Recall that 픏푛 is the WOT-closed ideal of 픏n generated by the set {퐿휔: |휔| = 푗}. Lemma (3.1.15)[156]:Let 훷 belong to 푅푒푝푘(ℒ푛). If 푇 ∶= (훷(퐿1) , … , 훷(퐿푛)) satisfies 푗 0,푗 ‖푇‖ = 푟 < 1, then ‖훷(퐴)‖ ≤ 푟 ‖퐴‖ for every 퐴 in ℒ푛 . 푗 Proof. Let 휓 be the 1 × 푛 row matrix with entries 퐿휔for |휔| = 푗. And let 휓(푇) denote the row matrix with entries 휓(푇) for |휔| = 푗. By Corollary (3.1.10)for 푛 < ∞ and Corollary

(3.1.13) for 푛 = ∞, we may factor 퐴 = 휔푋 for some 푋 in 퐶 푛푗 (픏푛). Notice that 휔 is an isometry, and therefore ‖퐴‖ = ‖푋‖. By the Frahzo-Bunce dilation result [82,43] for 푛 < ∞ and Popescu [210] for푛 = ∞, the 푛-tuple푟−1푇 dilates to an n-tuple of isometries 푆, and −1 therefore 휔푗(푟 푇)dilates to the isometry 휔푗(푆). Hence 푗 −1 푗 푗 ‖휔푗(푇)‖ = 푟 ‖휔푗(푟 푇)‖ ≤ 푟 ‖휔푗(푆)‖ = 푟 Then since 훷 is completely contractive, (1,푛푗) (1,푛푗) 푗 ‖훷(퐴)‖ = ‖훷 (휔)훷 (푋)‖ ≤ ‖휔푗(푇)‖‖푋‖ ≤ 푟 ‖퐴‖.

The first result generalizes the fact that there is a natural map of ℳ퐻∞ . onto the closed unit disk. The uniqueness result appears to be new even for 푛 = 1 when 푘 > 1. Recall that for 푛 = ∞, 𝔹∞ denotes theunit ball of Hilbert space with the weak topology. Theorem (3.1.16)[156]:For 푘 < ∞, there is a natural continuous projection 휋푛,푘of 푅푒푝푘(픏푛) onto the closed unit ball 𝔹̅̅̅푛̅̅,푘̅ given by 58

휋푛,푘(훷) = (훷(퐿1), … , 훷(퐿푛)) −1 For each 푇in 𝔹푛,푘, the open unit ball, there is a unique representation in휋푛,푘(푇). It is WOT- −1 continuous and is given by Popescu's functional calculus.The restriction of 휋푛,푘 to 𝔹푛,푘 is a homeomorphism. Proof: Since 훷 in Repk(픏n) is completely contractive, it follows that (1,푛) 푇 = 훷 (퐿) = [훷(퐿1) … 훷(퐿푛)] is a contraction. Hence 휋푛,푘 is a well defined map of 푅푒푝푘(픏푛)into 𝔹̅̅̅푛̅̅,푘̅. Since it is determined by evaluation at the points Li, this is a continuous map from 푅푒푝푘(픏푛)with the topology of pointwise convergence into the ball with the product topology. By Popescu's functional calculus, there is a representation 훷푇 for every 푇 in the interior 𝔹푛,푘 (and in fact, for every completely non-coisometric contraction). Since 푅푒푝푘(픏푛) is compact, the image is compact and therefore maps onto 𝔹̅̅̅푛̅̅,푘̅. When ‖푇‖ = 푟 < 1, the WOT-continuous representation 훷푇is defined as follows. Each 퐴 in 픏n is determined by 퐴휉1 = ∑휔 푎휔휉휔as a formal sum 퐴 = ∑휔 푎휔퐿휔. The image 훷푇(퐴) is determined as a norm convergent sum

훷푇(퐴) = ∑ 푎휔휔(푇) 휔 To see this, apply Lemma (3.1.15) for each 푗 ≥ 0 to obtain 1/2

푗 푗 2 ‖ ∑ 푎휔휔(푇)‖ ≤ 푟 ‖ ∑ 푎휔휔(퐿)‖ = 푟 ( ∑ |푎휔| ) |휔|=푗 |휔|=푗 |휔|=푗

Thus two partial sums of ∑휔 푎휔휔(푇)which both contain all words of lengthless than j will differ in norm by at most 푘 2 1/2 푘 푗 −1 ∑푘≥푗 푟 (∑|휔|=푘|푎휔| ) ≤ ∑푘≥푗 푟 ‖퐴‖ = 푟 (1 − 푟) ‖퐴‖,(2) which tends to zero as 푗 tends to infinity.Therefore this series is norm convergent.The fact that it is WOT-continuous was shown by Popescu in [209]. The proof of uniqueness follows similar lines. Let 훷 in 푅푒푝푘(픏푛) be a completely contractive representation of 픏n such that 휋푛,푘(훷) = 푇, where ‖푇‖ = 푟 < 1. We shall show that 훷 = 훷푇. So let 퐴 be an element of 픏n.Then by Corollary (3.1.10)for 푛 < ∞ and Corollary (3.1.13) for 푛 = ∞, A can be written uniquely as

퐴 = ∑ 푎휔퐿휔 + ∑ 퐿휔퐴휔 with 퐴휔 ∈ 픏푛 |휔|<푗 |휔|=푗 Therefore

훷(퐴) = ∑ 푎휔휔(푇) + ∑ 휔(푇)훷(퐴휔) |휔|<푗 |휔|=푗 |휔| Let ∑ (퐴) = ∑ (1 − ) 푎 퐿 denote the Cesaro sums. Recall that ‖∑ (퐴)‖ ≤ ‖퐴‖, 푘 |휔|<푘 푘 휔 휔 푘 and that they converge to 퐴 in the strong-* operator topology. For each integer 푗, there is an integer k sufficiently large that

|휔| ‖ ∑ 푎 퐿 ‖ < 푟푗‖퐴‖ 푘 휔 휔 |휔|<푗 |휔| Then 퐴 − ∑ (퐴) = 퐴 + ∑ 푎 퐿 where 퐴 belong to ℒ0,j. Clearly, ‖퐴 ‖ < 푘 1 ‖휔‖<푗 푘 휔 휔 1 n 1 (2 + 푟푗)‖퐴‖. Hence, using the fact that 훷 is contractive and Lemma (3.1.15), |휔| ‖훷(퐴) − 훷 (∑ (퐴))‖ ≤ ‖훷(퐴) − 훷 ( ∑ 푎휔퐿휔)‖ 푘 푘 |휔|<푗 푗 푗 푗 푗 +‖훷(퐴1)‖ ≤ 푟 ‖퐴‖ + 푟 (2 + 푟 )‖퐴‖ < 4푟 ‖퐴‖ Since 훷 and 훷푇 agree on polynomials in 퐿, it follows that

훷(퐴) = 푙푖푚 훷 (∑ (퐴)) = 푙푖푚 훷푇 (∑ (퐴)) = 훷푇(퐴). 푘⟶∞ 푘 푘⟶∞ 푘 Finally, we verify that the map sending 푇 to 훷푇 maps 𝔹푛,푘 homeomorphically onto the open −1 set πn,k(𝔹n,k). It is evident from the series representationof 훷푇 and estimate (2) above, that ′ if‖푇‖ ≤ 푟 < 1, ‖푇 ‖ ≤ 푟, and A is in 픏n, ′ 푗 −1 ‖훷푇(퐴) − 훷푇′(퐴)‖ ≤ ∑ |푎휔| ‖휔(푇) − 휔(푇 )‖ + 2푟 (1 − 푟) ‖퐴‖ |휔|≤푗 ′ Thus as 푇 converges to 푇, it follows that 훷푇′(퐴) converges to 훷푇(퐴). So this mapping of 𝔹푛,푘into 푅푒푝푘(픏푛) is continuous. Now we specialize to 1-dimensional representations. In this case, each fibre over a point on the boundary of the ball is homeomorphic to every other because the gauge automorphisms act on the ball by the unitary group, and thus is transitive on the boundary. Moreover, this fibre is always very large based on the fact that it is known to be very large for 푛 = 1. Theorem (3.1.17)[156]:There is a natural continuous projection πn,1of the space 푅푒푝1(픏푛) 푛 onto the closed unit ball 𝔹̅̅̅푛̅ in ℂ given by evaluation at the 푛-tuple(퐿1 , … , 퐿푛). −1 For each point λ in 𝔹푛, there is a unique multiplicative linear functionalin 휋푛,1(휆); and it is −1 given by 휑휆(퐴) = (퐴푣휆, 푣휆. The set 휋푛,1(𝔹푛)is homeomorphic to 𝔹n and the restriction of ∞ the Gelfand transform to this ball is a contractive homomorphism of ℒn into 퐻 (𝔹푛).The ball 𝔹푛 forms a Gleason part of 푅푒푝1(ℒ푛). These are the only weak-* continuous functionals on ℒn. −1 −1 For each point λ in 휕𝔹푛, 휋푛,1(휆) is homeomorphic to 휋푛,1(1,0, … 0). −1 There is a canonical surjection of πn,1(λ) onto the fibre 훭1 of 훭퐻∞ given by restricting 휑 in −1 푛 ̅ ∞ 휋푛,1 to 퐴푙푔(∑푖=1 휆푖퐿푖) ≃ 퐻 . This map has a continuous section. Proof: ByTheorem (3.1.15), the map 휋푛,1 maps 푅푒푝1(픏푛) onto 𝔹̅̅̅n̅ by evaluation at 퐿. For each −1 point 휆 in the open ball, there is a unique preimage 휋푛,1(휆)which is evidently 휑휆. Also, the preimage of 𝔹푛 is homeomorphic to the open ball. By Theorem (3.1.7), these are the only WOT-continuous multiplicative linear functional son 픏n. By Corollary (3.1.4), these coincide with the weak-* continuous ones.

̂ For each polynomial 푝(푧) = ∑ 푎휔휔 in ℂℱ푛, the Gelfand transform 푝(퐿)(휆) = 푝(휆)is ∞ evidently a contractive homomorphism of ℂℱ푛 into ℂ[푧] normed as a subset of 퐻 (𝔹푛). Suppose that 푝푛(퐿) is a WOT-Cauchy sequence in 픏n. Since the set {휑휆: ‖휆‖ ≤ 푟} is a compact ̂ set of WOT-continuous linear functionals for each 0 < 푟 < 1, the restriction of 푝푛(퐿) to this ∞ set converges uniformly. Thus the limit lies in 퐻 (𝔹푛). This shows that the Gelfand map ∞ yields a contractive homomorphism into ℋ (𝔹푛). which carries WOT-convergent sequences to sequences converging uniformly on compact subsets of the ball. Now recall that the Gleason part containing φ0 is the equivalence class {휑 ∈ 푅푒푝1(픏푛): ‖휑 − 휑0‖ < 2}. Consider the positive linear functional 훿휉(푇) = (푇휉, 휉)on ℬ(ℋ) for a unit vector 휉. Let 휁 be 휋 another unit vector with |(휉, 휁)| = 푐표푠 휃 for 0 ≤ 휃 ≤ it is a well known fact that 2 ‖훿휉 − 훿휁‖ = 푠푢푝 |(푇휉, 휉) − (푇휁, 휁)| = 2(1 − 푠푖푛 휃) ‖푇‖≤1 2 1⁄2 Since(푣0, 푣휆) = (1 − ‖휆‖ ) ≠ 0, it follows that ‖φ0 − φλ‖ < 2 for λ in 𝔹n. On the other 푛 ̅ hand, if ‖λ‖ = 1, then S = ∑푖=1 휆𝚤 퐿푖is a proper isometry inℒnsuch that 휑0(푆) = 0 and 푧−푟 휑 (푆) = 1. So the Möbius map 푏 (푧) = for 0 < 푟 < 1 can be used to obtain 휆 푟 1−푟2 휑0(푏푟(푆)) =-r and 휑휆(푏푟(푆)) = 1. Hence ‖휑0 − 휑휆‖ = 2. So the Gleason part of φ0 is precisely 𝔹n. ∞ Next consider the point 휆 = (1,0, … ,0)in ∂𝔹n. The algebra Alg(퐿1) is isomorphic to 퐻 . For −1 휑 in 휋푛,1(1,0, … ,0), let 휌(휑) be the restriction of 휑 to Alg(퐿1). Clearly, (휑) belongs to ℳ1, the fibre of ℳ퐻∞ over the point 1, and 휌 is continuous. We now produce a right inverse for 휌. ⊥ Let P be the projection onto the subspace span{휉 푘: 푘 ≥ 0}, and notice that 푃 ℋ is an 픏 - 푧1 푛 n ⊥ invariant subspace. So the map 휓(퐴) = 푃퐴|푃ℋ푛 is a homomorphism of 픏n. In fact, 푃 ℋ푛 is also ℜ푛 invariant. Thus the kernel of this homomorphism is 픍 = {퐴 ∈ ℒ푛: 푃퐴휉1 = 0}, which is the WOT-closed ideal generated by {퐿2, … , 퐿푛}. The range of 휓 is contained in the WOT-closed algebra generated by the operators PLiP, which are all 0 except for 푃퐿1푃 which is a unilateral shift.The map taking L1 to P퐿1푃 is isometric and ∞ WOT −continuous, and carries 퐴푙푔(퐿1) onto 풯(퐻 ). By composing 휓 with the isomorphism ∞ ∞ ∗ of Alg(퐿1)onto H , we may regard as a surjection of ℒn onto H .Let 훼 ≔ 휓 |ℳ1 be the restriction of the Banach space adjoint of 휓 to ℳ1. Clearly α is a continuous map; and if φ = α(ψ ), we have 휋푛,1(휑) = (휋1,1((휓), 0, … ,0) −1 So α maps 훭1into 휋푛,1(1,0, … ,0)and 휌 ∘ 훼(휓) = 휌(휓훹) for in 훭1. Therefore this is a continuous section, and ρ is surjective. In particular, this yields a −1 homeomorphism of 훭1 into πn,1(1,0, … ,0). For any other 휆 with ‖λ‖ = 1, choose a unitary 푈 = [푢푖푗] in ℳ푛 such that 푢푖푗 = 휆푗 . We will −1 −1 show that the gauge automorphism 훩푈 maps 휋푛,1(1,0, … ,0) onto 휋푛,1(휆).Indeed, for any φ in −1 휋푛,1(1,0, … ,0).

푛

휑훩푈(퐿푗) = 휑 (∑ 푢푖푗퐿푖) = 푢1푗 = 휆푗 푖=1 −1 It is evident that this map is continuous with inverse obtained by sending 휑 to휑훩푈 . So it −1 −1 induces a homeomorphism between 휋푛,1(1,0, … ,0).And휋푛,1(휆). The role of 퐿1 is played by −1 푛 ̅ 훩푈 (퐿1) = ∑푖=1 휆푖퐿푖. Example(3.1.18)[156]:This example is to illustrate some of the possibilities on the boundary when 푘 > 1. It is possible that the fibre over a boundary point is a singleton. Consider 푅푒푝3(픏2), the pair 0 0 0 0 0 0 푇1 = [1 0 0] and 푇2 = [0 0 0] 0 0 0 1 0 0 2 2 and a representation 훷 such that 훷(퐿푖) = 푇푖for 푖 = 1,2. Then since 푇1 = 푇2 = 푇1푇2 = 0,2 푇2푇1 = 0, it follows from Lemma (3.1.9) that 푘푒푟훷 contains the ideal ℒ2 . Every element 퐴 ′ ′ 0,2 in ℒ2can be represented uniquely as 퐴 = 푎0퐼 + 푎1퐿1 + 푎2퐿2 + 퐴 where 퐴 belongs to ℒ2 . Therefore 훷(퐴) = 푎0퐼 + 푎1푇1 + 푎2푇2is uniquely determined. On the other hand, the fibre over 푇may be very large indeed. Let 1 0 0 0 푇 = [ ] and 푇 = [ ] 1 0 0 2 1 0 −1 We consider a class of homomorphisms Φ of ℒ2in 휋2,2(푇). Let ζi denote the standard basis for 2 ℂ . Both Ti are lower triangular, so we will consider those representations 훷 which map 픏2into the algebra 풯2 of 2 × 2 lower triangular matrices. The functionals휑푖(퐴) = (훷(퐴)휁푖, 휁푖) are multiplicative since compression to the diagonal is −1 multiplicative on 풯2.Moreover, 휑1(퐿1) = 1and 휑1(퐿2) = 0, and hence 휑1lies in 휋2,2(1,0). Likewise,휑2(퐿1) = 휑2(퐿2) = 0. So 휑2 = 휑0 is evaluation at 0 by Theorem (3.1.17). Recall from Corollary (3.1.10) that every 퐴 in ℒ2 can be uniquely written as 퐴 = 푎0퐼 + 퐿1퐴1 + ∗ ∗ 퐿2퐴2, where 푎0 = 휑0(퐴) and 퐴푖 = 퐿푖(퐴 − 푎0퐼). Define δ(퐴) = 휑1(퐴2) = 휑1(퐿2(퐴 − 휑0(퐴)퐼). Then 훷(퐴) = 푎0퐼 + 훷(퐿1)훷(퐴1) + 훷(퐿2)훷(퐴2) 푎0 0 1 0 휑 (퐴 ) 0 0 0 휑 (퐴 ) 0 = [ ] + [ ] [ 1 1 ] + [ ] [ 1 2 ] 0 푎0 0 0 ∗ ∗ 1 0 ∗ ∗ 푎 + 휑 (퐴 ) 0 휑 (퐴) 0 = [ 0 1 1 ] = [ 1 ] 휑1(퐴2) 푎0 훿(퐴) 휑0(퐴) In order to have a representation, it remains to verify complete contractivity. An explicit family of such representations may be obtained as follows.

Let ℳ = 푠푝푎푛 {휉 푘: 푘 ≥ 0}and ℳ = 푠푝푎푛 {휉 푘: 푘 ≥ 0}, and setℳ = ℳ ⨁ℳ . Then 1 푧1 2 푧2푧1 1 2 ⊥ ℳ is invariant for 픏2 and ℜ2. Thus compression to ℳis a WOT-continuous homomorphism. ∞ The compression to ℳ1is a homomorphism onto H (L1), sending L1to the unilateral shift as we have discussed before. The compressions of both 퐿푖to ℳvanish on ℳ2, and L2maps ℳ1isometrically onto ℳ2. Hence the compressions are 62

푇 0 0 0 푃 퐿 | ≃ [ 푧 ] and 푃 퐿 | ≃ [ ] ℳ 1 ℳ 0 0 ℳ 2 ℳ 퐼 0 Thus Ψ maps 픏nonto the algebra of operators of the form 푇 0 ℎ0 ∞ [ ] for ℎ푖 ∈ 퐻 푇ℎ1 ℎ0(푂) Indeed, this shows that every element of 픏nmay be written uniquely as ′ ∞ ′ 퐴 = ℎ0(퐿1) + 퐿2ℎ1(퐿1) + 퐴 where ℎ푖 ∈ 퐻 and 푃ℳ퐴 = 0. ∞ Now let 휓 be any multiplicative linear functional on H in the fibre훭1. (2) Then 훷 = 휓 훹 is a completely contractive homomorphism of 픏2onto 풯2such that 훷(퐿푖) = 푇푖. Indeed, 푇ℎ 0 휓(ℎ ) 0 훷(퐴) = [ 0 ] = [ 0 ] ( ) 푇ℎ1 ℎ0 푂 휓(ℎ1) ℎ0(푂) −1 Hence we have shown that the fibre 휋2,2(푇) is very large. We analyze the automorphism group of 픏n. The automorphismsof the algebra 퐿1 = ∞ 퐻 are precisely the maps Θ풯(h) = h(풯) where 풯is a conformal automorphism of the unit disk. So Aut(퐿1) is isomorphic to 퐴푢푡(𝔹1), the group of conformal automorphisms of the unit disk. In particular, they are all norm and wot-continuous. See [118], where two proofs are given, both based on factorization of analytic functions. Our main result is Theorem (3.1.19), which is valid even for 푛 = ∞. Our original proof of Theorem (3.1.19) failed when 푛 = ∞. An automorphism of 픏nwill be called quasi-inner if it is trivial modulothe WOT-closed commutator ideal 픢̅(see Proposition (3.1.8)). Denote the set of all quasi-inner automorphisms by q-Inn(픏n). In particular, this contains the subgroup Inn(Ln픏n) of inner automorphisms. Theorem (3.1.19)[156]:There is a natural short exact sequence 풯 0 ⟶q-퐼푛푛(픏푛) ⟶Aut(픏푛) → Aut(𝔹푛) ⟶ (0) −1 1,푛 The map풯takes 훩to풯훩(휆) = (휑휆훩 ) (퐿)for 휆 ∈ 𝔹푛.Moreover, 풯has a continuous section kwhich carries Aut(𝔹푛)onto the subgroup Autu(픏푛)of unitarily implemented automorphisms. Thus Aut(픏푛)is a semi direct product. The proof will be carried out in stages. First we establish an automatic continuity result for automorphisms. Lemma (3.1.20)[156]:Every automorphism 훩of 픏n, for 푛 ≥ 2, is continuous. −1 Proof: The proof is a standard gliding bump argument. We define 퐵푖 = 훩 (퐿푖) and set Λ = 푚푎푥{1, ‖퐵1‖, ‖퐵2‖}. Suppose that Θ is not continuous . Then there is a sequence 퐴푘 in 픏푛 such that −푘 ‖퐴푘‖ ≤ (2Λ) and ‖Θ(퐴푘)‖ > 푘. Let 퐴 be defined by the norm convergent sum 푚 푘 푘 푚+1 푘 퐴 = ∑ 퐵2 퐵1퐴푘 = ∑ 퐵2 퐵1퐴푘 + 퐵2 ∑ 퐵2 퐵1퐴푚+1+푘: 푘≥1 푘=1 푘≥1 푘 Set 푋푚 = ∑푘≥0 퐵2 퐵1퐴푚+1+푘. Then for all 푘 > 0, 푚 ∗ ∗푚 ∗ ∗푚 푘 ∗ ∗푚 푚+1 ‖훩(퐴)푘‖ ≥ ‖퐿1퐿2 훩(퐴)‖ = ‖∑ 퐿1퐿2 퐿2퐿1(퐴푘) + 퐿1퐿2 퐿2 훩(푋푚)‖ = ‖훩(퐴푘)‖ > 푘. 푘=1 63

This is absurd, and consequently 훩 must be continuous. We show that every automorphism determines a special point in the ball. Proposition (3.1.21)[156]:Let 훩 be an automorphism of 픏n. Then there is a unique point 휆 in 0 −1 𝔹n such that 훩(픏푛) = 푘푒푟 휑휆. Indeed, 휑휆 = φ0Θ . Proof: Let (1;푛) 푆 = 훩 (퐿) ∶= [푆1 … 푆푛]: By Corollaries (3.1.10) and (3.1.13), 픏푛 = ℂ퐼 + 퐿풞푛(픏푛), and this decomposition is unique. Applying 훩 yields 픏푛 = ℂ퐼 + 푆풞푛(픏푛), and every A in 픏n has a unique decomposition as 퐴 = 훼퐼 + 푆퐵for some B in 풞n(픏n). Hence the continuous linear map 푇 from ℂ ⊕ 풞푛(픏푛)to픏n given by 푇(훼; 퐵) = 훼퐼 + 푆퐵 is a bijection. By Banach's isomorphism Theorem, 푇 is invertible. So there is a constant c > 0 so that 푐−1‖|훼|21 + 퐵∗퐵 ‖1/2 ≤ ‖훼1 + 푆퐵‖ ≤ 푐 ‖|훼|21 + 퐵∗퐵 ‖1/2: 0 Let 픍 = 푆퐶푛(픏푛) = 훩(픏푛). Since 푇 maps a subspace of codimension one onto ℑ, it also has codimension one. We claim that this ideal is WOT-closed. Suppose that 퐽훽 = 푆퐵훽 is a bounded net in 픍 which converges weak-∗ to an operator 푋 in 픏n. Then the net퐵훽 is bounded in 풞n(픏n) by the previous paragraph. Hence there is a cofinal subnet 퐵훽′ which converges weak-* to an operator 퐵 in 풞n(픏n). Consequently, it follows that X = SB belongs to 퐽. This shows that the intersection of 픏 with each closed ball is weak-* closed. by the Krein-Smulian Theorem (c.f. [73]), 픏 is weak-* closed. By Corollary (3.1.4), the weak-* and WOT topologies coincide on 픏n. Hence 퐽 is WOT − closed. − Consider the multiplicative linear functional휑 = 휑0훩 1, which yields the formula 휑(훼퐼 + 푆퐵) = 훼. Since 픏 is WOT-closed, this functional is WOT-continuous. Therefore by Theorem (3.1.7), there is a point λ in 𝔹n such that 휑 = 휑휆. We will show that automorphisms of 픏n are automatically WOT-continuous.First we establish a criterion for WOT-convergence in 픏n. Recall that Cn, 픏n is the ideal generated by {퐿휔: |휔| = 푘}. Lemma (3.1.22)[156]:For a bounded net 퐴훼 in 픏n, n < 1, the following are equivalent: (i) 푊푂푇-푙푖푚 퐴훼 = 0. 훼 (ii)푤 − 푙푖푚 퐴훼 휉1 = 0. 훼 0,푘 (iii) 푙푖푚 푑푖푠푡(퐴훼 ; 픏푛 ) = 0 for all 푘 ≥ 1. 훼 Proof: It is evident that (i) implies (ii). If (ii) holds, then write 훼 퐴훼 휉1 = ∑ 푎휔휉휔 : 휔 훼 0,푘 Then 퐴훼,푘 : = 퐴훼 ∑|휔|<푘 푎휔퐿휔 belongs to 픏푛 by Lemma (3.1.9) Condition 훼 (ii) Clearly implies that 푙푖푚푎휔 = 0 for every 휔. Hence 훼 0,푘 훼 2 lim sup dist(퐴훼 , 픏푛 ) ≥ lim sup ‖퐴훼 − 퐴훼,푘 ‖ ≤ lim sup ∑ |푎휔| = 0 훼 훼 훼 |휔|<푘

64 for every k ≥ 1. Now if (iii) holds, then 1 ⁄2 0,푘 0,푘 훼 2 0 = lim dist(퐴훼 , 픏푛 )lim sup dist(퐴훼 휉1; 픏푛 휉1) = ( ∑ |푎휔| ) 훼 훼 |휔|<푘 훼 A fortiori, 푙푖푚푎푤 = 0 for every w inℱn. Therefore 훼 ∗ 푙푖푚(퐴훼 휉푢; 휉푣) = 푙푖푚(퐴훼 휉1; 푅푢휉푣) = 0 훼 훼 for every pair of words 푢; 푣 in ℱ푛. These vectors span a dense subset of ℋn. As the net 퐴훼 is bounded, it converges WOT to 0. Theorem (3.1.23)[156]:Every automorphism Θof픏n is WOT-continuous. −1 Proof: By Lemma (3.1.22), there is a point λ in 𝔹n such that φ0Θ = φ휆. Thus 0 픍 = 훩(픏푛 ) = 푘푒푟휑휆: Hence 0,푘 0 푘 푘 (픏푛 ) = 훩(픏푛 ) = 픍 for all 푘 ≥ 1: 푘 Clearly ∩푘≤1 픍 = {0} since −1 푘 −1 푘 0,푘 훩 (∩푘≤1 픍 ) ⊂ 훩 (픍 ) = 픏푛 for all 푘 ≥ 1: Thus by Theorem (3.1.5), we have ̅̅̅푘̅̅̅̅̅ ∩푘≤1 픍 ℋ푛 = ‖0‖ (3) Set 휁휔 = 훩(퐿휔)휉1 for휔 ∈ ℱ푛. Fix an integer푗 and let ℳjand 풩jbe the closed linear spans of {휉휔: |휔| = 푗} and {휉휔: |휔| = 푗} respectively. If 훽 = ∑|휔|=푗 푏휔휉휔is a finite linear combination of the 휉휔, put B = ∑|휔|=푗 푏휔퐿휔.Then since 훩 is bounded,

‖ ∑ 푏휔휁휔‖ = ‖훩(퐵)휉1‖ ≤ ‖훩‖‖퐵‖ = ‖훩‖‖훽‖. |휔|=푗 Thus the map ∑|휔|=푗 푏휔휉휔 ⟼ ∑|휔|=푗 푏휔휁휔 extends to a bounded linear operator 푇푗 ∶ ℳ푗 ⟼ 풩푗 . 훼 훼 Now consider a bounded net퐵훼 = ∑|휔|=푗 푏휔휉휔 such that 푙푖푚 푏휔휉휔 = 0 for all 휔. Let 휔→0 훼 훽훼 = ∑|휔|=푗 푏휔휉휔. It follows that 휔 − 푙푖푚 훩( 퐵훼)휉1 = 휔 − 푙푖푚푇푗훽훼 = 푇푗휔 − 푙푖푚훽훼 = 0: 훼 훼 훼 Hence 훩(훽훼) converges WOT to 0.Again let 퐴훼be a bounded net converging WOT to 0 in 픏n and let Λ = 푠푢푝‖퐴훼‖, it suffices to show that 푙푖푚(훩( 퐴훼)휉1휁) = 0 훼 푘 ⊥ for 휁 in a dense subset of ℋ푛. A convenient choice is ∪푘≤1 (픍 ℋ푛) , which is dense by the 푘 ⊥ 2 equality (3). Choose ζ in (픍 ℋ푛) , and set 푝 = 푘 . Decompose 퐴훼 = 퐵훼 + 퐶훼where |휔| 퐵 = ∑(퐴 ) + ∑ 푎훼 퐿 and 퐶 = 퐴 − 퐵 ∈ 픏푘 . 훼 훼 푝 휔 휔 훼 훼 훼 푛 푝 |휔|=푗 Since the Cesaro mean ∑푝(퐴훼) is contractive, it follows that ∑푝(퐴훼) ≤ Λ. 훼 Also the terms 퐴훼,푗 = ∑|휔|=푗 푎휔퐿휔 are bounded by s, and thus 65

푘−1 −1 ‖퐵훼‖ ≤ Λ + 푝 ∑‖퐴훼,푗‖ ≤ 2Λ: 푗=1 Hence ‖퐶훼‖ ≤ 3Λ. Moreover each net 퐵훼 and 퐶훼 converge wot to 0. Now since 퐵훼 is supported on words of length less than 푝, we have seen that Θ(Bα) converges WOT to 0. Finally by construction, (훩(퐶훼)휉1; 휁) = 0 푘 since 훩(퐷훼)휉1 lies in 픍 ℋ푛, which is orthogonal to 휁. The weak-∗ topology on the unit ball of 퐵(ℋ푛) is metrizable, and the ball is compact. Thus it follows readily that a linear map훩 is weak-∗ continuous on a bounded convex set if and only if it takes every weak-∗ null sequence to a weak-∗ null sequence. Hence we see that 훩 is weak- ∗ continuous on every closed ball of 픏n. Therefore by the Krein-Smulian (c.f [73]), it follows that 훩 is weak-∗ continuous. By Corollary (3.1.4), the weak-픏n and WOT topologies coincide on 픏n. Thus 훩is WOT-continuous. The tools are now available to define the map τ given in Theorem (3.1.19). Using Theorem (3.1.17), we identify 𝔹n with 푅푒푝1(픏푛) by associating λ in 𝔹n with the multiplicative linear functional φ휆 in 푅푒푝1(픏푛). Theorem (3.1.24)[156]: For each 훩 in 퐴푢푡(픏푛), the dual map τΘ on 푅푒푝1(픏푛) given by −1 휏훩(휑) ∶= 휑 ∘ 훩 maps the open ball 𝔹푛 conformally onto itself. This determines a homomorphism of 퐴푢푡(픏푛) into the group Aut(𝔹n) of conformal automorphisms. If 휏훩(휑0) = 휑0, then there is a unitary matrix 푈 in 풰n such that 휏훩(휑휆) = 휑푈휆. −1 Proof: Since 훩 is WOT-continuous by Theorem (3.1.24), it follows that τΘ(φ휆) = φ휆 ∘ Θ is a WOT-continuous multiplicative linear functional. Hence by Theorem (3.1.7), this is a functional φμ. Thus τΘ maps 𝔹n into itself. We obtain an explicit formula for this map using (1;푛) the fact that 휑휆 (퐿) = 휆, whence −1 (1;푛) (1;푛) ̂ 휏훩(휆) = (휑휆훩 ) (퐿) = 휑휆 (푇) = 푇(휆); where −1 (1;푛) −1 −1 푇 = (훩 ) (퐿) = [훩 (퐿1) … 훩 (퐿푛)]. By Theorem (3.1.17), 푇̂ is analytic and thus so is 휏훩. Next notice that the map τ taking Θ to τΘ is a homomorphism. It is evident that 휏퐼푑 = 푖푑;that is, the identity automorphism induces the identity map on 𝔹푛. Suppose that Θjbelong to퐴푢푡(퐿푛), and 휏푗 = 휏(훩푗) for j = 1; 2.Then −1 (1;푛) −1 −1 (1;푛) −1 (1;푛) τ(훩1훩2)(휆) = (휑휆(훩1훩2) ) (퐿) = (휑휆훩2 훩1 ) (퐿) = (휑휏2(휆)(훩1 ) (퐿 (1;푛) = (휑휏1 (휏2(휆))) (퐿) = 휏1(휏2(휆)): Hence 휏(훩1훩2) = 휏(훩1) ∘ 휏(훩2). Consequently 휏훩휏훩−1 = 푖푑 = 휏훩−1휏훩; from which we deduce that 휏훩 is a bijection. Therefore 휏훩is a biholomorphic bijection (i.e. a conformal automorphism) of the ball. If 휏is a conformal automorphism of 𝔹푛. such that 휏 (0) = 0, then by Schwarz's Lemma, there is a unitary operator 푈 in 푈푛 such that 휏(휆) = 푈휆[256] for 푛 < ∞and [113] for 푛 = ∞. 66

The following corollary characterizes the quasi-inner automorphisms. Corollary (3.1.25)[156]: For 훩 in 퐴푢푡(픏푛), the following are equivalent: (i) 훩 belongs to 푘푒푟 휏 . (ii) 훩(퐿푖) − 퐿푖 belongs to 픢̅for 1 ≤ 푖 ≤ 푛. (iii) Θ(A) − A belongs to 픢̅ for every A in 픏n. Proof:If 훩 belongs to ker 휏 , then so does 훩−1; whence 휑휆(훩(퐿푖) − 퐿푖 = 휏훩−1(휆) − 휆 ( ) is zero for every휆in 𝔹n if and only if 훩 퐿푖 − 퐿푖 belongs to ⋂휆∈𝔹푛 푘푒푟 휑휆 for 1 ≤ 푖 ≤ 푛. But this set equals 픢 by Proposition (3.1.8). So (i) and (ii) are equivalent. Suppose that (ii) holds. As 픢 is an ideal, it readily follows that 훩(푝(퐿)) − 푝(퐿) belongs to 픢 for every polynomial in 퐿. Then because 훩 is wotcontinuous and 픢 is -closed, this extends to the WOT-closure of these polynomials, which is all of 픏n. This establishes (iii). Clearly (iii) implies (ii). To complete the picture, we need to construct explicit automorphisms to demonstrate that the map τ is surjective. In fact, much more will be established. An explicit section of τ will be found that maps 퐴푢푡(𝔹푛) onto the subgroup 퐴푢푡푢(픏푛) of unitarily implemented automorphisms. This will establish that 퐴푢푡(픏푛) actually has the structure of a semidirect product. A certain class of unitarily implemented automorphisms of 픏n are well known from quantum mechanics, and are called gauge automorphisms. Think of ℋn as being identified with the Fock space 퐹(퐻) with 푑푖푚퐻 = 푛. For any unitary 푈 on 퐻, form the unitary operator 푈̃ = ∑⊗ 푈⊗푘 푘≥0 which acts on Fock space by acting as the 푘-fold tensor product of 푈 on the 푘-fold tensor product of ℋ. It is evident that 푈̃ℓ(휁) 푈̃∗ = ℓ(푈휁) for 휁 ∈ ℋ: therefore 훩푈 = 퐴푑 푈̃determines an automorphism of 픏n. If 푈 = [푢푖푗 ] is an 푛 × 푛 unitary matrix, this automorphism can also be seen to be given by 푛

훩푈(퐿푗) = ∑ 푢푖푗퐿푖 for 1 ≤ 푗 ≤ 푛. 푖=1 An easy calculation shows that 훩푈훩푉 = 훩푈푉; so this is a homomorphism of the unitary group 푈푛 into the automorphism group 퐴푢푡(픏푛). It follows from Lemma (3.1.27) below that 휏훩푈 = 푈̅, the coordinatewise conjugate of 푈.So τ maps the group of gauge automorphisms onto the unitary group. In [299], Voiculescu constructed a larger subgroup of automorphisms of the Cuntz- Toeplitz algebra ℰ푛 which turn out to be the one we want. He starts with the group 푈(1, 푛)consisting of those (푛 + 1) × (푛 + 1) matrices 푋 such that 푋∗퐽푋 = 퐽, where 퐽 = 1 0 푥 휂∗ [ ]. One may compute that these matrices have the form 푋 = [ 0 1] where the 0 퐼푛 휂2 푋1 coefficients satisfy the (redundant) relations: 2 2 2 (i) ‖휂1‖ = ‖휂2‖ = ‖푥0‖ − 1 67

∗ (ii) 푋1휂1 = 푥̅̅0̅휂2 and 푋1 휂2 = 푥0휂1 ∗ ∗ ∗ (iii) 푋1 푋1 = 퐼푛 + 휂1휂1 and 푋1푋1 = 퐼푛 + 휂2휂2. 푛 푛 Let us write 퐿휁 = ∑푖=1 휁푖 퐿푖 for 휁 ∈ ℂ . Voiculescu shows that there is a (unique) automorphism 훩푋 of ℇn such that the restriction to the generators is given by −1 〈 〉 훩푋(퐿휁) = (푥0퐼 퐿휂2) (퐿푋1휁_ 휁, 휂1 퐼): It is easy to verify that the kernel of this map consists of the scalar matrices 푥0퐼푛+1 for x0 in the circle 핋. Moreover Voiculescu constructs a unitary operator 푈푋 by −1 푈푋(퐴휉1) = 훩푋(퐴)(푥0퐼 퐿휂2) 휉1 for all 퐴 ∈ 픏푛 ∗ so that 훩푋(퐴) = 푈푋퐴푈푋 for 퐴 in 픏n. It is apparent that the norm-closed (nonself-adjoint) algebra 프n generated by 푓퐿푖 ∶ 1 ≤ 푖 ≤ 푛 is mapped into itself by this map. Since this is a group homomorphism, it maps 프n onto itself. Then because 훩푋 is unitarily implemented, it is WOT-continuous and thus determines an automorphism of 픏n. We will provide discussion below to indicate another method of obtaining these automorphisms that fits into our framework some what better. There is also a natural map from 푈(1, 푛) onto 퐴푢푡(𝔹푛) by fractional linear maps. This result must be well known. We do not have a reference, but the results of Phillips [211] on simplectic automorphisms of the ball of ℬ(퐻) may be modified to apply to the ball of ℬ(ℋ; 풦) for Hilbert spaces ℋand풦. Taking ℋ = ℂn and 풦 = ℂ yields our map. 푛 Lemma (3.1.27)[156]: For 푋 in 푈(1, 푛), define a map 휃푋: 𝔹푛 → ℂ by 푋1휆 + 휂2 휃푋(휆) = for 휆 ∈ 𝔹푛. 푥0 + 〈휆, 휂1〉 Then 휃푋 belongs to 퐴푢푡(𝔹푛) and the associated map 휃: 푈(1, 푛) → 퐴푢푡(𝔹푛) is a surjective homomorphism with kernel equal to the scalars. Proof:First one computes using (i) and (ii) above: 2 2 |푥0 + 〈휆, 휂1〉| − ‖〈푋1 − 휆, 휂2〉‖ 2 2 2 2 = |푥0| + |〈휆, 휂1〉| − ‖푋1휆‖ − ‖휂2‖ + 2 푅푒(〈휆, 푥0휂1〉 − 〈푋1휆, 휂2〉) 2 2 2 2 ∗ = (|푥0| − ‖휂2‖ ) − (‖푋1휆‖ − |〈휆, 푥0휂1〉| ) + 2 푅푒〈휆, 푥0휂1〉 − 〈푋1 휂2〉 = 1 − |휆|2. Thus this map carries 𝔹n onto itself, and so belongs to 퐴푢푡(𝔹푛). A straightforward calculation shows that this map is a group homomorphism.Again the kernel 1 0 of this map is the circle of scalar matrices in 푈(1, 푛). The unitary operator 푋 = [ ] is sent 0 푈 −1 to 푈. Now 휃푋(0) = 푥0 휂2 is an arbitrary point in the ball. Hence the range of 휃 is a transitive subgroup of 퐴푢푡(𝔹푛) containing the unitary group. By Schwarz's lemma [256, 113], the range is the whole group of conformal automorphisms. To see the relationship between Θ and θ, we make the following computation.

Lemma (3.1.27)[156]:휏(훩푋̅) = 휃(푋̅ ) for all 푋 in 푈(1, 푛). 푥 휂∗ Proof: Compute for푋 = [ 0 1] that 휂2 푋1

∗ −1 ∗ 푥̅0 −휂1 푋 = 퐽푋 퐽 = [ ∗ ] . −휂2 푋1 n Therefore if eiform the standard basis for ℂ , then −1 ( )( ) [ −1 ] ∗ 〈 〉 휏 훩푋 휆 = 휑휆훩푋 (퐿푖 ) = 휑휆 ((푥̅0퐼 + 퐿휂1) (퐿푋1 푒푖 + 푒푖, 휂2 )) 푛 −1 ( 〈 〉) 〈 ∗ 〉 〈 〉 = 푥̅0 + 휆, 휂1̅ ∑( 휆, 퐿푋1 푒푖 + 푒푖, 휂2 )푒푖 푖=1 푛 −1 −1 = (푥̅0 + 〈휆, 휂1̅ 〉) ∑(〈푋̅1휆, 푒푖〉 + 〈휂2, 푒푖, 〉)푒푖 = (푥̅0 + 〈휆, 휂1̅ 〉) (푋̅1휆 + 휂̅2) 푖=1 = 훩푋̅(휆): Theorem (3.1.28)[156]: The restriction of τ to the subgroup Autu(픏n) of unitarily implemented automorphisms is an isomorphism onto Aut(𝔹n). Proof: Define a map 풦: 퐴푢푡(𝔹푛) → 퐴푢푡푢(픏푛) as follows: given αα in 퐴푢푡(𝔹푛),pick 푋 ∈ −1 푈(1, 푛) belonging to 휃 (훼)and set 풦(α) = Θ푋̅. This is a well defined monomorphism because 훩 and 휃 have the same kernel and complex conjugation is an automorphism of 푈(1, 푛).By the previous lemma, it follows that τ풦 is the identity on 퐴푢푡(𝔹푛). In particular, τ restricted to 퐴푢푡푢(픏푛) is a surjective homomorphism. To prove that this map is injective, suppose that 훩 is a unitarily implemented automorphism such that 휏(훩) = 푖푑. A fortiori, Θ is contractive. But 훩(퐿푖) = 퐿푖 + 퐶푖 where 퐶푖 ∈ ℯ̅whence 2 2 1 ≥ ‖훩‖2 ≥ (‖퐿푖 + 퐶푖)휉1‖ = 1 + ‖퐶푖휉1‖ . Consequently,퐶푖휉1 = 0 which implies that 퐶푖 = 0. Therefore 훩 = 퐼푑 andour map is an isomorphism. We record an immediate consequence of the proof. Corollary (3.1.29)[156]: Every contractive automorphism of 픏n is unitarily implemented. In particular, it is completely isometric. It would be interesting to know if automorphisms of 픏n are automatically completely bounded. All the necessary parts for Theorem (3.1.19) have now been accumulated. The homomorphism τ is now known to be surjective, with kernel 푞 − 퐼푛푛(픏푛) and a continuous section k onto 퐴푢푡(픏푛) as required. −1 Notice that if 훩 is unitarily implemented, then 푆푖 = 훩 (퐿푖) will be isometries with pairwise orthogonal ranges. They generate the ideal 푛 −1 0 ∑ 푆푖 픏푛 = 훩 (픏푛). 푖=1 This is a WOT-closed two-sided ideal of codimension one, and thus by Theorem (3.1.5) its range is a 픏n픎n invariant subspace of codimension one. The complement is a one-dimensional 0 invariant subspace for 픏n, and thus by Theorem (3.1.3) is spanned by νλ for some λ in 𝔹푛. It is easy to check that 휏훩(0) = 휆. Conversely, given λ, we can construct such isometries. By Theorem (3.1.3), the subspace ⊥ {휈휆} is 픎n invariant and has an n-dimensional wandering space λ. Let ξi for 1 ≤ 푖 ≤ 푛 be 69 an orthonormal basis for Wλ. Then by [68], the operators Si = Lξi are isometries in 픏n with ⊥ ranges summing to {휈휆} . We will sketch how to construct the automorphism _ which takes 퐿푖 to Si for 1 ≤ 푖 ≤ 푛. The first step is to show that 휈휆 is cyclic for the WOT-closed subalgebra 프 generated by {푆1, … , 푆푛}. This is established by showing that 휁휔 = (푆)휈휆, 휔 ∈ ℱ푛, is an orthonormal basis ∗ for ℋn. This immediately yields a unitary operator 푊 such that 푊퐿푖푊 = 푆푖 such that 퐴푑푊 is an endomorphism of 픏n. The second step is to show that 풴 = 픏n. Since it is contained in 픏n, we see that 휈0 = 휉1 ∗ is an eigenvalue for 프 . Since 프 is unitarily equivalent to 픏n, there is a non-zero μ such that ′ 푊휈휇 = 휉1. Apply the argument again to obtain a second unitary W′ so that A푑푊′푊(퐿푖) = 푆푖 = ′ ⊥ 퐿 ′ where ζ form an orthonormal basis for the wandering space of {휉 } . But then (when n < 휁퐼 I 1 ′ 1) there is a unitary 푈 in 푈푛 such that 휁퐼 = 푈푒푖 = 푈휉푧푖 . Unfortunately, this argument fails for 푛 = ∞. Consequently, it follows that 퐴푑푊′푊 = 훩푈. Thus the two endomorphisms 퐴푑푊 and 퐴푑푊′ must have been automorphisms. Section (3.2) Cmmutative Banach Algebras of Teoplitz Operators Recall first that the C∗-algebras generated by Toeplitz operators which are commutative on each weighted Bergman space over the unit disk were completely classified in [262]. Under some technical assumption on “richness” of a class of generating symbols the result was as follows. A C∗-algebra generated by Toeplitz operators is commutative on each weighted Bergman space if and only if the corresponding symbols of Toeplitz operators are constant on cycles of a pencil hyperbolic geodesics on the unit disk, or if and only if the corresponding symbols of Toeplitz operators are invariant under the action a maximal commutative subgroup of the Möbius transformations of the unit disk. We note that the commutativity on each weighted Bergman space was crucial in the part “only if” of the above result. Generalizing this result to Toeplitz operators on the unit ball, it was proved in [245, 251] that, given a maximal commutative subgroup of biholomorphisms of the unit ball, the C∗- algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each weighted Bergman space. The geometric description of corresponding symbols in terms of so-called Lagrangian foliations (which generalize the notion of a pencil of hyperbolic geodesics to multidimensional case) was also given. It turned out that for the unit ball of dimension n there are 푛 + 2 essentially different “model” commutative C∗- algebras, all others are conjugated with one of them via biholomorphisms of the unit ball. It was firmly expected that the above algebras exhaust all possible algebras of Toeplitz operators on the unit ball which are commutative on each weighted Bergman space. We present here a quite unexpected result. There exist other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C∗-algebra, and for 푛 = 1 all of them collapse to the C∗-algebra, which is generated by Toeplitz operators with radial symbols. Let 𝔹n be the unit ball in ℂn, that is, 푛 푛 2 2 2 𝔹 = {푧 = (푧1, . . . , 푧푛) ∈ ℂ : |푧| = |푧1| +. . . +|푧푛| < 1},

70 and let 핊n be the corresponding unit sphere, the boundary of the unit ball 𝔹푛. In what follows we will use the notation 휏(𝔹푚) for the base of the unit ball 𝔹푚, considered as a Reinhard domain, i.e., 푚 2 2 2 휏 (𝔹 ) = {(푟1, . . . , 푟푚) = (|푧1|, . . . , |푧푚|) ∶ 푟 = 푟1 + . . . + 푟푚 ∈ [0, 1)}. 푛 Given a multi-index 훼 = (훼1, 훼2, . . . , 훼푛) ∈ ℤ+ we will use the standard notation, |훼| = 훼1 + 훼2 + . . . + 훼푛, 훼! = 훼1! 훼2! ··· 훼푛!, 훼 훼1 훼2 훼푛 푧 = 푧1 푧2 ··· 푧푛 . Two multi-indices α and β are called orthogonal, 훼 ⊥ 훽, if 훼. 훽 = 훼1훽1 + 훼2훽2+ . . . + 훼푛훽푛 = 0. (4) Denote by d푉 = 푑푥1푑푦1. . . 푑푥푛푑푦푛, where 푧푙 = 푥푙 + 푖푦푙, 푙 = 1, 2, . . . , 푛, the standard 푛 Lebesgue measure in ℂ ; and let 푑푆 be the corresponding surface measure on 핊푛. We introduce the one-parameter family of weighted measures, 훤(푛 + 휆 + 1) 푑푣 (푧) = (1 − |푧|2)휆 푑푉 (푧), 휆 > −1, 휆 휋푛훤(휆 + 1) which are probability ones in 𝔹푛; and recall two known equalities (see, for example, [150]) 푛 훼 −훽 2휋 훼! ∫ 휉 휉 푑푆(휁) = 훿훼,훽, (5) 핊푛 (푛 – 1 + |훼|)!, ( ) 훼 −훽 훼! 푛 + 휆 + 1 ∫ 푧 푧 푑푣휆(푧) = 훿훼,훽 (6) 𝔹푛 훤(푛 + |훼| + 휆 + 1). 푛 We introduce the weighted space 퐿2(𝔹 , 푑푣휆) and its subspace, the weighted Bergman space 2 2 푛 푛 퐴휆 = 퐴휆(𝔹 ), which consists of all functions analytic in 𝔹 . The (orthogonal) Bergman 푛 푛 2 푛 projection 𝔹 of 퐿2(𝔹 , 푑푣휆)onto 퐴휆(𝔹 ) is known to have the following integral form 휑(휁) 푑푣 (휁) (퐵 휑)(푧) = ∫ 휆 . 휆 ̅ 푛+휁+1 𝔹푛 (1 − 푧 · 휁) n Finally, given a function 푎(푧) ∈ 퐿∞(𝔹 ), the Toeplitz operator 푇푎 with symbol a acts on 2 푛 퐴휆(𝔹 )as follows 2 푛 2 푛 푇푎: 휑 ∈ 퐴휆(𝔹 ) → 퐵휆(푎휑) ∈ 퐴휆(𝔹 ). Let푘 = (푘1, . . . , 푘푚) be a tuple of positive integers whose sum is equal to 푛: 푘1+. . . +푘푚 = 푛. The length of such a tuple may obviously vary from 1, for 푘 = (푛),to n, for푘 = (1, . . . , 1). 푛 Given a tuple 푘 = (푘1, . . . , 푘푚), we rearrange the n coordinates of 푧 ∈ 𝔹 in푚 groups, each one of which has 푘푗 , 푗 = 1, . . . , 푚, entries and introduce the notation 푧(1) = (푧1,1, . . . , 푧1,푘1), 푧(2) = (푧2,1, . . . , 푧2,푘2), . . . , 푧(푚) = (푧푚,1, . . . , 푧푚, 푘푚). 푘푗 We represent then each 푧(푗) = (푧푗,1, . . . , 푧푗, 푘푗 ) ∈ 𝔹 in the form

2 2 푘푗 푧(푗) = 푟푗휉(푗), where푟푗 = √|푧푗,1| + . . . |푧푗, 푘푗 | and 휉(푗) ∈ 핊 . 푛 Given a tuple 푘 = (푘1, . . . , 푘푚), a bounded measurable function 푎 = 푎(푧), 푧 ∈ 𝔹 , will be called k-quasi-radial if it depends only on 푟1, . . . , 푟푚.

Varying 푘 we have a collection of the partially ordered by inclusion sets ℛk of 푘-quasiradial functions. The minimal among these sets is the set ℛ(푛) of radial functions and the maximal one is the set ℛ(1,...,1) of separately radial functions. There is some ambiguity in the above definition. Indeed given a tuple k there are many corresponding sets ℛk which differ by perturbation of coordinates. At the same time each perturbation of coordinates of z is a biholomorphism, say 푘, of the unit ball 𝔹n, which generates the unitary equivalence of the Toeplitz operators Ta and Ta∘k Thus it is sufficient, in fact, to consider only one of these perturbation different sets. To avoid all possible repetitions and ambiguities in what follows we will always assume first, that 푘1 ≤ 푘2 ≤ . . . ≤ 푘푚, and second, that

푧1,1 = 푧1, 푧1,2 = 푧2, … , 푧1, 푘1 = 푧푘1 , 푧2,1 = 푧푘1+1, … , 푧2, 푘2 = 푧 , … , 푧 , 푘 = 푧 . (7) 푘1+푘2 푚 푚 푛 Given 푘 = (푘1, . . . , 푘푚)and any 푛-tuple 훼 = (훼1, . . . , 훼푛), we define α(1) = (α1, . . . , αk1), α(2) = (αk1+1, . . . , αk1+k2), . . . , α(m) = (αn−km+1, . . . , αn). as each set ℛkis a subset of the set ℛ(1,...,1) of separately radial functions, the Toeplitz operator 푇푎 with symbol 푎 ∈ ℛ푘, by [245], is diagonal with respect to the standard monomial basis in 2 푛 𝒜휆 (𝔹 ). The exact form of the corresponding spectral sequence gives the next lemma. Lemma (3.2.1)[193]: Given a 푘 −quasi-radial function 푎 = 푎(푟1, . . . , 푟푚), we have 훼 훼 푛 푇푎푧 = 훾푎,푘,휆(훼) 푧 , 훼 ∈ ℤ+, Where 훾푎,푘,휆(훼) = 훾푎,푘,휆(|훼(1)|, . . . , 훼(푚)) 2푚 (푛 + |_| + _ + 1) = 푚 훤(휆 + 1) ∏푗=1(푘푗 − 1 + |훼(푗)|) ! 푚 2 2|훼(푗)|+2푘푗−1푗 × ∫ 푎(푟1, . . . , 푟푚) (1 − |푟| )휆 ∏ 푟푗 푑푟푗 휏(𝔹푚) 푗=1 Proof: We calculate ( ) 훼 훼 훼 훼 훤 푛 + 휆 + 1 훼 −훼 2 휆 〈푇푎푧 , 푧 〉 = 〈푧 , 푧 〉 = 푛 × ∫ 푎(푟1, . . . , 푟푚) 푧 , 푧 (1 − |푧| ) 푑푉(푧). 휋 훤(휆 + 1) 𝔹푚 푘푗 Changing the variables 푧(푗) = 푟푗휉(푗), where 푟푗 ∈ [0, 1] and 휉(푗) ∈ 핊 , 푗 = 1, . . . , 푚, we have 푚 ( ) 훼 훼 훤 푛 + 휆 + 1 2 휆 2|훼(푗)|+2푘푗−1푗 〈푧 , 푧 〉 = 푛 ∫ 푎(푟1, … , 푟푚) (1 − |푧| ) ∏ 푟푗 푑푟푗 휋 훤(휆 + 1) 푚 𝔹 푗=1 푚 훼(푗) −훼(푗) × ∏ ∫ 휉(푗) 휉(푗) 푑푆(휉(푗)) 푘푗 푗=1 핊 푚 푚( | | ) 2 푛 + 훼 + 훼 + 1 2 2|훼(푗)|+2푘푗−1푗 = 푚 ∫ 푎(푟1, . . . , 푟푚) (1 − |푟| )휆 ∏ 푟푗 푑푟푗 . 훤(휆 + 1) ∏푗=1(푘푗 – 1 + |훼(푗)|) ! 휏(𝔹푚) 푗=1 Then the result follows by (11).

Given 푘 = (푘1, . . . , 푘푚) we use the representations 푧(푗) = 푟푗휉(푗), 푗 = 1, . . . , 푚,to define the vector 푘1 푘2 푘푚 휉 = (휉(1), 휉(2), … , 휉(푚)) ∈ 핊 × 핊 × … × 핊 . we introduce now an extension of 푘 −quasi-radial functions, which may be called following [125, 129, 317] the quasi-homogeneous functions. A function 휑(푧) is called quasi- homogeneous (or k −quasi-homogeneous) function if it has the form 푠 푠(1) 푠(2) 푠(푚) 휑(푧) = 휑(푧(1), 푧(2), … , 푧(푚)) = 푎(푟1, 푟2, … , 푟푚)휉 = 푎(푟1, 푟2, … , 푟푚)휉(2) 휉(2) … 휉(푚) , 푛 where a(푟1, 푟2, . . . , 푟푚) ∈ ℛ푘and 푠 ∈ ℤ . After separating positive and negative entries in s, it admits the unique representation 푠 = 푝 − 푛 푠 n 푞, where 푝, 푞 ∈ ℤ+and 푝 ⊥ 푞. Then 휉 , for s ∈ ℤ , is always understood as ξs = ξpξ−q, n where 푠 = 푝 − 푞, with p, q ∈ ℤ+ and 푝 ⊥ 푞. We will call the pair (p, q) the quasi- 푝 −푞 homogeneous degree of the 푘-quasi-homogeneous function 푎(푟1, 푟2, . . . , 푟푚)휉 휉 . 푝 −푞 Lemma (3.2.2)[193]: The Toeplitz operator 푇푎휉 휉 with k-quasi-homogeneous symbol 푝 −푞 훼 푛 푎휉 휉 acts on monomials 푧 , 훼 ∈ ℤ+ as follows 훼 0 푖푓 ∃ 푙 푠푢푐ℎ 푡ℎ푎푡 훼푙 < 푞푙 − 푝푙 푇푎휉푝휉−푞푧 = { 훼+푝−푞 훾̃푎,푘,푝,푞,휆(훼)푧 , 푖푓 ∀ 푙 훼푙 ≥ 푞푙 − 푝푙, where 2푚 (푛 + |훼| + 훼 + 1) 훾̃푎,푘,푝,푞,휆(훼) = 푚 훤(휆 + 1) ∏푗=1(푘푗 − 1 + |훼(푗)|)(훼 + 푝 − 푞) ! 푚 2 2|훼(푗)+푝(푗)−푞(푗)|+2푘푗−1 × ∫ 푎(푟1, . . . , 푟푚) (1 − |푟| )휆 ∏ 푟푗 푑푟푗 (8) 휏(𝔹푚) 푗=1 푛 Proof: For each two multi-indices 훼, 훽 ∈ ℤ+, we calculate 훼 훽 푝 ̅푞 훼 훽 〈푇푎휉푝휉̅ 푞푧 , 푧 〉 = 〈훼휉 휉 푧 , 푧 〉 훤(푛 + 휆 + 1) 푝 ̅푞 훼 훽 2 휆 = 푛 ∫ 푎(푟1, . . . , 푟푚) 휉 휉 푧 , 푧̅ (1 − |푧| ) 푑푉(푧). 휋 훤(휆 + 1) 𝔹푛 푘푗 Changing the variables 푧(푗) = 푟푗휉(푗), where 푟푗 ∈ [0, 1) and 휉(푗) ∈ 핊 , 푗 = 1, … , 푚 we have 〈휉푝휉̅푞푧훼, 푧훽〉 푚 ( ) 훤 푛 + 휆 + 1 2 휆 2|훼(푗)+푝(푗)−푞(푗)|+2푘푗−1 = 푛 × ∫ 푎(푟1, . . . , 푟푚) (1 − |푟| ) ∏ 푟푗 푑푟푗 휋 훤(휆 + 1) 푛 𝔹 푗=1 푚 훼 (푗)+푝(푗) ̅훽(푗)+푝(푗) × ∏ ∫ 휉(푗) 휉(푗) 푑푆(휉(푗)) 푘푗 푗=1 핊

2푚 (푛 + |훼| + 훼 + 푝) = 훿훼+푝,훽+푞 푚 훤(휆 + 1) ∏푗=1(푘푗 − 1 + |훼(푗)+푝(푗)|) ! 푚 2 휆 |휆(푗)+훽(푗)|+2푘푗−1 × ∫ 푎(푟1, … , 푟푚) (1 − |푟| ) ∏ 푟푗 푑푟푗 휏(𝔹푚) 푗=1

The last integral is non zero if and only if 훼 + 푝 = 훽 + 푞 and 훼푙 + 푝푙 − 푞푙 ≥ 0, for each 푙 = 1, 2, . . . , 푛. Now for β = α + p − q, with 훼푙 + pl − ql ≥ 0, for each 푙 = 1, 2, . . . , 푛, we have by (11),

(훼 + 푝 − 푞)! 훤(푛 + 휆 + 1) 〈푧훽, 푧훽〉 = 〈푧훼+푝−푔, 푧훼+푝−푔〉 = , 훤(푛 + |훼 + 푝 − 푞| + 휆 + 1) and the result follows. A particular case of the next theorem when 푘 = (푛) and 휆 = 0 was proved in [317]. Theorem (3.2.3)[193]: Let 푘 = (푘1, 푘2, . . . , 푘푚) and 푝, 푞 be a pair of orthogonal multi-indices. Then for each pair of non identically zero k-quasi-radial functions 푎1 and 푎2, the Toeplitz 2 푝 −푞 푛 operators 푇푎1 and 푇푎2휉 휉 commute on each weighted Bergman space 𝒜휆 (𝔹 ) if and only if |푝(푗)| = |푞(푗)| for each 푗 = 1, 2, . . . , 푚. Proof: For those multi-indices α with αl + pl − ql ≥ 0, for each 푙 = 1, 2, . . . , 푛, by Lemmas(3.2.1) and (3.2.2) we have

훼 푇 푝̅ 푞푇 푧 푎2휉 휉 훼푙 2푚(푛 + |훼 + 푝 + 푞| + 휆 + 1)(훼 + 푝)! = 푚 훤(휆 + 1) ∏푗=1(푘푗 − 1 + |훼(푗)| + 푝(푗))! (훼 + 푝 − 푞) ! 푚 2 휆 |2훼(푗)+푝(푗)+푞(푗)|+2푘푗−1 × ∫ 푎2(푟1, … , 푟푚) (1 − |푟| ) ∏ 푟푗 푑푟푗 휏(𝔹푚) 푗=1 2푚 (푛 + |훼| + 휆 + 1) × 푚 훤(휆 + 1) ∏푗=1(푘푗 − 1 + |훼(푗)|) ! 푚 2 휆 2|훼(푗)|+2푘푗−1 훼+푝−푞 × ∫ 푎2(푟1, … , 푟푚) (1 − |푟| ) ∏ 푟푗 푑푟푗 푧 휏(𝔹푚) 푗=1 and

훼 훼 That is 푇 푇 푝̅ 푞푧 = 푇 푇 푝̅ 푞푧 if and only if |p | = |q | for each 푗 = 1, 2, . . . , 푚 푎2 푎1휉 휉 푎1 푎2휉 휉 (j) (j)

We note that under the condition 푝(푗)| = |푞(푗)| , for each 푗 = 1, 2, . . . , 푚, formula (13) reads as 2푚 (푛 + |훼| + 훼 + 1) 훾̃푎,푘,푝,푞,휆(훼) = 푚 훤(휆 + 1) ∏푗=1(푘푗 − 1 + |훼(푗)+푝(푗)|)! (훼 + 푝 − 푞) ! 푚 2 휆 2|훼(푗)|+2푘푗−1 × ∫ 푎(푟1, … , 푟푚) (1 – |푟| ) ∏ 푟푗 푑푟푗 . 휏(𝔹푚) 푗=1 푚 ∏푗=1(푘푗 − 1 + |훼(푗)|)! (훼 + 푝)! = 푚 훾훼,푘휆(훼) ∏푗=1(푘푗 − 1 + |훼(푗)+푝(푗)|)! (훼 + 푝 − 푞) 푚 (푘푗 − 1 + |훼(푗)|)! (훼(푗)+푝(푗))! = ∏ [ ] 훾훼,푘휆(훼) (9) 푗=1 (푘푗 − 1 + |훼(푗)+푝(푗)|)! (훼(푗)+푝(푗) − 푞(푗))! As surprising corollaries we have: Corollary (3.2.4)[193]:Given 푘 = (푘1, 푘2, . . . , 푘푚), for each pair of orthogonal multi-indices p and q with |푝(푗)| = |푞(푗)|, for all 푗 = 1, 2, . . . , 푚, and each 푎(푟1, 푟2, . . . , 푟푚) ∈ ℛ푘, we have 푇푎푇휉푝휉̅ 푞 = 푇휉푝휉̅ 푞푇푎 = 푇푎휉푝휉̅ 푞. Given푘 = (푘1, 푘2, . . . , 푘푚), and a pair of orthogonal multi-indices p and q with |푝(푗)| = |푞(푗)|,, for all 푗 = 1, 2, . . . , 푚, let 푝̃(푗) = (0, . . . , 0, 푝(푗), 0, . . . , 0) and 푞̃(푗) = (0, . . . , 0, 푞(푗), 0, . . . , 0). Then, of course, 푝 = 푝̃(1) + 푝̃(2) + . . . + 푝̃(푚) and 푞 = 푝̃(1) + 푞̃(2)+ . . . + 푞̃(푚). For each j = 1, 2, . . . , m, we introduce the Toeplitz operator 푇 = 푇 푝̃ 푝̃ . 푗 휉 (푗)휉̅ (푗) Corollary (3.2.5)[193]:The operators 푇푗, 푗 = 1, 2, . . . , 푚, mutually commute.Given an ℎ- tuple of indices (푗1, 푗2, . . . , 푗ℎ), where 2 ≤ ℎ ≤ 푚, let

푝̃ℎ = 푝̃(푗1) + 푝̃(푗2)+ . . . + 푝̃( 푗ℎ) and푞̃ℎ = q̃ (j1) + q̃ (j2)+ . . . +q̃ ( jh).

Then ℎ

푝̅ 𝑔̅ℎ ∏ 푇푖𝑔 = 푇휉 ℎ휉̅ 푔=1 In particular, 푚

∏ 푇푖 = 푇휉푝휉̅ 푞. 푗=1 Given 푘 = (푘1, 푘2, . . . , 푘푚), we consider any two bounded measurable 푘 −quasi- 푝 ̅푞 푢 ̅푣 homogeneous symbols 푎(푟1, 푟2, . . . , 푟푚)휉 휉 and 푏(푟1, 푟2, . . . , 푟푚)휉 휉 , which satisfy the conditions of Theorem (3.2.4) , i.e., 푎(푟1, 푟2, . . . , 푟푚) and 푏(푟1, 푟2, . . . , 푟푚)are arbitrary k-quasi- radial functions, 푝 ⊥ 푞, 푢 ⊥ 푣, and|푝(푗)| = |푞(푗)| and |푢(푗)| = |푣(푗)|, for all 푗 = 1, 2, . . . , 푚. 푝 ̅푞 푢 ̅푣 Theorem (3.2.6)[193]:Let (푟1, 푟2, . . . , 푟푚)휉 휉 and 푏(푟1, 푟2, . . . , 푟푚)휉 휉 , be as above. Then the 2 푛 Toeplitz operators 푇푎휉푝휉̅ 푞 and 푇푏휉푢휉̅ 푣, commute on each weighted Bergman space 𝒜휆 (𝔹 ) if and only if for each 푙 = 1, 2, . . . , 푛 one of the next conditions is fulfilled (i) 푝푙 = 푞푙 = 0; (ii) 푢푙 = 푣푙 = 0; (iii) 푝푙 = 푢푙 = 0; (iv) 푞푙 = 푣푙 = 0.

Varying α it is easy to see that the last equality holds if and only if for each 푙 = 1, 2, . . . , 푛 one of the next conditions is fulfilled (i) 푝푙 = 푞푙 = 0; (ii) 푢푙 = 푣푙 = 0; (iii) 푝푙 = 푢푙 = 0; (iv) 푞푙 = 푣푙 = 0. To finish the proof we mention that under either of the above conditions both quantities α 훼 Taξp휉̅ 푞Tbξuξ̅푣z and 푇푏휉푢휉̅ 푣푇푎휉푝휉̅ 푞푧 are zero or non zero simultaneously only. Example (3.2.7)[193]:Let 푛 = 7 and k = (2, 5). Then by Theorem (3.2.4) the Toeplitz 푝 ̅푞 operators with symbols 푎(푟1, 푟2) ∈ ℛ푘 and 푏 휉 휉 , where 푏(푟1, 푟2) ∈ ℛ푘, 푝 = (1, 0, 0, 3, 0, 1, 0), 푞 = (0, 1, 1, 0, 1, 0, 2), commute. We mention that here 푝(1) = (1, 0), 푝(2) = (0, 3, 0, 1, 0) and 푞(1) = (0, 1), 푞(2) = (1, 0, 1, 0, 2). As easy to see, all pairs (푢, 푣) of orthogonal multi-indices such that (by Theorem (3.2.8)) the Toeplitz operators with k-quasi-homogeneous symbols having that quasi-homogeneous degrees mutually commute, and commute with both Ta and Tbξp휉̅ 푞 are of the form 푢 = (푢1, 0, 0, 푢4, 0, 푢6, 0), 푣 = (0, 푣2, 푣3, 0, 푣5, 0, 푣7), (10) where 푢1, 푢4, 푢6 ∈ ℤ+, 푣2, 푣3, 푣5, 푣7 ∈ ℤ+, and 푢1 = 푣2, 푢4 + 푢6 = 푣3 + 푣5 + 푣7. (11) that is, the Banach algebra generated by all Toeplitz operators푇푎휉푢휉−푣 , where 푎(푟1, 푟2) ∈ ℛ푘, and the orhogonal multi-indices 푢 and 푣 of the form (10) satisfy the condition (11), is commutative. We formalize the above example as follows. First, to avoid the repetition of the unitary equivalent algebras and to simplify the classification of the (non unitary equivalent) algebras, in addition to (7), we can rearrange the variables zl and correspondingly the components of multi-indices in p and q so that

(i) for each 푗 with 푘푗 > 1, we have

푝(푗) = (푝푗,1, … , 푝푗,ℎ푗 , 0, … , 0) and 푞(푗) = (0, … , 0, 푞푗,ℎ푗+1, … , 푞푗,푘푗) ; (12) (ii) if 푘푗′ = 푘푗′′ with푗′ < 푗′′, then ℎ푗′ ≤ ℎ푗′′. Now, given 푘 = (푘1, . . . , 푘푚), we start with 푚-tuple ℎ = (ℎ1, . . . , ℎ푚), where ℎ푗 = 0 if kj = 1 and 1 ≤ ℎ푗 ≤ 푘푗 − 1 if kj ≥ 1; in the last case, if 푘푗′ = 푘푗′′ with 푗′ < 푗′′, then ℎ푗′ ≤ ℎ푗′′. . We denote by ℛ푘(ℎ) the linear space generated by all k-quasi-homogeneous functions 푝 −푞 푎(푟1, 푟2, . . . , 푟푚) 휉 휉 , where 푎(푟1, 푟2, . . . , 푟푚) ∈ ℛ푘, and the components 푝(푗) and 푞(푗), 푗 = 1, 2, . . . , 푚, of multi- indices p and q are of the form (12) with

푝푗,1+ . . . + 푝푗,ℎ푗 = 푞푗,ℎ푗+1+ . . . + 푞푗,푘푗 , 푝푗,1, . . . , 푝푗,ℎ푗 , 푞푗,ℎ푗+1 + 1, . . . , 푞푗,푘푗 ∈ ℤ+. We note that ℛk ⊂ ℛk(h) and that the identity function e(z) ≡ 1 belongstoℛ푘(ℎ). We have the following result. Corollary (3.2.8)[193]:The Banach algebra generated by Toeplitz operators with symbols from ℛ푘(ℎ)is commutative. We would like to emphasize the following features of such algebras: (i) For different 푘 and ℎ these algebras are not conjugated via biholomorphisms of the unit ball; (ii) These algebras are just Banach and not C∗-algebras; extending them to C∗-algebras they become non commutative; (iii) Given 푘 ≠ (1, 1, . . . , 1), there is a finite number of different m-tuples ℎ and thus a finite number of different corresponding commutative algebras; 2 푛 (iv) These algebras remain commutative for each weighted Bergman space 𝒜휆 (𝔹 ), with 휆 > −1, (v) For 푛 = 1 all of them collapse to the single C∗-algebra generated by Toeplitz operators with radial symbols. We finish presenting another application of Theorems (3.2.4) and(3.2.8),Studying commutativity properties of Toeplitz operators on the Bergman space on the unit disk I. Louhichi and N. V. Rao [124] conjectured that if two Topelitz operators commute with a third one, none of them being the identity, then they commute with each other. As next example shows, this conjecture is wrong when formulated for Toeplitz operatorson the unit ball (𝔹n), with n > 1. Example (3.2.9)[193]:Given 푛 > 1, let 푘 = (2, 1, . . . , 1). Consider the following three symbols (1,0) ̅(0,1) 푎0 = 푎(푟1, 푟2, … , 푟푛−1), 푎1 = 푏(푟1, 푟2, … , 푟푛−1)휉(1) 휉(1) , (1,0) ̅(0,1) 푎2 = 푐(푟1, 푟2, … , 푟푛−1)휉(1) 휉(1) where a, b, c ∈ ℛk.

Then by Theorem (3.2.3) 푇푎0 commutes with both 푇푎1 and 푇푎2, while by Theorem (3.2.6) the operators 푇푎1 and 푇푎2 do not commute.

Chapter 4 Eigenvalue Inequalities and on the Eigenvalues of Normal Matrices Using techniques from algebraic topology we derive linear inequalities which relate the 푛×푛 spectrum of a set of Hermitian matrices 퐴1, . . . , 퐴푟 ∈ ℂ with the spectrum of the sum 퐴1 + · · · +퐴푟. There results are a direct generalization of a theorem of Wielandt on the eigenvalues of the sum of two normal matrices. Characterizations of eigenvalues of normal matrices using the lexicographical order in ℂ are presented, with some applications. Section (4.1) Schubert Calculus

Consider real 푛 × 푛 diagonal matrices 퐷1, . . . , 퐷푟with diagonal elements 휆1(퐷1) ≥ 휆2(퐷1) ≥ . . . ≥ 휆푛(퐷1), 푙 = 1, . . . , 푟. In this section we are concerned with geometric properties of the set of possible spectrums of the matrices 푟 ∗ {∑ 푈푙 푈푙: 푈푙 are unitary}. (1) 푙=1 Equivalently we are interested in the following question: 푛×푛 Given Hermitian matrices 퐴1, . . . , 퐴푟 2 ∈ ℂ each with a fixed spectrum 휆1(퐴푙) ≥ ⋯ 휆푛(퐴푙), 푗 = 1, … , 푟 and arbitrary else. Is it possible to find then linear inequalities which describe the possible spectrum of the matrix 퐴1 + ··· + 퐴푟? For 푟 = 1 this question is of course trivial. For 푟 = 2 the question is classical and very well studied (compare with [139, 10, 263, 34, 239, 237, 225, 108]). An early example of an eigenvalue inequality for a sum of two Hermitian matrices is that of Weyl [108,112,158,33,78]. A generalization of the Weyl inequalities to k −fold partial sums of eigenvalues of Hermitian matrices 퐴, 퐵 and 퐴 + 퐵 is due to Freede and Thompson [225]. Still more general is the class of eigenvalue inequalities described by Horn [10,32] for sums of two eigenvalues. We will present a systematic geometric approach to obtain such eigenvalue inequalities. Although our main results are in the case of two matrices, where 푟 = 2, the approach works equally well in the case of r-fold sums 퐴1 + ⋯ + 퐴푟 of Hermitian matrices 퐴1, … , 퐴푟. Our interest in this problem originates in the observation by Thompson [239, 237] who indicates that most of the known inequalities for the case 푟 = 2 can be derived using methods from algebraic topology, i.e. by the Schubert calculus of complex Grassmann manifolds. As this topological approach is described only in a rudimentary form in [239, 237,35,132] we first present a rigorous development of the Schubert calculus technique towards eigenvalue inequalities. We then show that it is also possible to derive with the same method a large set of inequalities for the case 푟 > 2 as well. The algebraic topology approach to solving inverse eigenvalue problems is by no means limited to the task of finding eigenvalue inequalities for sums of Hermitian matrices. In fact, the technique has been already successfully applied to solve an outstanding inverse eigenvalue problem arising in control theory, i.e. the pole placement problem for multivariable linear systems by static output feedback. See. [248, 138].

The minmax principles of Wielandt and Hersch-Zwahlen are reviewed, which characterize in geometric terms partial sums of eigenvalues of a Hermitian matrix. We review the relevant results from the Schubert calculus of Grassmann manifolds. We apply the technique and state the main results. We show how the inequalities of Weyl [108], [34] and Freede-Thompson [225] follow from the main theorem. In the last section we describe a large ∗ 푛 set of nonzero products in the cohomology ring 퐻 (퐺푘ℂ ), 풵) of the Grassmann manifold, leading to a new class of inequalities for sums of eigenvalues of Hermitian matrices 퐴1,···, 퐴푟. Let 퐴 ∈ ℂ푛×푛 be a complex Hermitian matrix with eigenvalues 휆1(퐴)≥휆2(퐴) ≥ . . . ≥ 휆푛(퐴). (2) The classical Courant-Fischer minmax principle then asserts that (compare e.g. [222]): Theorem (4.1.1)[289]: For 1 ≤ 푖 ≤ 푛: ∗ 휆1(퐴) = 푚푎푥 푚푖푛 푡푟(퐴푥푥 ) (3) 푑푖푚푉=푖 푥∈푉 = 푚푎푥 푚푖푛 푡푟(퐴푥푥∗) (4) 푑푖푚푊=푛−푖+1 푥∈푊 ‖푥‖=1 Amore general version of the minmax principle is due to Wielandt [109] and Hersch- Zwahlen [138] and characterizes partial sums of eigenvalues via flags of subspaces of ℂ푛. To state their result we first recall some basic notions and definitions from geometry: The complex projective space ℂℙn is defined as the set of all one-dimensional complex subspaces of ℂ푛+1, i.e. as the set of all complex lines passing through the origin 0 휖 ℂ푛+1. More 푛 generally, the complex Grassmann manifold 퐺푘(ℂ ) is defined as the set of all k-dimensional complex linear subspaces of ℂ푛. In particular for 푘 = 1 one has the complex projective space 푛 푛−1 퐺1(ℂ ) = ℂℙ . The Grassmannian is a smooth, compact manifold of real dimension 2푘(푛 − 푘). 푛 Equivalently, the Grassmannian 퐺푘(ℂ ) may be defined as the set of all Hermitian projection operators 푃: ℂ푛 → ℂ푛 of rank 푘. A Hermitian projection operator of ℂ푛 is a Hermitian matrix 푃 ∈ ℂ푛×푛satisfying 푃∗ = 푃, 푃2 = 푃, and rank 푃 = 푘. (5) 푛 푛 푛 For any k-dimensional complex linear subspace 퐿 ⊂ ℂ let 푃퐿: ℂ → ℂ be the uniquely determined projection operator satisfying ⊥ 푖푚(푃퐿) = 퐿, 푘푒푟(푃퐿) = 퐿 , (6) where 퐿⊥ denotes the orthogonal complement of 퐿 in ℂ푛 with respect of the standard Hermitian 푛 ⊥ 푛×푘 inner product. Thus 푃퐿 is the orthogonal projection of ℂ onto 퐿 along 퐿 . If 푋 ∈ ℂ is any full rank matrix whose columns form a basis of 퐿, then one has ∗ −1 ∗ 푃퐿 = 푋(푋 푋) 푋 (7) Conversely, for any full rank matrix 푋 ∈ ℂ푛×푘, the operator defined by (7) is a rank 푛 ⊥ 푛 푘 Hermitian projection operator on ℂ . Thus the map 퐿 → 푃퐿 is a bijection of 퐺푘(ℂ ) onto the set {푃 ∈ ℂ푛×푛: 푃 ∗ = 푃, 푃2 = 푃, 푎푛푑 푟푎푛푘푃 = 푘}. 푛 푛 푛 Given any 푘-dimensional linear subspace 퐿 ⊂ ℂ let 푃퐿: ℂ → ℂ denote the associated Hermitian projection operator. We then define ∗ −1 ∗ 푡푟(퐴|퐿 ): = 푡푟(푃퐿퐴푃퐿) = 푡푟(퐴푃퐿) = 푡푟(퐴푋(푋 푋) 푋 ), (8)

푛×푘 where 푋 ∈ 퐶 is any full rank matrix whose columns form a basis of 퐿. Note that 푡푟(퐴|퐿 )) is the trace of a Hermitian operator and therefore a real number. Definition (4.1.2)[289]: The smooth map n RA: Gk(ℂ ) → ℝ L → 푡푟(퐴|퐿 ) (9) 푛 is called the Rayleigh quotient of A on 퐺푘(ℂ ). If 푘 = 1 the map 푅퐴 coincides with the classical Rayleigh quotient < 퐴푥, 푥 > 푅 (푥) = (10) 퐴 < 푥, 푥 > The extremal principles for the partial sums of eigenvalues of a Hermitian matrix 퐴 of Wielandt, Hersch-Zwahlen and Riddel are now stated as follows: Theorem (4.1.3)[289]: (Wielandt [19]) For 1 < 푖1 < . . . < 푖푘 < 푛: 휆 (퐴) + ⋯ + 휆 (퐴) = 푚푎푥 푚푖푛 푡푟(퐴| ) (11) 푖1 푖푘 푛 퐿 푉1⊂⋯⊂푉푘 푉1∈퐺푘(ℂ ) 푑푖푚푉 =푖 푗 푗 푑푖푚(퐿∩푉푗)≥푖푗 = min max 푡푟(퐴| ) (12) n 퐿 dimWj=n−ij+1 L∈Gk(ℂ ) dimWj=n−ij+1 dim(L∩Vj)≥j In particular, for 푘 = 1, Theorem (4.1.3) specializes to the Courant-Fischer minmax principle as formulated in Theorem (4.1.1). Note that ,it can be shown (see [222]) that the maximal value of (11) is assumed at a “partial flag of eigenspaces”, i.e. at a flag (푉1, . . . , 푉푘) having the property that ( ) 푑푖푚(푉푗) = 푖푗 and 푉푗 ⊂ 푘푒푟 휆1퐼 − 퐴 ⊕ … ⊕ 푘푒푟 (휆푖푗퐼 − 퐴),for 푗 = 1, . . , 푘 We conclude with the following result from Hersch-Zwahlen [138]: Theorem (4.1.4)[289]: Let 퐴 be a Hermitian matrix with eigenvalues 휆1(퐴) ≥ . . . ≥ 휆1(퐴) and a corresponding orthogonal set of eigenvectors 푣1, . . . , 푣푛. Denote with 푉푚 ≔ 푠푝푎푛 (푣1, . . . , 푣푛), 푚 = 1, … , 푛. (13) Let 1 ≤ 푖1 < ⋯ < 푖푘 ≤ 푛 then one has: 휆 (퐴) + ⋯ + 휆 (퐴) = 푚푖푛 {푡푟(퐴| ): 푑푖푚(퐿 ∩ 푉) ≥ 푗, 푗 = 1, … , 푘}. (14) 1 푖푘 푛 퐿 퐿∈퐺퐾(ℂ ) Thus the result of Hersch-Zwahlen just says that the sum of eigenvalues 휆1(퐴) + ⋯ + 휆푖푘(퐴) is characterized as the minimal value of the trace function 푡푟(퐴|퐿)when evaluated on a Schubert 푛 subvariety of 퐺푘(ℂ ). 푛 Consider again the Grassmann manifold 퐺푘(ℂ )consisting of k-dimensional linear subspaces n 푛 of the vector space C . Using the Plü cker embedding 퐺푘(ℂ )can be embedded into the 푛! projective space ℂ푛ℙ푁 of dimension 푁 = − 1. Under this embedding 퐺 (ℂ푛) is a 푘!(푛−푘)! 푘 projective variety described by a famous set of quadratic relations (see e.g. [203]). Definition (4.1.5)[289]: A flag ℱ is a sequence of nested subspaces 푛 {0} ⊂ 푉1 ⊂ 푉2 ⊂ ⋯ ⊂ 푉3 = ℂ (15) where we assume that dim 푉푖 = 푖 for 푖 = 1, . . . , 푛. Let 푖 = (푖1, . . . , 푖푘) denote a sequence of numbers having the property that 푖 ≤ 푖1 ≤ . . . ≤ 푖푘 ≤ 푛. (16) Definition (4.1.6)[289]: For each flag ℱ and each multiindex i define 81

( 푛) 퐶(푖; ℱ) ≔ {푊 ∈ 퐺푘 ℂ : 푑푖푚(푤 ∩ 푉푖푠 ) = 푠} is called a Schubert cell and ( 푛) 푆(푖; ℱ) ≔ {푊 ∈ 퐺푘 ℂ : 푑푖푚(푤 ∩ 푉푖푠) = 푠} is called a Schubert variety. We emphasize that the Schubert cell 퐶(푖; ℱ)is indeed a cell, i.e. isomorphic to theaffine space 푁 푘 ℂ where 푁: = ∑푗=1 푖푗 − 푗 is the dimension of the cell 퐶(푖; ℱ). (Compare with [203].) Moreover the Zariski closure of the cell 퐶(푖; ℱ)is the variety 푆(푖; ℱ)which is a projective 푛 algebraic subvariety of 퐺푘(ℂ ). The following results are well known and we refer e.g. to [309, 203]. Theorem (4.1.7)[289]: For every fixed flag ℱ the Schubert cells퐶(푖; ℱ) decompose the 푛 Grassmann variety 퐺푘(ℂ ) into a finite cellular 퐶푊–complex. The integral homology 푛 퐻2푚(퐺푘(ℂ ), 푍) has no torsion and is freely generated by the fundamental classes of the Schubert varieties 푆(푖; ℱ)of real dimension 2푚. Consider a fixed Schubert variety 푆(푖; ℱ)Its homology class is independent of the choice of the flag ℱ and therefore depends only on the numbers 푖1, . . . , 푖푘. we will use the symbol (푖1, . . . , 푖푘) to denote this homology class. The Poincar´e-dual of the class (푖1, . . . , 푖푘) will be denoted by ∗ 푛 {휇1, . . . , 휇푘} ≔ {푛 − 푘 − 푖1 + 1, 푛 − 푘 − 푖2 + 2, … , 푛 − 푖푘} ∈ 퐻 (퐺푘(ℂ ), 푍) (17) At this point we want to mention that our notation was already used by Schubert (compare with the book of Fulton [309,306,258]) and is slightly different to the one used in [203, 103,46,173,30]. The cohomology ring 푘(푛−푘) ∗ 푛 2푚 푛 퐻 (퐺푘(ℂ ), 푍) ≔ ⨁ 퐻 (퐺푘(ℂ ), 푍) (18) 푚=0 has in a natural way the structure of a graded ring. From Poincar´e-duality and Theorem (4.1.8) 2푚 푛 it follows in particular that each graded component 퐻 (퐺푘(ℂ ), 푍) is a free 푍-module with basis the set of Schubert cocycles {휇1, . . . , 휇푘} where 푛 ≥ 푘 ≥ 휇1 ≥ . . . ≥ 휇푘 ≥ 0and 푘 ∑푗=1 푖푗 = 푗 = 푚. Before we describe the multiplicative structure of this ring we formulate the following proposition which establishes the crucial link between geometric intersection properties of ∗ 푛 Schubert varieties and algebraic properties of the ring 퐻 (퐶푘(퐶 ), 푍). A proof of this as well as more general theorems can be found e.g. in [308, 202]. Proposition (4.1.8)[289]:Consider 푟 Schubert varieties 푆(푖푙; ℱ푙), 푙 = 1, . . . , 푟. I 푟+1

∏{푛 − 푘 − 푖푗푙 + 1, … , 푛 − 푖푘푙} ≠ 0, (19) 푙=1 then the intersection 푟

⋂ 푆(푖푙; ℱ푙) ≠ 0. (20) 푙=1

∗ 푛 The multiplicative structure of 퐻 (퐺푘(ℂ ), 푍) is described by the classical formulas of Pieri and Giambelli. For this denote with 휎푗 ≔ {푘, 0, … , … ,0}, 푗 = 1, … , 푛 − 푘. (21) n In fact σj is the j − th Chern class of the universal (classifying) bundle over Gk(ℂ ). In the following we describe the formulas of Pieri and Giambelli. Giambelli’s formula expresses a general Schubert cocycle {휇1, . . . , 휇푘} as a polynomial in the special Schubert cocycle 휎푗 and Pieri’s formula expresses the product of a general Schubert cocycle with a special Schubert cocycle. Pieri’s formula:

{휇1, … , 휇푘}. 휎푗 = ∑ {푉1, … , 푉푘} (22) 휇푖−1≥푉푖≥휇푖 푘 푘 ∑푖=1 푉푖=(∑푖=1 휇푖)+푗 Giambelli’s formula: … 휎휇 휎휇1 휎휇1+1 1+푘−1 휎 휎 ⋮ {휇 , . . . , 휇 } = 푑푒푡 (휎 ) = 푑푒푡 휇2−1 휇2 (23) 1 푘 휇푖+푗−푖 ⋮ ⋱ ⋮

(휎휇푘−푘+1 … 휎휇푘 ) Note that Giambelli’s formula implies that the Chern classes σj generate the ring ∗ 푛 퐻 (퐺푘(ℂ ), 푍). ∗ 푛 There is a deep relationship between the ring 퐻 (퐺푘(ℂ ), 푍)and the ring of symmetric 푆푘 functions 푍[푥1, … , 푥푘] , where 푆푘 denotes the group of permutations, acting on 푘 letters. To explain this relationship we consider a special set of symmetric functions called Schur functions. (See e.g. [123, 247,]). For this let 휇: = (휇1, … , 휇푘) and define 푑푒푡[퓍휇푗+푘−푗] 푠 ≔ ; 푖, 푗 = 1, … , 푘. (24) 휇 푑푒푡[퓍푘−푗] Note that 푠휇 is the quotient of two alternating functions and therefore a symmetric function, called a Schur function. As explained in detail in [123,109,29] the set of Schur functions

{푠휇: 휇1 ≥ 휇2 ≥ ⋯ ≥ 휇푘 > 0 푎푛푑 ∑ 휇푖 = 푞} (25) is an additive basis of the space of symmetric functions of degree q. As explained in [103, 136, 247] one has a ring epimorphism 푆푘 ∗ 푛 휓: 푍[푥1, . . . , 푥푘] → 퐻 (퐺푘(ℂ ), 푍) 푠휇 → {휇1, . . . , 휇푘} (26) The kernel of this map has as an additive basis the set of Schur functions sμ with sμ > 푛 − 푘. ∗ 푛 Using this epimorphism any calculation in the ring 퐻 (퐺푘(ℂ ), 푍) can be formally done in the 푆푘 ring 푍[푥1, . . . , 푥푘] .We want to mention the rule of Littlewood and Richardson which explains how to additively expand a product of Schur functions in terms of Schur functions: Consider two Schur functions 푠휇 and 푠푣. The product 푠휇푠푣 is a symmetric functionof degree ∑ 휇푖 + ∑ 푣푖 and has therefore an expansion in terms of Schur functions: 휆 푠휇푠푣 = ∑ 퐶휇,푣푠휆 (27) 휆

휆 The appearing coordinates 퐶휇,푣 are usually called the Littlewood Richardson coefficients [123,107,247]. In order to give a combinatorial characterization of those coefficients let 휇 = (휇1, . . . , 휇푘) be a partition of n representing the Schur function푠휇. In other words we assume 푘 that 푛 ≥ 푘 ≥ 휇1 ≥ 휇2 ≥ . . . ≥ 휇푘 ≥ 0and ∑푖=1 휇푖 = 푛 If the integer μi is repeated 푟푖–times in 푟1 푟푡 the partition 휇, the abbreviated notation 휇 = (휇1 , . . . , 휇푡 ) will be used. The number |휇| ∶= 푘 ∑푖=1 휇푖 is sometimes called the weight of the partition μ and the numbers μi are called the parts of the partition. It is usual to present a partition by a left based array of boxes which has exactly 휇푖 boxes in the 푖– 푡ℎ row. Such an array is sometimes called a tableau. Example (4.1.9)[289]: Two partitions with corresponding diagrams are illustrated: (3,2,1) (32, 1)

Let 휆 = (휆1, . . . , 휆푘) be a second partition. One writes휆 ≥ 휇 if 휆푖 ≥ 휇푖, 푖 = 1, . . . , 푘. If 휆 ≥ 휇 one defines the skew tableau 휆/휇 as the tableau obtained from the tableau 휆 by removing the first 휇푖 boxes in the row i of the tableau 휆. Example (4.1.10)[289]:휆 = (5, 4, 2, 2), 휇 = (3, 2, 1) then 휆/휇 is given

We are now in a position to formulate the theorem of Littlewood and Richardson. The following formulation as well as the subsequent example can be found in the article of Stanley [247,228,236,249,87]. Theorem (4.1.11)[289]: Let 푠휇 and 푠푣be two Schur functions represented by two partitions 휆 휇, 푣. Then the Littlewood Richardson coefficient 퐶휇,푣 of 푠휆 in the expansion of the product 푠휇푠푣 is zero unless 휆 ≥ 휇. In this case the coefficient is equal to the number of ways of inserting 푣1 1’푠, 푣2 2’푠, 푣3 3’푠, . .. into the skew tableau 휆/휇 subject to the conditions: (i) The numbers are weakly increasing in each row and strictly increasing in each column. (ii) If 훼1, 훼2, . .. is the set of numbers obtained when reading of the numbers inserted in λ/μ from right to left then for any i, j the numbers of 푖’푠 among 훼1, 훼2, . . . , 훼푗 is not less than the number(푖 + 1)’푠 among the numbers 훼1, 훼2, . . . , 훼푗. The following example given in [247]: Example(4.1.12)[289]: Let 휆 = (5, 4, 2, 2), 휇 = (3, 2, 1) and 푣 = (4, 2, 1). Then the following skew diagrams 휆/휇 are the only ones which satisfy (i). and (ii). In particular the coefficient of sλ in the expansion of the product 푠휇푠푣is equal to

1 1 1 1 1 1 1 2 2 2 1 2 1 1 2 2 3 1 3 1 3

∗ 푛 Using the Littlewood Richardson rule together with the description of the ring 퐻 (퐺푘(ℂ ), 푍) ∗ 푛 as given in (26) we are in a position to multiply arbitrary cocycles in 퐻 (퐺푘(ℂ ), 푍)The following example illustrates the procedure:

∗ 6 Example(4.1.13)[289]: Consider the elements {3, 2, 0} and {2, 1, 0} in 퐻 (퐺3(ℂ ), 푍)Then {3, 2, 0}{2, 1, 0} = {5, 3, 0} + {5, 2, 1} + {4, 4, 0} + 2{4, 3, 1} + {4, 2, 2} + {3, 3, 2} (28) We conclude with the Poincar´e duality theorem of cocycles. For this consider a cocycle ∗ 푛 {휇1, . . . , 휇푘}. The dual cocycle in 퐻 (퐺푘(ℂ ), 푍) is defined as the cocycle 휆: = {푛 – 푘 − 휇푘, . . . , 푛 − 푘 − 휇1}.Using this notation one has: Theorem (4.1.14)[289]: {휇1, . . . , 휇푘} {푣1, . . . , 푣푘} = {푛 − 푘, … , 푛 − 푘} Proof: Apply Theorem (1.3.12)of Littlewood and Richardson together with the description of ∗ 푛 퐻 (퐺푘(ℂ ), 푍) induced by the representation (26). In order to derive result we will use the following simple lemma,. Lemma (4.1.15)[289]: Suppose the eigenvalues of a Hermitian 푛 × 푛 matrix 퐴 are ordered as 휆1(퐴) ≥ . . . ≥ 휆푛(퐴). Then for any 1 ≤ 푖1 < . . . < 푖푘 ≤ 푛 one has: 푘 ( ) ( ) 휆푖1 −퐴 + ⋯ + 휆푖푘 −퐴 = − ∑ 휆푛−푖푗+1(퐴) (29) 푗=1 푛×푛 In the following we will consider Hermitian matrices 퐴1, . . . , 퐴푟+1 ∈ ℂ with corresponding eigenvalues ( ) ( ) 휆푖1 퐴푙 ≥ . . . ≥ 휆푛 퐴푙 , 푙 = 1, . . . , 푟 + 1 (30) and corresponding orthogonal sets of eigenvectors 푣1푙, . . . , 푣푛푙. Assume that 퐴푟+1 = 퐴1 + · · · + 퐴푟. (31) For each Hermitian operator 퐴푙, 푙 = 1, . . . , 푟 + 1 construct a flag of eigenspaces 푛 ℱ푙: {0} ⊂ 푉1푙 ⊂ 푉2푙 ⊂ . . . ⊂ 푉푛푙 = ℂ (32) defined through the property: 푉푚푙: = 푠푝푎푛(푣1푙, . . . , 푣푚푙), 푚 = 1, . . . , 푛. (33) The following result, which has been first proved by Thompson [225] for the case 푟 = 2, establishes the crucial relationship between matrix spectral inequalities and the Schubert calculus. Lemma (4.1.16)[289]: Let 퐴1, . . . , 퐴푟be complex Hermitian n × n matrices and denote with ℱ푙, . . . , ℱ푟+1 the corresponding flags of eigenspaces defined by (33). Assume 퐴푟+1 = 퐴1 + · · · + 퐴푟. and let 푖푙 = (푖1푙, . . . , 푖푘푙) be 푟 + 1 sequences of integers satisfying 1 ≤ 푖1푙 < . . . < 푖푘푙 ≤ 푛, 푙 = 1, . . . , 푟 + 1. (34) n Suppose the intersection of the 푟 + 1 Schubert subvarieties of Gk(ℂ ) is nonempty, i.e.:

푆 (푖푙; ℱ푙) ⋂ … ⋂ 푆(푖푟+1; ℱ푟+1) ≠ 0 (35) Then the following matrix eigenvalue inequalities hold: 푘 푟 푘

∑ 휆푛−푖푗,푟+1(퐴1 + ⋯ + 퐴푟) ≤ ∑ ∑ 휆푛−푖푗푙+1(퐴푙) (36) 푗=1 푙=1 푗=1 85

푘 푟 푘

∑ 휆푛−푖푗,푟+1+1(퐴1 + ⋯ + 퐴푟) ≥ ∑ ∑ 휆푖푗푙(퐴푙) (37) 푗=1 푙=1 푗=1

n Proof: Consider L ∈ Gk(C ) with 푟+1

퐿 ∈ ⋂ 푆 (푖푙; ℱ푙) ≠ 0. (38) 푙=1 Then, by using the Hersch-Zwahlen extremal principle (Theorem (4.1.4)) one has: 0 = 푡푟((퐴1 + · · · + 퐴푟 − 퐴푟+1)|퐿) (39) 푘

= ∑ 푡푟(퐴푙|퐿)– 푡푟(퐴푟+푙|퐿) (40) 푙=1 푘

≥ ∑ 푚푖푛{푡푟(퐴푙|퐿): 퐿 ∈ 푆(푖푟+1; ℱ푟+1)} + 푚푖푛{푡푟(−퐴푟+1|퐿): 퐿 ∈ 푆(푖푟+1; ℱ푟+1)} (41) 푗=1 푟 푘 푘 ( ) ( ) = ∑ ∑ 휆푖푗푙 퐴푙 + ∑ 휆푖푗,푟+1 −퐴푟+1 . (42) 푙=1 푗=1 푗=1 Thus by Lemma (4.1.15) one has: 푘 푟 푘

∑ 휆푛−푖푗,푟+1+1(퐴푟+) ≥ ∑ ∑ 휆푖푗푙(퐴푙) (43) 푗=1 푙=1 푗=1 which proves (36). The inequality (37) follows from (36) by replacing the matrices 퐴푙 by −퐴푙, 푙 = 1, . . . , 푟 + 1and using Lemma (4.1.15). This completes the proof. In general it will be difficult to verify the intersection property (35) as it assumes the knowledge of the eigenspaces of 퐴1, . . . , 퐴푟 and of 퐴푟+1 = 퐴1 + · · · +퐴푟. By combining Lemma (4.1.16) with the intersection theoretic result of Proposition (4.1.8) we obtain a result with a more easily verifiable hypothesis. Theorem (4.1.17)[289]:Let 푖푙 = (푖1푙, . . . , 푖푘푙) be 푟 + 1 sequences of integers satisfying 1 ≤ 푖1푙 < . . . < 푖푘푙 ≤ 푛, 푙 = 1, . . . , 푟 + 1. (44) ∗ 푛 Let {푛 − 푘 − 푖1푙 + 1, . . . , 푛 − 푖푘푙} ∈ 퐻 (퐺푘(ℂ ), 푍)denote the Schubert cocycle that is the Poincar´e dual of the fundamental homology class of the Schubert variety 푆(푖1; ℱ푙) for 푙 = ∗ 푛 1, . . . , 푟 + 1. If the (푟 + 1) −fold product of the Schubert cocycles in 퐻 (퐺푘(ℂ ), 푍) ) 푟+1

∏{푛 − 푘 − 푖1푙 + 1, . . . , 푛 − 푖푘푙} ≠ 0 (45) 푙=1 then the eigenvalue inequality (36) and (37) holds for any set of Hermitian matrices 1 푛×푛 퐴 , . . . , 퐴푟 ∈ ℂ .

Corollary (4.1.18)[289]:Let 푖: = (푖1, . . . , 푖푘), 푗: = (푗1, . . . , 푗푘), 푝: = (푝1, . . . , 푝푘),be sequences satisfying 1 ≤ 푖1 < . . . < 푖푘 ≤ 푛, 1 ≤ 푗1 < . . . < 푗푘 ≤ 푛 and1 ≤ 푝1 < . . . < 푝푘 ≤ 푛. If the triple product {푛 − 푘 − 푖1 + 1, . . . , 푛 − 푖푘}{푛 − 푘 − 푗1 + 1, . . . , 푛 − 푗푘} × {푛 − 푘 − 푝1 + 1, . . . , 푛 − 푝푘} ≠ 0 (46) is nonzero then for any pair of complex Hermitian matrices 퐴, 퐵 ∈ ℂ푛×푛 the following eigenvalue inequalities hold: 푘 푘 푘 ( ) ∑ 휆푛−푝푣+1(퐴 + 퐵) ≥ ∑ 휆푖푣 퐴 + ∑ 휆푗푣(퐵) (47) 푣=1 푣=1 푣=1 푘 푘 푘 ( ) ( ) ( ) ∑ 휆푝푣 퐴 + 퐵 ≤ ∑ 휆푛−푝푣+1 퐴 + ∑ 휆푛−푝푣+1 퐵 . (48) 푣=1 푣=1 푣=1 We conclude with a simple example. ∗ 푛 Example (4.1.19)[289]: In 퐻 (퐺푘(ℂ ), 푍) ) the following nonzero products exist: {1, 0}{1, 0}{2, 0} = {2, 2} (49) {1, 0}{1, 0}{1, 1} = {2, 2} (50) {1, 0}{1, 0}{1, 0}{1, 0} = 2{2, 2}. (51) By Theorem (4.1.16) and Corollary (4.1.17) the following eigenvalue inequalities hold for arbitrary 4 × 4 Hermitian matrices: 휆1(퐴 + 퐵) + 휆4(퐴 + 퐵)휆1(퐴) + 휆3(퐴) + 휆1(퐵) + 휆3(퐵) (52) 휆2(퐴 + 퐵) + 휆3(퐴 + 퐵) ≤ 휆1(퐴) + 휆3(퐴) + 휆1(퐵) + 휆3(퐵), (53) 휆2(퐴 + 퐵 + 퐶) + 휆4(퐴 + 퐵 + 퐶) ≤ 휆1(퐴) + 휆3(퐴) + 휆1(퐵) + 휆3(퐵) + 휆1(퐶) + 휆3(퐶). (54) We apply the preceding results to verify some classical eigenvalue inequalities. The first inequality is given in [108]. Weyl Inequality [108]: (4.1.20)[289]: For any indices 1 ≤ 푖, 푗 ≤ 푛 with1 ≤ 푖 + 푗 − 1 ≤ 푛 and any Hermitian matrices A, B ∈ ℂn×n one has: 휆푖+푗−1(퐴 + 퐵) ≤ 휆푖(퐴) + 휆푗(퐵). (55) 푛 푛−1 ∗ 푛−1 푛 Proof: Here 푘 = 1, 퐺1(ℂ ) = ℂℙ and 퐻 (ℂℙ ), Z) = 푍[푥]/(푥 ) is a truncated polynomial ring. Using this classical description of the cohomology ring of the projective space, the Schubert cocycles are {푖} = 푥푖, 푖 = 0, . . . , 푛 − 1. (56) Let 푖1, 푗1 and 푝1 defined by: 푖1: = 푛 − 푖 + 1, 푗1: = 푛 − 푗 + 1, 푝1 ≔ 푖 + 푗 − 1. (57) Then (46) reduces to {푖 − 1}{푗 − 1}{푛 − 푖 − 푗 + 1} = {푛 − 1}. (58) But since 푥푛−1 generates 퐻2(푛−1)(ℂℙ푛−1), 푍) ≡ 푍 one has {푛 − 1} ≠ 0. Thus the Weyl inequality follows immediately from Corollary (4.1.18). Lidskii Inequality: (4.1.21)[289]:

푛×푛 For 1 ≤ 훼1 < . . . < 훼푘 ≤ 푛and for any Hermitian matrices 퐴, 퐵 ∈ ℂ one has the matrix eigenvalue inequality: 푘 푘 푘

∑ 휆훼푗 (퐴 + 퐵) ≤ ∑ 휆훼푗 (퐴) + ∑ 휆푗 (퐵) (59) 푗=1 푗=1 푗=1 Proof:Consider 푖 ≔ (푛 − 훼푘 + 1, . . . , 푛 − 훼1 + 1), 푗: = (푛 − 푘 + 1, . . . , 푛), 푝 ≔ (훼1, . . . , 훼푘). Then the product in condition (46) of Corollary (4.1.18) is given by {훼푘 − 푘, . . . , 훼1 − 1}{0, . . . , 0}{푛 − 푘 − 훼1 + 1, . . . , 푛 − 훼푘}. (60) ∗ 푛 Since {0, . . . , 0} = 1 ∈ 퐻 (퐺푘(ℂ ), 푍)) and {n − k − α1 + 1, . . . , n − αk} is Poincarè dual to {훼푘 − 푘, . . . , 훼1 − 1} the above triple product is equal to {푛 − 푘, . . . , 푛 – 푘} and hence nonzero. This completes the proof of the Lidskii Inequality. Thus both the Weyl and the Lidskii inequality are direct consequences of the Poincar´e duality 푛−1 푛 of the projective space ℂℙ and of the Grassmannian 퐺푘(ℂ )respectively. A proof of the next inequality requires a more subtle topological argument. Freede-Thompson Inequality [225]: (4.1.22)[289]: For any 1 ≤ 훼1 < . . . < 훼푘 ≤ 푛, 1 ≤ 푏1 < . . . < 푏푘 ≤ 푛 with 훼푘 + 푏푘 − 푘 ≤ 푛 and Hermitian matrices 퐴, 퐵 ∈ ℂ푛×푛 one has: 푘 푘 푘

∑ 휆훼푣+푏푣−푣 (퐴 + 퐵) ≤ ∑ 휆훼푣 (퐴) + ∑ 휆푏푣 (퐵). (61) 푣=1 푣=1 푗=1 Proof: Consider 푖 ≔ (푛 − 훼푘 + 1, . . . , 푛 − 훼1 + 1), 푗: = (푛 − 푏푘 + 1, . . . , 푛 − 푏1 + 1), 푝 ∶= (훼1 + 푏1 − 1, . . . , 훼푘 + 푏푘 − 푘). Then the product in condition (46) of Corollary (4.1.18) is given by {훼푘 − 푘 − 훼1 + 1}{푏푘 − 푘 − 푏1 − 1}{푛 − 푘 − 훼1 + 2, . . . , 푛 + 푘 − 훼푘 − 푏푘}. (62) By assumption one has αk + bk − 2k ≤ n − k. From the Littlewood Richardson rule it follows that the product of the first two factors is of the form: 휆 {훼푘 − 푘 − 훼1 + 1}{푏푘 − 푘 − 푏1 − 1}{훼푘 − 푏푘 − 2푘, . . . , 훼1 + 푏1 − 2} + ∑ 푐휇푣 (63) 휆 λ where cμv are again the Littlewood Richardson coefficients and the sum is taken overall partitions 휆, 휆 ≠ 훼푘 − 푏푘 − 2푘, . . . , 훼1 + 푏1 − 2. Now the result follows from the observation that the cocycle 훼푘 − 푏푘 − 2푘, . . . , 훼1 + 푏1 − 2 is (compare with Theorem (4.1.14)) dual to the cocycle {푛 − 푘 − 훼1 − 푏1 + 2, . . . , 푛 + 푘 − 훼푘 − 푏푘}, i.e. the product (62) is nonzero and Theorem (4.1.18) applies. ∗ 푛 It is a consequence of Theorem (4.1.17) that any nonzero product in 퐻 (퐺푘(ℂ ), 푍)implies an eigenvalue inequality of the form (36) and an inequality of the form (37). We describe a large class of nonzero products. In particular we will describe all maximal nonzero products in ∗ 푛 ∗ 푛 퐻 (퐺푘(ℂ ), 푍) and we will describe all maximal nonzero products in 퐻 (퐺푘(ℂ ), 푍)consisting of 3 factors. The following lemmas prepare for those results. Lemma (4.1.23)[289]: Assume 휇: = {휇1, . . . , 휇푘} and 푣: = {푣1, . . . , 푣푘} are two cocycles in ∗ 푛 퐻 (퐺푘(ℂ ), 푍) which are complimentary in dimension, i.e. there weights satisfy |휇| + |푣| = 푘(푛 − 푘). Then μv ≠ 0 if, and only if μ and v are dual to each other, i.e. 푣 = {푛 − 푘 − 휇푘, . . . , 푛 − 푘 − 휇1}. 88

Proof: See also [203] for a different proof based on Poincar’e-duality. From the description of ∗ 푛 푘 퐻 (퐺푘(ℂ ), 푍) in (25) it is clear that 휇푣 ≠ 0 exactly when the coefficient of {(푛 − 푘) } = {푛 − 푘, . . . , 푛 − 푘} in the expansion μv is nonzero. Applying the rule of Littlewood and Richardson to the skew tableau (푛 − 푘)푘/휇 one verifies that there is only one possibility to fill this tableau with 푣1 1’푠, 푣2 2’푠, . . . , 푣푘 푘’푠, and in this case one necessarily has 푣1 = 푛 − 푘 − 휇푘, . . . , 푣푘 = 푛 − 푘 − 휇1.

푟 Lemma (4.1.24)[289]:Assume 휇푙 = {휇1푙, . . . , 휇푘푙}, 푙 = 1, . . . , 푟, are cocycles with∑푙=1 휇1푙 ≤ ∗ 푛 푛 − 푘. Then the following identity holds in 퐻 (퐺푘(ℂ ), 푍): 푟 푟 푟

{푛 − 푘 – ∑ 휇푘푙 , . . . , 푛 − 푘 − ∑{휇1푙} ∏{휇1푙, . . . , 휇푘푙} 푙=1 푙=1 푙=1 = {푛 − 푘, . . . , 푛 − 푘}. (64) Proof: Using inductively Littlewood Richardson’s rule it follows that 푟 푟 푟

∏{휇1푙, . . . , 휇푘푙} = {∑ 휇푘푙 , . . . , ∑ 휇푘푙} + ∑ 퐶휇 {휇1푙, . . . , 휇푘푙}. (65) 푙=1 푙=1 푙=1 휇 푟 푟 (Compare with (63)). Because {푛 − 푘 − ∑푙=1 휇푘푙 , . . . , 푛 − 푘 − ∑푙=1 휇1푙} is the Poincarè dual of the first term after the equality sign the result follows from the previous Lemma. ∗ 푛 In the next Lemma we will identify the Schubert symbol {푥1, 푥2} ∈ 퐻 (퐺푘(ℂ ), 푍)with zero for 푥1 > 푛 − 2. ∗ 푛 Lemma (4.1.25)[289]: If{훼1, 훼2}, {푏1, 푏2} are two cocycles in 퐻 (퐺푘(ℂ ), 푍 and 푚 ≔ 푚푖푛{(훼1 − 훼2), (푏1 − 푏2)} (66) then one has 푚

{훼1, 훼2}{푏1, 푏2} = ∑{훼1 + 푏1– 푖, 훼2 + 푏2 푖}. (67) 푖=0 Proof: Direct consequence of the Littlewood Richardson rule. (Compare with [107].) For the following Lemma let [푥] denote the largest integer smaller or equal to x. ∗ 푛 Lemma (4.1.26)[289]: If {α1l, α2l} ∈ 퐻 (퐺푘(ℂ ), 푍), l = 1, . . . , r, } are r Schubert cocycles with 훼11 − 훼21 ≥ · · · ≥ 훼1푟 − 훼2푟 (68) and 푚 푚 1 푚 ∶= 푚푖푛 {[ ∑(푎 – 푎 )] , ∑(푎 – 푎 )} (69) 2 1푙 2푙 1푙 2푙 푙=1 푙=2 then there are positive nonzero integers ci such that 푟 푚 푟 푚

∏{푎1푙, 푎2푙} = ∑ 푐푖 {∑ 푎1푙 − 푖, ∑ 푎2푙 + 푖}. (70) 푙=1 푖=0 푙=2 푖=0 푚 In particular if ∑푖=0 푐푖 푎1푙 ≤ 푚 + 푛 − 2 at least one summand is nonzero and therefore the whole product is nonzero. Proof: Let 훼 ∈ {2, . . . , 푟} be the largest integer with the property that 89

푚

(푎11 − 푎21) ≥ ∑(푎1푙 − 푎2푙). (71) 푙=0 푚 Denote with 푚̃: = ∑푙=2(푎1푙 − 푎2푙). Using inductively Lemma (4.1.25) one sees that 휎 푚̃ 훼 훼

∏{푎1푙, 푎2푙} = ∑ 푐푖̃ {∑ 푎1푙 − 푖, ∑ 푎2푙 + 푖}. (72) 푙=1 푖=0 푙=1 푙=0 with positive, nonzero constants 푐̃𝚤. In particular if 훼 = 푟 then 푚 = 푚̃ and the result is proven. 1 If 훼 < 푟 then (푎 − 푎 ) < ∑푚 (푎 − 푎 ) and therefore 푚 = [ ∑푚 (푎 − 푎 )]. 1푙 2푙 푙=2 1푙 2푙 2 푙=1 1푙 2푙 Multiplying inductively expression (71) with the factors {푎1푙 , 푎2푙}, 푙 = 훼 + 1, . . . , 푟 one deduces also in this case, using the fact that all Littlewood Richardson coefficients are positive, 휎 푚 푚 that ∏푙=1{푎1푙 − 푎2푙} = ∑푖=0 퐶푖 {∑푙=1 퐶{푥푖, 푦푖}, where 푟 푟 푟 푟

∑ 푎1푙 − 푚 ≤ 푥푖 ≤ ∑ 푎1푙 푎푛푑 ∑ 푎2푙 ≤ 푦푖 ∑ 푎2푙 + 푚. (73) 푖=0 푙=1 푙=1 푙=1 푟 In particular, if ∑푙=1 푎1푙 − 푚 ≤ 푛 − 2, the product is nonzero, which completes the proof. As a direct consequence of this Lemma we obtain a description of all maximal nonzero products ∗ 푛 in 퐻 (퐺푘(ℂ ), 푍) ∗ 푛 Theorem (4.1.27)[289]: Assume {푎1푙 , 푎2푙} ∈ 퐻 (퐺푘(ℂ ), 푍), 푙 = 1, . . . , 푟, are r cocycles with 푟

∑(푎1푙 , 푎2푙) = 2(푛 − 2) (74) 푙=1 r Then ∏l=1{a1l , a2l} ≠ 0 if, and only if

(푎1푙 , 푎2푙) ≤ ∑ (푎1푙 , 푎2푙), 푗 = 1, … , 푟. (75) 푙∈(1,…,푗−1,푗+1,…,푟 Proof: After a possible reindexing we can assume that 푎1푙 − 푎2푙 ≥ · · · ≥ 푎1푙 − 푎2푙. (76) 1 Because of assumption (74), m = [ ∑r (a − a )]. Because of the description of 2 l=1 1l 2l ∗ 푛 퐻 (퐺푘(ℂ ), 푍) in (25) it is clear that the product is nonzero if, and only if the coefficient of 2(푛−2) 푛 {푛 − 2, 푛 − 2} ∈ 퐻 (퐺푘(ℂ ), 푍) in the product expansion is nonzero. By the last Lemma 푟 this is the case iff ∑푙=1 푎1푙 ≤ 푚 + 푛 − 2. Moreover because of (73) the number 1 ∑푟 (푎 − 푎 )] is an integer. But then ∑푟 푎 ≤ 푚 + 푛 − 2. is equivalent to ∑푟 (푎 + 2 푙=1 1푙 2푙 푙=1 1푙 푙=1 1푙 푎2푙)] ≤ 2(푛 − 2)which is true by assumption (74). Combining Theorem (4.1.27) with Theorem (4.1.17) one finally has: Theorem (4.1.28)[289]: Let (푖1푙, 푖2푙) be 푟 + 1 pairs of integers with: 1 ≤ 푖1푙 < 푖2푙 ≤ 푛, 푙 = 1, . . . , 푟 + 1 (77) 푟+1

푟(2푛 − 1) + 3 ≤ ∑(푖1푙 + 푖2푙) (78) 푙=1

푖2푗 – 푖1푗 ≤ 1 − 푟 + ∑ (푖1푙 + 푖2푙) 푗 = 1, … , 푟 + 1. (79) 푙∈{1,…,푗−1,푗+1,…,푟+1 90

푛×푛 Then for any set of Hermitian matrices 퐴1, . . . , 퐴푟+1 ∈ ℂ satisfying the relation 퐴푟+1 = 퐴1 + · · · + 퐴푟 the following eigenvalue inequalities hold: 푟 ( ) ( ) 휆푛−푖1,푟+1+1 퐴푟+1 + 휆푛−푖2,푟+1+1 퐴푟+1 ≥ ∑(퐴푙) + 휆푖2푙(퐴푙)) (80) 푙=1 푟

휆푖1,푟+1(퐴푟+1) + 휆푖2,푟+1(퐴푟+1) ≤ ∑(휆푛−푖1푙+1(퐴푙) + 휆푛−푖2푙+1(퐴푙)) (81) 푙=1 Proof: Denote with 푎1푙 = 푛 − 푖1푙 − 1and푎2푙 = 푛 − 푖2푙. Then condition (78) is equivalent to 푟+1 the condition ∑푙=1 (푎1푙 + 푎2푙) ≤ 2(푛 − 2) and condition (79) is equivalent to inequality (75). 푟+1 The product ∏푙=1 {푛 − 푖1푙 − 1, 푛 − 푖2푙}is nonzero and the result follows once again from Theorem (4.1,17) . In order to illustrate the theorem in the case 푟 = 2, let 퐴 = 퐴1, 퐵 = 퐴2 and let (푖1,1, 푖2,1) = (푛 – 푎2 + 1, 푛 – 푎1 + 1), (82) (푖1,2, 푖2,2) = (푛 – 푏2 + 1, 푛 – 푏1 + 1), (83) (푖1,3, 푖2,3) = (푐1, 푐2). (84) Then we obtain Corollary (4.1.29)[289]: Let 1 ≤ 푎1 < 푎2 ≤ 푛, 1 ≤ 푏1 < 푏2 ≤ 푛 and 1 ≤ 푐1 < 푐2 ≤ 푛 satisfy the system of linear inequalities 푎1 + 푎2 + 푏1 + 푏2 ≤ 푐1 + 푐2 + 3 (85) 푎1 + 푎2 + 푏1 + 푏2 ≤ 푐1 + 푐2 − 1 (86) 푏2 − 푏1 ≤ 푎2 − 푎1 + 푐2 − 푐1 − 1 (87) 푐2 − 푐1 ≤ 푎2 − 푎1 + 푏2 − 푏1 − 1. (88) Then the eigenvalue inequality 휆푐1(퐴 + 퐵) + 휆푐2(퐴 + 퐵) ≤ 휆푎1(퐴) + 휆푎2(퐴) + 휆푏1(퐵) + 휆푏2(퐵) (89) holds for any pair of Hermitian 푛 × 푛 matrices A, B. We would like to remark that the assumptions in Corollary (4.1.29) imply the assumptions in Horn [10]. It is also possible to derive the inequality (89) by the methods developed in [10]. ∗ 푛 We describe all maximal nonzero products of 퐻 (퐺푘(ℂ ), 푍consisting of 3 factors. The results are based on a description of the Littlewood Richardson coefficients as given by Schlosser in [107]. In the following we explain his description and simultaneously adapt the notation for our purposes. Let 휇: = (휇1, . . . , 휇푘), 푣: = (푣1, . . . , 푣푘)and휆: = ( 휆1, . . ., 휆푘)be partitions. We are interested in 휆 conditions when the Littlewood Richardson coefficient 푐휇,푣 is nonzero.We will use the 휆 combinatorial description of 푐휇,푣 as given in Theorem (4.1.11) and the following parameterization by Schlosser [107]. Consider the tableau λ and denote with 푝ℎ푖the number of boxes in the skew tableau 휆/휇 with label i in the ℎ −th row. This gives us the following description forthe tableau λ:

Row (90)

1 휇1 푝11 휆1

2 휇2 푝21 푝22 휆2 ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ 푝 … 푝 K 휇푘 푝푘1 푘2 푘푘 휆푘

|휇| 푣1 푣푘2 … 푣푘 Total

Of course not all configurations of numbers 푝ℎ푖will result in a filling compatible with the rule of Littlewood and Richardson. On the other hand, as shown in [107], one can iteratively fill the skew tableau 휆 /휇, starting with pk1 and proceeding inductively with 푝ℎ푖, ℎ = 푘, . . . , 푖 + 1, 푖 = 1, . . . , 푘 − 1, subject to the following inequalities: 푀푎푥(ℎ, 푖; (푣) ≤ 푝ℎ푖 ≤ 푀푖푛(ℎ, 푖; (푣), (휇)) (91) Where 푘 푘

푀푎푥(ℎ, 푖; (푣)) = 푚푎푥{0, 푣푖 − 푣푖 − 1 − ∑ 푝푗푖 + ∑ 푝푗, 푖 + 1} 푘=ℎ+1 푘=ℎ−1 푖−1 푘 ( ) ( ) 푀푖푛(ℎ, 푖; 푣 , 휇 ) = 푚푖푛{휇ℎ−1 − 휇ℎ + ∑(푝ℎ−1,푗 − 푝ℎ,푗), 푣푖 − ∑ 푝푗푖} 푗=1 푗=ℎ+1 and 푘

푝푗푖 = 푣ℎ − ∑ 푝ℎ푖, 푖 = 1, … , 푘 (92) 푗=ℎ+1 we assume that v0 = 0, p0,j = 0, ph,0 = 0. (93) For our purposes, which is stated in similar form in [107], is: Theorem (4.1.30)[289]: Let μ, v be partitions and let phi be iteratively described through (91) and (92). Denote with ℎ

휆ℎ ≔ 휇ℎ + ∑ 푝ℎ푖, ℎ = 1, … , 푘. (94) 푖=1 휆 Then 휆: = (휆1, . . . , 휆푘)describes a tableau and the Littlewood Richardson coefficient 푐휇,푣 is nonzero

Corollary (4.1.31)[289]:Let 휇, 푣 be partitions and let 휆 satisfy the inequalities induced by the iterative scheme (91) and (92). Then {휇}{푣}{푛 − 푘– 휆푘, . . . , 푛 − 푘 – 휆1} ≠ 0 (95) Proof:The cocycle {푛 − 푘 − 휆푘, . . . , 푛 − 푘 − 1}is the Poincarè dual of the cocycle {휆} and λ because the Littlewood Richardson coefficient cμ,vis nonzero the results follows from Lemma (4.1.23). Corollary (4.1.32)[289]:Let 퐴, 퐵 be complex Hermitian 푛 × 푛 matrices. Let 휇, 푣 be partitions and let 휆 satisfy the inequalities induced by (90) and (91). Let 푎1 ∶= 휇푘 + 1, . . . , 푎푘: = 휇1 + 푘 (96) 푏1: = 푣푘 + 1, . . . , 푏푘: = 푣1 + 푘 (97) 푐1: = 휆푘 + 1, . . . , 푐푘: = 휆1 + 푘 (98) Then 푘 푘 푘

∑ 휆푐푣 (퐴 + 퐵) ≤ ∑ 휆푎푣 (퐴) + ∑ 휆푏푣 (퐵). (99) 푣=1 푣=1 푣=1 Section (4.2) Wielant’s Theorem with Spectral sets and Banach Algebra The classes of Hermitian and unitary matrices have a rich structure and much is known about the eigenvalues of these types of matrices. The more general class of normal (i.e. unitarily diagonalizable) complex matrices is less well understood. And not much is known about spectral problems involving normal matrices even with their eigenvalues being described in terms of those of their Hermitian and skew-Hermitian parts. The dierence between Hermitian and general normal matrices is that the latter can have as eigenvalues arbitrary complex numbers. ℂ, of course ℂ is not an ordered field. But it turns out that the simple fact that ℂ can be totally ordered as a vector space over the reals is enough to obtain useful information on spectra of normal matrices using Hermitian matrices as an inspiration. A total order in ℂ compatible with addition of complex numbers and multiplication by positive reals is the lexicographic order. It is characterized by its positive cone 퐻 = {푎 + 푖푏: 푎 > 0 표푟, 푖푓 푎 = 0, 푏 > 0}. Compatibility with addition means 퐻 + 퐻 ⊆ 퐻, and compatibility with multiplication by positive reals means휆퐻 ⊆ 퐻 for휆 > 0 . The order being total means 퐻 ∪ −퐻 = ℂ ∖ {0}. The lexicographic order is not Archimedian and, apart from rotations of the positive cone, is the only total order in ℂ compatible with the above mentioned operations. We shall use the lex lex iθ notation ≤ for it, and, for real θwe use ≤θ for the total order with positive cone e H. Let 퐴 be an 푛 × 푛 complex normal matrix. Let 훼1, … . , 훼푛be its eigenvalues ordered so that 푙푒푥 푙푒푥 훼1 ≥ … ≥ 훼푛and let 푣1, … , 푣푛 be corresponding orthonormal eigenvectors of A. For 푗 = ′ 1, … , 푛 denote by 퐸푗and 퐸푗 the subspaces of ℂn spanned by 푣1, … , 푣푗 and 푣푗, … , 푣푛respectively. Applying the argument used to obtain the corresponding result for Hermitian matrices, we get: Theorem (4.2.1)[143]: For 푗 − 1, … , 푛 we have ∗ ∗ 훼푗 = 푚푖푛 푥 퐴푥 = 푚푎푥 푥 퐴푥 ‖ ‖ ′ 푥∈퐸푗, 푥 =1 푥∈퐸푗,‖푥‖=1 In addition, we have 93

∗ ∗ 훼푗 = 푚푎푥 푚푖푛 푥 퐴푥 = 푚푖푛 푚푎푥 푥 퐴푥 푑푖푚퐻=푗 푥∈퐸푗,‖푥‖=1 푎푑푖푚퐻=푛−푗+1 푥∈퐸푗,‖푥‖=1 (Here max and min are used in the lexicographic sense). 푙푒푥 Analogous characterizations hold for any order of the type ≤휃 either using the same proof or applying the theorem to the normal matrix e−iθA. Note how these results make immediately visible the fact that the numerical range 푊(퐴) = {푥∗퐴푥 ∶ ‖푥‖ = 1} of a normal matrix 퐴 is the convex hull of its eigenvalues any straight line moving in the plane parallel to itself must touch 푊(퐴) first at an eigenvalue of 퐴. From the above theorem we immediately obtain, again repeating the Hermitian argument, a result concerning principal normal submatrices of normal matrices: 푙푒푥 푙푒푥 Theorem (4.2.2)[143]:Let 퐴 be an 푛 × 푛 normal matrix with eigenvalues 훼1 ≥ … ≥ 훼푛 푙푒푥 푙푒푥 If B is a principal 푘 × 푘 normal submatrix of 퐴 with eigenvalues 훽1 ≥ … ≥ 훽푘, we have 푙푒푥 푙푒푥 훼푗 ≥ 훽푗 ≥ 훼푗 + 푛 − 푘, 푗 = 1, … , 푘. lex An analogous result holds for any order of the type ≤θ . For other interlacing results in this setting see [154,160,126,135], [50]. The result in [154] shows that for a n × n normal matrix to have a principal (푛 − 1) × (푛 − 1) normal principal submatrix is a highly restrictive condition, essentially forcing the matrix apart from a rotation and a translation, to be Hermitian. It seems plausible that one can obtain this from Theorem (4.2.2) above. In [50,139] an interlacing result is presented for the arguments of eigenvalues of a normal matrix and a normal principal submatrix a relation with Theorem (4.2.2) above is unclear. and then there is the general interlacing theorem for singular values which for normal matrix and submatrix yields a statement whose relation with the above result is again unclear. Note also that Theorem (4.2.2). does not follow directly from the interlacing theorem for Hermitian matrices applied to the Hermitian and skew-Hermitian parts of 퐴 and 퐵. A generalization of the first part of Theorem (4.2.1) can be obtained by mimicking the corresponding result for Hermitian matrices [139]. 푛 Take a sequence 푉 = (푉1, … , 푉푛) of subspaces of ℂ with 푉1 ⊂ ⋯ ⊂ 푉푛 and 푑푖푚(푉푖) = 푖 for 푖 = 1, … , 푛_Given a sequence 퐼 = (푖1, … , 푖푟), with i ≤ i1 < ⋯ < ir ≤ n, the Schubert variety 푛 associated to V and I is ΩI(V) = {퐿subspace of ℂ 푑푖푚(퐿) = 푟, 푑푖푚(퐿 ∩ 푉푖푑 ≥ 푑, 푑 = 1, … , 푟} . Keep the notation and write 퐸 = (퐸1, … , 퐸푛), ′ ′ ′ 퐸′ = (퐸푛, … , 퐸1). Put also 퐼 = (푛 − 푖푟 + 1, … , 푛 − 푖1 + 1) If 퐿 is a subspace of dimension 푟 and 푥1, … , 푥푟 is an orthonormal basis of L, the Rayleigh trace of Awith respect to 퐿 is 푟 ∗ 푡푟(퐴|퐿) = ∑ 푑푑퐴푥푑. 푑=1 (This does not depend on the basis) 푙푒푥 푙푒푥 Theorem (4.2.3)[143]:If the eigenvalues of a normal matrix 퐴 are 훼1 ≥ … ≥ 훼푛 one has ( | ) ( | ) 훼푖1 + ⋯ + 훼푖푟 = 푚푖푛 푡푟 퐴 퐿 = 푚푎푥 푡푟 퐴 퐿 퐿∈훺1(퐸) 퐿∈훺1′(퐸′) where again max and min are used in the lexicographic sense. 94

푙푒푥 This characterization of course also valid for any order of the type ≤휃 can be applied to obtaining inequalities for the eigenvalues of a sum of two normal matrices if this sum is itself normal. 푙푒푥 푙푒푥 Let 퐴 and 퐵 be 푛 × 푛 normal matrices with eigenvalues 훼1 ≥ … ≥ 훼푛and 푙푒푥 푙푒푥 훽1 ≥ … ≥ 훽푘, respectively. Suppose that 퐴 + 퐵 is normal, with eigenvalues 푙푒푥 푙푒푥 훾1 ≥ … ≥ 훾푛. Let 퐸, 퐸′, 퐹, 퐹′ and 퐺, 퐺′′ be sequences of subspaces built from the eigenvectors of 퐴, 퐵 and 퐴 + 퐵,as before. Let 퐼, 퐽and 퐾 be sequences of 푟 indices: 퐼 = (푖1, … , 푖푟), 1 ≤ 푖1 < ⋯ < 푖푟 ≤ 푛, 퐽 = (푗1, … , 푗푟), 1 ≤ 푗1 < ⋯ < 푗푟 ≤ 푛, 퐾 = (푘1, … , 푘푟), 1 ≤ 푘1 < ⋯ < 푘푟 ≤ 푛. Then, using the characterizations of Theorem (4.2.3), it is easy to see that Theorem (4.2.4) [143]:If ′ ′ 훺퐾(퐺) ∩ 훺퐼′(퐸 ) ∩ 훺퐽′(퐹 ) ≠ 0, Then 푙푒푥 훾푘1 + ⋯ + 훾푘푟 ≤ 훼푖1 + ⋯ + 훼푖푟 + 훽푗1 + ⋯ 훽푗1푟. For the Hermitian case this appears in [263], [289] So a geometric condition, nonempty intersection of the three Schubert Varieties, implies a linear inequality between the eigenvalues of the three normal matrices 퐴, 퐵 and 퐴 + 퐵. We abreviate this inequality to 푙푒푥 ∑ 훾푘 ≤ 훼퐼 + ∑ 훽퐽. For the Hermitian case,by Klyachko, has shown that the inequalities arising from all such geometric conditions actually yield a complete list of restrictions for the eigenvalues of a sum of two Hermitian matrices in terms of the eigenvalues of the summands, For recent surveys on this see [306], [13]. Klyachko’s results, coupled with the combinatorial work of Knutson and Tao [11] imply the classical Horn conjecture, on eigenvalues of Hermitian Matrices, which we now recall. For two real ordered spectra 훼 and 훽 denote by 퐸(훼, 훽) the set of all possible ordered spectra of sums of two Hermitian matrices with spectra 훼 and 훽. For each 푟-tuple 퐼 = (푖1 … 푖푟) with 1 ≤ 푖1 < ⋯ < 푖푟 ≤ 푛 define 휌(퐼) = (푖푟 − 푟, … , 푖2 − 2, 푖1 − 1). Then Horn’s conjecture, now proved, can be presented as the following recursive description of the set E:

퐸(훼, 훽) = {훾: ∑ 훾 = ∑ 훼 + ∑ 훽 and ∑ 훾푘 = ∑ 훼퐼 + ∑ 훽퐽 wherever 휌(퐾) ∈ 퐸[휌(퐼), 휌(퐽)], 1 ≤ 푟 < 푛}. ′ By the Schubert calculus (see for example [287]), the geometric condition 훺퐾(퐺) ∩ 훺퐼′(퐸 ) ∩ ′ 훺퐽′(퐹 ) ≠ 0 is equivalent to 휌(퐾) ∈ 퐿푅[휌(퐼), 휌(퐽)], meaning that the 푟-tuple ρ(K) can be obtained from ρ(I) and ρ(J)using the combinatorial Littlewood-Richardson rule. From the results in [1] and [11] it turns out that it is also equivalent to 휌(퐾) ∈ 퐸[휌(퐼), 휌(퐽)].

Return now to normal matrices 퐴 with spectrum 훼, 훽 with spectrum β and 퐴 + 퐵 with spectrum ′ ′ γ with notations as above. As we have seen, the condition ΩK(G) ∩ ΩI′(E ) ∩ ΩJ′(F ) ≠ 0 implies ρ(K) ∈ LR[ρ(I), ρ(J)]. Therefore, bearing in mind the results quoted, we can now state: Theorem (4.2.5)[143]:For 1 ≤ 푟 < 푛, whenever one has 휌(퐾) ∈ 퐸[휌(퐼), 휌(퐽)]the inequality 푙푒푥 ∑ 훾퐾 ≤ ∑ 훼퐼 + ∑ 훽퐽 holds for the eigenvalues of the normal matrices 퐴, 퐵 and 퐴 + 퐵. And the same, of course, for any order of the type≤푙푒푥. In [113], Helmut Wielandt proved an interesting result which gave regions in the complex plane which contain all the eigenvalues of the sum of two normal matrices 퐴 and 퐵 in terms of the spectra of 퐴 and 퐵. We give a generalization of Wielandt’s result to Banach algebras and we also give a multiplicative version of Wielandt’s theorem. Before statingWielandt’s theorem, we need to review some geometric concepts in elementary complex function theory. Definition (4.2.6)[244]: A generalized circle is either a circle ({푧 ∈ ℂ ∶ |푧 − 휅| = 푟} where 휅 ∈ ℂ and 푟 > 0) or a straight line ({푧 ∈ ℂ: 푅푒(훼푧) = 훽} where 훼 ∈ ℂ\{0} and훽 ≥ 0)in the complex plane. Definition (4.2.7) [244]: A Mobius transformation is a function of the form 푓 (푧) = 푎푧+푏 where 푎, 푏, 푐, 푑 ∈ ℂ and 푎푑 − 푏푐 ≠ 0. 푐푧+푑 We note that a Mobius transformation maps generalized circles to generalized circles. Definition (4.2.8) [244]: A circular region is a subset of the complex plane of the form푓 (퐾) where 푓 is a Möbius transform and 퐾 is either the open unit disk {푧 ∈ ℂ: |푧| < 1} or the closed unit disk {푧 ∈ ℂ: |푧| ≤ 1}. A subset of a complex plane is a circular region if it is either an open or closed disk, a complement of an open or closed disk or a half-plane. The boundary of a circular region is always a generalized circle. We can now state Wielandt’s theorem. We let 휎(푀) denote the spectrum of the matrix 푀. If 푆 and 푇 are non-empty subsets of the complex plane, then 푆 + 푇 = {푠 + 푡: 푠 ∈ 푆, 푡 ∈ 푇} and 푆 · 푇 = {푠푡: 푠 ∈ 푆, 푡 ∈ 푇}. Theorem (4.2.9)[244]: [113]. Let 퐴 and 퐵 be two n by n normal matrices. Let 퐾 be a circular region which contains all of the eigenvalues of 퐵, then 휎(퐴 + 퐵) ⊆ 휎(퐴) + 퐾. We will give a generalization of this result to Banach algebras. We will assume no knowledge of normed algebras and Banach algebras beyond their definitions. (See [311]. All of the normed algebras and Banach algebras in this section will be automatically assumed to be complex and unital. We denote the unit of a unital normed algebra as 1. If a is an element in a unital normed algebra 𝒜, then the spectrum of a is the set {λ ∈ ℂ: (λ1 − a) is not invertible}. The spectrum of a Banach algebra is always non-empty and compact. We note that 퐵(ℋ), the set of all bounded linear operators on a Hilbert space is an example of a Banach algebra; if further ℋ is finite dimensional then 퐵(ℋ)is the algebra of 푛 by 푛 complex matrices where 푛 = 푑푖푚(ℋ). We have the following.

Proposition (4.2.10)[244]:Let 𝒜 be a Banach algebra then 1 + 푥 is invertible whenever 푥 ∈ 𝒜 with ‖x ‖ < 1. While this result may fail if𝒜 is an incomplete normed algebra, there are incomplete normed algebras for which the above result also holds as noted in [242]. All of our results below hold if we replace Banach algebra with complex unital normed algebra for which the conclusion of Proposition (4.2.10) holds. Since the proof that every element of a complex Banach algebra has compact spectrum uses Proposition (4.2.10) rather than using completeness directly, if A is a complex unital normed algebra for which the conclusion of Proposition (4.2.10) holds all its elements will have non-empty compact spectrum. The term spectral set has several different meanings in mathematics. We will always use the term in the following sense: Definition (4.2.11)[244]: Let 𝒜 be a complex unital Banach algebra and let 푎 ∈ 𝒜. A closed subset S of the complex plane which contains the spectrum of a is called a spectral set of a if ‖푟(푎)‖ ≤ 푠푢푝푧∈푆|푟(푧)|for all rational functions 푟 which have no poles in 푆. This concept is due to von Neumann [138] who gave the definition in the special case where the Banach algebra is 퐵(ℋ),. Many different subsets of the complex plane may be the spectral set for the same element 푎 ∈ 𝒜. In general, the intersection of two spectral sets is not necessarily a spectral setand hence there is no minimal spectral set of a unless the spectrum of a is itself a spectral set of 푎. Any set which contains a spectral set must itself be a spectral set. We note that the rational functions form amonoid under composition; the invertible elements of thismonoid are the Mobius transformations. This observation leads to the following proposition: Proposition (4.2.12)[244]: Let 𝒜 be a Banach algebra and let 푎 ∈ 𝒜. Let 푚(푧) be a Mobius transformation whose pole lies outside the spectrum of a. If 푆 is a spectral set for a, then 푚(푆) is a spectral set for 푚(푎). Spectral sets play an especially important role in the special case where the Banach algebra 𝒜 = 퐵(ℋ) where ℋ is a Hilbert space. We note that in his original work [138], von Neumann gave necessary and sufficient conditions for a closed circular region in the complex plane to be a spectral set of a bounded linear operator. Proposition (4.2.13)[244]: [138]. Let ℋ be a complex Hilbert space and let 퐴 ∈ 퐵(ℋ). Let κ ∈ ℂ and 푟 > 0, then the circular region {푧 ∈ ℂ: |푧 − 휅| ≤ 푟} is a spectral set for 퐴 if and only if ‖퐴 − 휅퐼‖ ≤ 푟. Proposition (4.2.14)[244]:[138] Let ℋbe a complex Hilbert space and let 퐴 ∈ 퐵(ℋ). Let 휅 ∈ ℂand푟 > 0, then the circular region {푧 ∈ ℂ: |푧 − 휅| ≥ 푟} is a spectral set for 퐴 if and only if ‖(퐴 − 휅퐼)−1‖ ≤ 푟 − 1. We note that the forward implications of the two previous propositions follow immediately from the definition of a spectral set and apply to general Banach algebras. The backwards implications of the two previous propositions are false for general Banach algebras. Proposition (4.2.15)[244]: [138]. Let ℋ be a complex Hilbert space and let 퐴 ∈ 퐵(ℋ). Let 훼 ∈ ℂ\{0} and 훽 ≥ 0, then the circular region {푧 ∈ ℂ ∶ 푅푒(훼푧) ≥ 훽} is a spectral set for A if 퐴+퐴∗ and only if α( ) ≥ 훽퐼. 2

Since a spectral set of an operator must contain its spectrum; the smallest possible candidate to be a spectral set of an operator is the spectrum. The class of operators whose spectrum is itself a spectral set are important enough to be named (in honour of von Neumann fittingly enough). Definition (4.2.16)[244]: Let ℋ be a Hilbert space and let 퐴 ∈ 퐵(ℋ).퐴 is said to be a von Neumann operator if the spectrum of 퐴 is a spectral set of 퐴. The class of von Neumann operators are an important class of operators; important subsets of von Neumann operators include the normal operators and the subnormal operators [140]. If ℋ is finite dimensional, then 퐴 ∈ 퐵(ℋ) is a von Neumann operator if and only if it is normal. We now extend the definition of a von Neumann operator to Banach algebras in the obvious way. Definition (4.2.17)[244]: Let 𝒜 be a unital Banach algebra and let 푎 ∈ 𝒜. Then a is called a von Neumann element if the spectrum of 푎 is itself a spectral set of 푎. Theorem (4.2.18)[244]: Let 𝒜 be a unital Banach algebra and let a, b ∈ 𝒜. Let 푆푎 and Sb be spectral sets of 푎 and 푏 respectively. If 푆푎 and 푆푏 are separated by a generalized circle then 푎 − 푏 is an invertible element of 퐴. Proof: Let 퐺 be a generalized circle which separates 푆푎 and Sb. If G is a circle, then 퐺 = {푧 ∈ ℂ: |푧 − 휅| = 푟} for some 휅 ∈ ℂ and 푟 > 0. One of Sa and Sb is contained in the set {z ∈ ℂ: |z − κ| < 푟} and the other is contained in the set {푧 ∈ ℂ: |푧 − 휅| > 푟}. WLOG let 푆푎 ⊆ {푧 ∈ ℂ: |푧 − 휅| > 푟} and 푆푏 ⊆ {푧 ∈ ℂ: |푧 − 휅| < 푟}. From this it follows that 푎 − 휅1 is invertible, ‖(푎 − 휅1)−1‖ < 푟−1 and ‖푏 − 휅1‖ < 푟.Therefore ‖(푎 − 휅1)−1(푏 − 휅1)‖ < 1 and 푎 − 푏 = (푎 − 휅1) − (푏 − 휅1) = (푎 − 휅1)[1 − (푎 − 휅1)−1(푏 − 휅1)] is invertible. Now suppose 퐺 is a line.Now choose 휏 ∈ ℂ such that 휏 ∉ 퐺 ∪ 휎푎 ∪ 휎푏. Then 푎 − 휏 1 and 푏 − 휏 1 are two invertible elements of the Banach algebra having spectral sets 푆푎 − 휏and 푆푏– 휏 respectively which are separated by a line 퐺 − 휏 which does not contain the origin. Hence −1 −1 −1 (푎 − 휏퐼) and (푏 − 휏퐼) have spectral sets (Sa − τ) and (Sb − τ) − 1 which are separated by a circle(퐺 − 휏)−1. Hence (푎 − 휏 1)−1 − (푏 − 휏 1)−1 is invertible which means 푎 − 푏 = (푎 − 휏 1) − (푏 − 휏 1) = (푎 − 휏 1)[(푏 − 휏 1)−1 − (푎 − 휏 1)−1](푏 − 휏 1) is invertible. Corollary (4.2.19)[244]:. Let 𝒜 be a unital Banach algebra and let 푎, 푏 ∈ 𝒜. Let 푆푎 nd 푆푏 be spectral sets of 푎 and 푏 respectively and let 퐾 any circular region which contains Sb. Then 휎(푎 + 푏) ⊆ 푆푎 + 퐾. Proof: Suppose 휆 ∈ 휎(푎 + 푏), then 푏 − (휆1 − 푎) = 푎 + 푏 − 휆1 is not invertible. The intersection of 휆 − 푆푎and K must be non-empty by Theorem (4.2.18), since the boundary of 퐾 is a generalized circle. Now let 휇 ∈ (휆 − 푆푎) ∩ 퐾. We note that since 휆 − 휇 ∈ 푆푎, 휇 ∈ 퐾and 휆 = (휆 − 휇) + 휇, our result follows. Our generalization of Wielandt’s theorem now follows immediately. Theorem (4.2.20)[244]:. Let 𝒜 be a unital Banach algebra and let 푎, 푏 be von Neumann elements in 𝒜. Let K be a circular region containing σ(b). Then 휎(푎 + 푏) ⊆ 휎(푎) + 퐾. We also have a multiplicative version of Corollary (4.2.19).

Theorem (4.2.21)[244]: Let 𝒜 be a unital Banach algebra and let 푎, 푏 ∈ 𝒜 with at least one of 푎 or 푏 being invertible. Let 푆푎 and Sb be spectral sets of a and b respectively and let 퐾 be any circular region which contains Sb. Then 휎(푎푏) ⊆ 푆푎 · 퐾. Proof: If a invertible then 휆1 − 푎푏 = 푎(휆푎−1 − 푏). Suppose λ cannot be expressed as a −1 product of a number in 푆푎 and a number in 퐾. Then {휆/푧 ∶ 푧 ∈ 푆푎)}is a spectral set for 휆푎 which lies entirelyoutside K. Then by Theorem (4.3.13), 휆푎−1 − 푏 and hence 휆1 − 푎푏 is invertible. The proof where 푏 is invertible is similar. The special case where 푎 and 푏 are von Neumann elements of the Banach algebra is a multiplicative version of Wielandt’s theorem. Theorem (4.2.22)[244]: Let 𝒜 be a unital Banach algebra and let 푎 and 푏 be von Neumann elements of 𝒜with at least one of 푎 or 푏 being invertible. Let 퐾 be any circular region which contains the spectrum of 푏. Then 휎(푎푏) ⊆ 휎(푎) · 퐾.

Chapter 5 Riesz and Szeg퐨̈ Type Factorizations with Helson -Szeg퐨̈ Theorem We show the contractivity of the underlying conditional expectation on 퐻푝(퐴) for 푝 < 1.We introduce noncommutative Hardy-Lorentz spaces and give the Szegöand inner – outer type factorizations of these spaces.We then proceed to use Helson Szegö theorem to haracterise the symbols of invertible Toeplitz operators on the noncommutative Hardy spaces associated to subdiagonal subalgebras. Section (5.1) Factorizations for Noncommutative Hardy Spaces We deal with the Riesz and Szegö type factorizations for noncommutative Hardy spaces associated with a finite subdiagonal algebra in Arveson’s sense [300]. Let M be a finite von Neumann algebra equipped with a normal faithful tracial state τ. Let 퐷 be a von Neumann subalgebra of M, and let 훷: 푀 → 퐷 be the unique normal faithful conditional expectation such that 휏 ∘ 훷 = 휏. 퐴 finite subdiagonal algebra of M with respect to 훷 (or 퐷) is a 휔*-closed subalgebra 퐴 of 푀 satisfying the following conditions (i) 퐴 + 퐴∗is 휔∗-dense in 푀; (ii) 훷 is multiplicative on A, i.e., 훷(푎푏) = 훷(푎)훷(푏) for all a, 푏 ∈ 퐴; (iii) 퐴 ∩ 퐴∗ = 퐷. We should call to fact that 퐴∗denotes the family of the adjoints of the elements of 퐴, i.e., 퐴∗ = {푎∗: 푎 ∈ 퐴}. The algebra 퐷 is called the diagonal of 퐴. It is proved by Exel [240] that a finite subdiagonal algebra 퐴 is automatically maximal in the sense that if 퐵 is another subdiagonal algebra with respect to 훷 containing 퐴, then 퐵 = 퐴. This maximality yields the following useful characterization of 퐴: 퐴 = {푥 ∈ 푀: 휏(푥푎) = 0, ∀ 푎 ∈ 퐴0}, (1) where 퐴0 = 퐴 ∩ 푘푒푟훷 (see [300]). Given 0 < 푝 ≤ ∞ we denote by 퐿푝(푀) the usual noncommutative 퐿푝-space associated with (푀, 휏). Recall that 퐿∞(푀) = 푀, equipped with the operator norm. The norm of퐿푝(푀) will 푝 푝 be denoted by ‖・‖푝. For 푝 < ∞ we define 퐻 (퐴) to be the closure of 퐴 in 퐿 (푀), and for 푝 = ∞ we simply set 퐻∞(퐴) = 퐴 for convenience. These are the so-called Hardy spaces associated with 퐴. They are noncommutative extensions of the classical Hardy spaces on the torus 푇. On the other hand, the theory of matrix-valued analytic functions provides an important noncommutative example. We see [300] and [89]for more examples. We will use the following 푝 standard notation in the theory: If 푆 is a subset of 퐿 (푀), [푆]푝 will denote the closure of 푆 in 푝 ∗ 푝 퐿 (푀) (with respect to the 휔 -topology in the case of 푝 = ∞). Thus 퐻 (퐴) = [퐴]푝. Formula (1) admits the following 퐻푝(퐴) analogue proved by Saito [153]: 푝 푝 퐻 (퐴) = {푥 ∈ 퐿 (푀): 휏(푥푎) = 0, ∀ 푎 ∈ 퐴0}, 1 ≤ 푝 < ∞. (2) Moreover, 퐻푝(퐴) ∩ 퐿푝(푀) = 퐻푞(퐴), 1 ≤ 푝 < 푞 ≤ ∞. (3) These noncommutative Hardy spaces have received a lot of attention since Arveson’s pioneer work. We refer Marsalli- West [176] and Blecher-Labuschagne [66, 58, 56], whereas more references on previous works can be found in the survey [89]. Most results on the classical Hardy spaces on the torus have been established in this noncommutative setting. Here we 100 mention only two of them directly related with the objective. The first one is the Szegö factorization theorem. Already in the fundamental work [300], Arveson proved the following factorization theorem: For any invertible 푥 ∈ 푀 there exist a unitary 푢 ∈ 푀 and 푎 ∈ 퐴 such that 푥 = 푢푎 and 푎−1 ∈ 퐴. This theorem is a base of all subsequent works on noncommutative Hardy spaces. It has been largely improved and extended. The most general form up to date was newly obtained by Blecher and Labuschagne [66]: Given 푥 ∈ 퐿푝(푀) with 1 ≤ 푝 ≤ ∞ such that 훥(푥) > 0 there exists ℎ ∈ 퐻푝(푀) such that |푥| = |ℎ|. Moreover, h is outer in the sense 푝 that [ℎ퐴]푝 = 퐻 (푀). Here 훥(푥) denotes the Fuglede-Kadison determinant of 푥 and |푥| = (푥∗푥)1/2 denotes the absolute value of x. We should emphasize that this result is the (almost) perfect analogue of the classical Szeg표̈ theorem which asserts that given a positive measurable function w on the torus there exists an outer function ' such that 푤 = |휑| iff logw is integrable. The second result we wish to mention concerns the Riesz factorization, which asserts that 퐻푝(퐴) = 퐻푞(퐴). 퐻푟(퐴) for any 1 ≤ 푝, 푞, 푟 ≤ ∞ such that 1/푝 = 1/푞 + 1/푟. More precisely, given 푥 ∈ 퐻푝(퐴) and 휀 > 0there exist 푦 ∈ 퐻푞(퐴) and 푧 ∈ 퐻푟(퐴) such that 푥 = 푦푧 and ‖푦‖푞‖푧‖푟 ≤ ‖푥‖푝 + 휀. This result is proved in [153] for 푝 = 푞 =2, in [176] for 푟 = 1and independently in [164] and in [89] for the general case as above. Recall that in the case of the classical Hardy spaces the preceding theorems hold for all positive indices. The problem of extending these results to the case of indices less than one was left unsolved in these works. (We mentioned this problem for the Riesz factorization explicitly in [89]). The main purpose is to solve the problem above. As a byproduct, we also extend all results on outer operators in [66] to indices less than one. A major obstacle to the solution of the previous problem is the use of duality, often in a crucial way, on noncommutative Hardy spaces. For instance, duality plays an important role in proving formulas (2) and (3), which are key ingredients for the Riesz factorization in [89]. In a similar fashion, we will see that their extensions to indices less than one will be essential for our proof of the Riesz factorization for all positive indices. Our key new tool is the contractivity of the conditional expectation 훷 on 퐴 with respect 푝 to ‖. ‖푝for 0 < 푝 < 1. Consequently, Φ extends to a contractive projection from 퐻 (퐴) onto 퐿푝(퐷). This result is of independent interest and proved . We devoted to the Szegö and Riesz type factorizations. In particular, we extend to all positive indices Marsalli-West’s theorem quoted previously. It contains some results on outer operators, notably those in 퐻푝(퐴) for 푝 < 1. see [66]. We devoted to a noncommutative Szegö formula, which wasobtained in [66] with the additional assumption that 푑푖푚퐷 < ∞. In particular, A will always denote a finite subdiagonal algebra of (M, τ) with diagonal 퐷. It is well-known that extends to a contractive projection from 퐿푝(푀) onto 퐿푝(퐷) for every 1 ≤ 푝 ≤ ∞. In general, 훷 cannot be, of course, continuously extended to 퐻푝(퐴) for 푝 < 1. Surprisingly, 훷 denotes extend to a contractive projection on 퐻푝(퐴). Theorem (5.1.1)[279]: Let 0 < 푝 < 1. Then ∀ 푎 ∈ 퐴 , ‖훷(푎)‖푝 ≤ ‖푎‖푝 . (4) Consequently, 훷 extends to a contractive projection from 퐻푝(퐴). onto 퐿푝(퐷).The extension will be denoted still by훷. 101

Inequality (4) is proved by Labuschagne [163] for 푝 = 2−푛 and for operators a in 퐴 which are invertible with inverses in 퐴 too. Labuschagne’s proof is a very elegant and simple argument by induction. It can be adapted to our general situation. Proof: Since {푘2−푛: 푘, 푛 ∈ 푁, 푘 ≥ 1} is dense in (0, 1), it suffices to prove (4) for 푝 = 푘2−푛. Thus we must show −푛 −푛 휏(|훷(푎)|푘2 ) ≤ 휏(|푎|푘2 ) , ∀ 푎 ∈ 퐴 . (5) This inequality holds for 푛 = 0 because of the contractivity of Φ on 퐿푘(푀). Now suppose its validity for some 푘 and 푛. We will prove the same inequality with 푛 + 1 instead of 푛. To this end fix 푎 ∈ 퐴 and 휀 > 0. Define, by induction, a sequence (푥푚) by −푛 1 −푛 푥 = (|푎| + 휀)푘2 and 푥 = [푥 + (|푎| + 휀)푘2 푥−1]. 1 푚+1 2 푚 푚 * Observe that all 푥푚 belong to the commutative C -subalgebra generated by |푎|. Then it is an 푘2−푛−1 easy exercise to show that the sequence (xm) is nonincreasing and converges to (|푎| + 휀) uniformly (see [158]). We also have 1 −푛 1 −푛 휏(푥 ) = [휏(푥 ) + 휏(푥−1/2 (|푎| + 휀)푘2 푥−1/2)] ≥ [휏(푥 ) + 휏(푥−1/2 |푎|푘2 푥−1/2)] 푚+1 2 푚 푚 푚 2 푚 푚 푚 1 −푛 = [휏(푥 ) + 휏( |푎|푘2 푥−1)]. 2 푚 푚 Now applying Arveson’s factorization theorem to each 푥푚, we find an invertible 푏푚 ∈ 퐴 with −1 푏푚 ∈ 퐴 such that 2푛/푘 |푏푚| = 푥푚 . Let p = 푘2−푛. Then 1 −1 −1 −1 ∗ −1 푝 −푝 푝 ‖푎푏푚 ‖푝 = ‖|푎| 푏푚 ‖푝 = ‖|푎| |(푏푚 ) |‖푝 = ‖|푎||푏푚| ‖푝 = (휏(|푎| |푏푚| )) 푝 −1 1/푝 = (휏(|푎| 푥푚 )) where we have used the commutation between |푎| and |푏푚| for the next to the last equality. Therefore, by the induction hypothesis and the multiplicativity of 훷 on 퐴 1 −푛 −푛 휏(푥 ) ≥ [휏(|푏 |푘2 ) + 휏(|푎푏−1 |푘2 )] 푚+1 2 푚 푚 1 −푛 −푛 ≥ [휏(훷|푏 |푘2 ) + 휏(훷|(푎)훷(푏 )−1 |푘2 )]. 2 푚 푚 However, by the Hölder inequality 2 푘2−푛−1 −1 푘2−푛 푘2−푛 (휏(훷|(푎) | )) ≤ 휏(|훷((푎)훷(푏푚) | )|휏(푏푚)| ). It thus follows that 1 −푛 −푛−1 2 −푛 −1 휏(푥 ) ≥ [휏(|훷(푏 )|푘2 ) + (휏(|훷(푎)훷 |푘2 )) 휏(훷|(푏 )|푘2 ) ] 푚+1 2 푚 푚 −푛−1 ≥ 휏(|훷(푎)|푘2 ).

푘2−푛−1 Recalling that 푥푚 → (|푎| + 휀) as 푚 → ∞, we deduce −푛−1 −푛−1 휏((|푎| + 휀)푘2 ≥ 휏|훷(푎)|푘2 ). Letting 휀 → 0 we obtain inequality (5) at the (푛 + 1) − 푡h step.

102

Corollary (5.1.2)[279]:Φis multiplicative on Hardy spaces. More precisely, 훷(푎푏 = 훷(푎)훷(푏)for푎 ∈ 퐻푝(퐴) and 푏 ∈ 퐻푞(퐴)with0 < 푝, 푞 ≤ ∞. Proof: Note that 푎푏 ∈ 퐻푟(퐴) for any 푎 ∈ 퐻푝(퐴) and b ∈ 퐻푞(퐴), where r is determined by 1/푟 = 1/푝 + 1/푞. Thus 훷(푎푏) is well defined. Then the corollary follows immediately from the multiplicativity of 훷 on 퐴 and Theorem (5.1.1). The following is the extension to the case 푝 < 1 of Arveson-Labuschagne’s Jensen inequality (cf. [300, 163]). Recall that the Fuglede-Kadison determinant 훥(푥) of an operator x ∈ 퐿푝(푀) (0 < 푝 ≤ ∞)can be defined by ∞

훥(푥) = 푒푥푝(휏(푙표푔 |푥|)) = 푒푥푝 (∫ 푙표푔 푡 푑푣|푥|(푡)) , 0 where 푑푣|푥| denotes the probability measure on ℝ+ which is obtained by composing the spectral measure of |푥| with the trace 휏. It is easy to check that 훥(푥) = 푙푖푚‖푥‖푝 . 푝→0 As the usual determinant of matrices, Δ is also multiplicative: 훥(푥푦) = 훥(푥)훥(푥). We refer for information on determinant to [26, 300] in the case of bounded operators, and to [157, 76] for unbounded operators. Corollary (5.1.3)[278]:For any 0 < 푝 ≤ ∞ and 푥 ∈ 퐻푝(퐴) we have 훥(훷(푥)) ≤ 훥(푥). Proof: Let푥 ∈ 퐻푝(퐴).Then 푥 ∈ 퐻푞(퐴)too for 푞 ≤ 푝. Thus by Theorem (5.1.1) ‖훷(푥)‖푞 ≤ ‖ 푥‖푞 . Letting 푞 → 0 yields 훥(훷(푥)) ≤ 훥(푥). The following result is a Szegö type factorization theorem. It is stated in [89]. We take this opportunity to provide a proof. It is an improvement of the previous factorization theorems of Arveson [300] and Saito [153]. As already quoted in the introduction, Blecher and Labuschagne newly obtained a Szegö factorization for any ω ∈ Lp(M) with 1 ≤ p ≤ ∞ such that Δ(ω) > 0. Note that the property that ℎ−1 ∈ 퐻푞(퐴) whenever 휔−1 ∈ 퐿푞(푀) will be important for our proof of the Riesz factorization below. Let us also point out that although not in full generality, this result has hitherto been strong enough for applications. See Theorem (5.1.13) below for an improvement. Theorem (5.1.4)[279]:Let 0 < 푝, 푞 ≤ ∞. Let 휔 ∈ 퐿푝(푀) be an invertible operator such that 휔−1 ∈ 퐿푞(푀). Then there exist a unitary 푢 ∈ 푀 and ℎ ∈ 퐻푝(퐴) such that 휔 = 푢ℎ and ℎ−1 ∈ 퐻푞(퐴). Proof:We first consider the case 푝 = 푞 = 2. The proof of this special case is modelled on Arveson’s original proof of his Szeg¨o factorization theorem (see also [153]). Let 푥 be the orthogonal projection of w in [푤퐴0]2; and set 푦 = 푤 − 푥. Thus y ⊥ [wA0]2; whence 푦 ⊥ [푦퐴0]2. It follows that ∗ ∀ 푎 ∈ 퐴0, 휏(푦 푦푎) = 0. ∗ 1 ∗ ∗ Hence by (2), 푦 푦 ∈ 퐻 (퐴) = [퐴]1, and 푦 푦 ∈ [퐴 ]1 too. On the other hand, it is easy to see ∗ 1 ∗ ∗ that [퐴]1 ∩ [퐴 ]1 = 퐿 (퐷). Indeed, if 푎 ∈ [퐴]1 ∩ [퐴 ]1, then τ(푎푏) = 0 for any 푏 ∈ 퐴0 + 퐴0; so 휏(푎푏) = 휏(훷(푎)푏) for any 푏 ∈ 퐴 + 퐴∗. It follows that 푎 = 훷(푎) ∈ 퐿1(퐷). Consequently, 푦∗푦 ∈ 퐿1(퐷), so|푦| ∈ 퐿2(퐷). 103

Regarding 푀 as a von Neumann algebra acting on 퐿2(푀) by left multiplication, we claim that y is cyclic for M. This is equivalent to showing that 푦 is separating for the commutant of 푀. However, this commutant coincides with the algebra of all right multiplications on 퐿2(푀)by the elements of 푀. Thus we are reduced to prove that if 푧 ∈ 푀 is such that 푦푧 = 0, then 푧 = 0. We have: ∗ ∗ 2 ∗ 2 2 ∗ 2 2 0 = 휏(푧 푦 푦푧) = 휏(|푦| |푧 | ) = 휏(|푦| 훷(|푧 | )) = ‖푦푑‖2, ∗ 2 1/2 where 푑 = 훷(|푧 | ) ∈ 퐷; whence 푦푑 = 0. Choose a sequence (푎푛) ⊂ 퐴0 such that 푥 = lim 휔푎푛. (6) Then (recalling that 휔−1 ∈ 퐿2(푀)) −1 −1 0 = 휏(휔 푦푑) = 푙푖푚 휏 (휔 (휔 − 휔푎푛)푑) = 휏(푑) − 푙푖푚 휏(푎푛푑) = 휏(푑) 푛 푛 It follows that 푑 = 0, so by virtue of the faithfulness of 훷, 푧 = 0 too. This yields our claim. 2 Therefore, [푀푦]2 = 퐿 (푀). It turns out that the right support of y is 1. Since 푀 is finite, the 2 left support of y is also equal to 1, so 푦 is of full support. Consequently[푀푦]2 = 퐿 (푀)too. Let 푦 = 푢|푦|be the polar decomposition of 푦. Then 푢 is a unitary in 푀. Let ℎ = 푢∗푤. We are going to prove that ℎ ∈ 퐻2(퐴). To this end we first note the following orthogonal decomposition of 퐿2(푀): 2 ∗ 퐿 (푀) = [푦퐴0]2 ⊕ [푦퐷]2 ⊕ [푦퐴0 ]2 (7) Indeed, for any 푎 ∈ 퐴 and 푏 ∈ 퐴0 we have 〈푦푎, 푦푏∗〉 = 휏(푏푦∗푦푎) = 휏(|푦|2푎푏) = 0; ∗ so [푦퐴0]2 ⊕ [푦퐷]2 ⊕ [푦퐴0 ]2 is really an orthogonal sum. On the other hand, by the previous paragraph, we see that 2 ∗ 퐿 (푀) = [푦푀]2 ⊂ [푦퐴0]2 ⊕ [푦퐷]2 ⊕ [푦퐴0 ]2 . therefore, decomposition (7) follows. Applying 푢∗ to both sides of (7), we deduce 2 ∗ ∗ ∗ ∗ 퐿 (푀) = [푢 푦퐴0]2 ⊕ [푢 푦퐷]2 ⊕ [푢 푦퐴0 ]2 ∗ = [|푦|퐴0]2 ⊕ [|푦|퐷]2 ⊕ [|푦|퐴0]2 . 2 Since |푦| ∈ 퐿 (퐷), [|푦|퐴0]2 ⊂ [퐴0]2, and similarly for the two other terms on the right. Therefore, 2 ∗ ∗ 2 퐿 (푀) = [|푦|퐴0]2 ⊕ [|푦|퐷]2 ⊕ [|푦|퐴0]2 ⊂ [퐴0]2 ⊕ [퐷]2 ⊕ [퐴0 ]2 = 퐿 (푀) . Hence ∗ ∗ [|푦|퐴0]2 = [퐴0]2, [|푦|퐷]2 = [퐷]2, [|푦|퐴0]2 = [퐴0 ]2 (8) Passing to adjoints, we also have ∗ ∗ [퐴0|푦|]2 = [퐴0]2, [퐷|푦|]2 = [퐷]2, [퐴0|푦|]2 = [퐴0 ]2 . ∗ 2 ∗ Now it is easy to show that ℎ = 푢 휔 ∈ 퐻 (퐴). Indeed, since 푦 ⊥ [휔퐴0], 휏(푦 휔푎) = 0 for all ∗ 푎 ∈ 퐴0; so 휏(푎|푦|푢 휔) = 0. However, [퐴0|푦|]2 = [퐴0]2. Thus 2 ∀ 푎 ∈ 퐻0 (퐴), 휏(푎ℎ) = 0. Hence by (1), ℎ ∈ 퐻2(퐴), as desired. It remains to show that ℎ−1 ∈ 퐻2(퐴).To this end we first observe that 훷(ℎ)훷(ℎ−1) = 1. Indeed, given 푑 ∈ 퐷 we have, by (6) 휏((ℎ)훷(ℎ−1)|푦|푑) = 휏(ℎ−1|푦|푑훷(ℎ)) = 휏(휔−1푢|푦|푑훷(ℎ)) −1 ∗ ∗ = 푙푖푚 휏 (휔 (휔 − 휔푎푛)푑훷(ℎ)) = 휏푑훷(ℎ)) = 휏(ℎ푑) = 휏(푢 휔푑) = 휏(푢 푦푑) 푛 = 휏(|푦|푑), 104 where we have used the fact that ∗ ∗ 휏(푢 푥푑) = 푙푖푚 훷 (푢 휔푎푛푑) = 푙푖푚 휏 (ℎ푎푛푑) = 0. 푛 푛 2 Since [|푦|퐷]2 = 퐿 (퐷), we deduce our observation. Therefore, 훷(ℎ) is invertible and its inverse is 훷(ℎ − 1). On the other hand, by (6) ∗ ∗ 훷(ℎ) = 푙푖푚 훷 (푢 (푦 + 휔푎푛)) = 훷(|푦|) + 푙푖푚 훷 (ℎ푎푛) = 푢 푦. 푛 푛 Hence, 푢 = 푦훷(ℎ)−1 = 푦훷(ℎ−1). Now let 푎 ∈ 퐴0. Then −1 −1 −1 −1 휏(ℎ 푎) = 휏(휔 푢푎) = 휏(휔 푦훷(ℎ − 1)푎) = 푙푖푚 휏 휔 (휔 − 휔푎푛)훷(ℎ − 1)푎) = 0. 푛 It follows that ℎ−1 ∈ 퐻2(퐴).Therefore, we are done in the case 푝 = 푞 = 2. The general case can be easily reduced to this special one. Indeed, if 푝 ≥ 2and 푞 ≥ 2, then given 휔 ∈ 퐿푝(푀) with 휔−1 ∈ 퐿푞(푀), we can apply the preceding part and then find a unitary 푢 ∈ 푀 and ℎ ∈ 퐻2(퐴) such that 휔 = 푢ℎ andℎ−1 ∈ 퐻2(퐴). Then ℎ = 푢∗휔 ∈ 퐿푝(푀), so 휔 ∈ 퐻2(퐴) ∩ 퐿푝(푀) = 퐻푝(퐴) by (3). Similarly, ℎ−1 ∈ 퐻푞(퐴). Suppose 푚푖푛(푝, 푞) < 2. Choose an integer 푛 such that 푚푖푛(푛푝, 푛푞) ≥ 2. Let 휔 = 푣|휔| be the polar decomposition of 휔. Note that 푣 ∈ 푀 is a unitary. Write 1 1/푛 1/푛 휔 = 푣|휔| |휔|푛 … |휔| = 휔1휔2 … 휔푛, 1/푛 1/푛 푛푝 −1 where 휔1 = 푣|휔| and 휔푘 = |휔| for2 ≤ 푘 ≤ 푛. Since 휔푘 ∈ 퐿 (푀) and 휔푘 ∈ 퐿푛푞(푀), by what is already proved we have a factorization 휔푛 = 푢푛ℎ푛 푛푝 −1 푛푞 with 푢푛 ∈ 푀a unitary, ℎ푛 ∈ 퐻 (퐴)such that ℎ푛 ∈ 퐻 (퐴). Repeating this argument, we again get a same factorization for 휔푛−1푢푛: 휔푛−1푢푛 = 푢푛−1ℎ푛−1 ; and then for 휔푛−2푢푛−1, and so on. In this way, we obtain a factorization: 휔 = 푢ℎ1 … ℎ푛, 푛푝 −1 푛푞 where 푢 ∈ 푀 is a unitary, ℎ푘 ∈ 퐻 (퐴) such that ℎ푘 ∈ 퐻 (퐴). Setting ℎ = 푢ℎ1 … ℎ푛, we then see that 휔 = 푢ℎ is the desired factorization. Hence the proof of the theorem is complete. Remark(5.1.5) [279]:Let 휔 ∈ 퐿2(푀) be an invertible operator such that 휔−1 ∈ 퐿2(푀). Let 휔 = 푢ℎ be the factorization in Theorem (5.1.4). The preceding proof shows that [ℎ퐴]2 = 2 퐻 (퐴). Indeed, it is clear that[푦퐴]2 ⊂ [휔퐴]2. Using decomposition (7), we get ∗ [휔퐴]2 ⊖ [푦퐴]2 = [휔퐴]2 ∩ [푦퐴0 ]2 . Now for any 푎 ∈ 퐴 and 푏 ∈ 퐴0, 〈휔푎, 푦푏∗〉 = 휏(푦∗휔푎푏) = 0 since 푦 ⊥ [휔퐴0]. It then follows that [휔퐴]2 ⊖ [푦퐴]2 = {0}, so[휔퐴]2 = [푦퐴]2. Hence, by (8) ∗ ∗ 2 [푦퐴]2 = [푢 휔퐴]2 = [푢 푦퐴]2 = [|푦|퐴]2 = 퐻 (퐴).

We turn to the Riesz factorization. We first need to extend (3) to all indices. Proposition (5.1.6)[279]: Let 0 < 푝 < 푞 ≤ ∞. Then 푝 푞 푞 푝 푞 푞 퐻 (퐴) ∩ 퐿 (푀) = 퐻 (퐴) and 퐻 (퐴) ∩ 퐿 (푀) = 퐻0 (퐴), 105

푝 where 퐻0 (퐴) = [퐴0]푝. Proof: It is obvious that 퐻푞(퐴) ⊂ 퐻푝(퐴) ∩ 퐿푞(푀). To prove the converse inclusion, we first consider the case q = ∞. Thus let 푥 ∈ 퐻푝(퐴) ∩ 푀. Then by Corollary (5.1.2), ∀ 푎 ∈ 퐴0 , 훷(푥푎) = 훷(푥)훷(푎) = 0. Hence by (1), 푥 ∈ 퐴.

Now consider the general case. Fix an 푥 ∈ 퐻푝(퐴) ∩ 퐿푞(푀). Applying Theorem (5.1.4) to 휔 = (푥∗푥 + 1)1/2, we get an invertible ℎ ∈ 퐻푞(퐴)such that ℎ∗ℎ = 푥∗푥 + 1 and ℎ−1 ∈ 퐴. Since ℎ∗ℎ ≤ 푥∗푥 , there exists a contraction 푣 ∈ 푀 such that 푥 = 푣ℎ. Then 푣 = 푥ℎ−1 ∈ 퐻푝(퐴) ∩ 푀, so 푣 ∈ 퐴. Consequently, 푥 ∈ 퐴. 퐻푞(퐴) = 퐻푞(퐴). Thus we proved the first 푝 equality. The second is then an easy consequence. For this it suffices to note that 퐻0 (퐴) = {푥 ∈ 퐻푝(퐴): 훷(푥) = 0}. The later equality follows from the continuity of 훷 on 퐻푝(퐴). Theorem (5.1.7)[279]: Let 0 < 푝, 푞, 푟 ≤ ∞ such that 1/푝 = 1/푞 + 1/푟. Then for 푥 ∈ 푝 푞 푟 퐻 (퐴) and 휀 > 0 there exist 푦 ∈ 퐻 (퐴) and 푧 ∈ 퐻 (퐴) such that 푥 = 푦푧 and ‖푦‖푞 ‖푧‖푟 ≤ ‖푥‖푝 + 휀. Consequently, 푞 푟 ‖푥‖푝 = 푖푛푓{‖푦‖푞 ‖푧‖푟: 푥 = 푦푧, 푦 ∈ 퐻 (퐴), 푧 ∈ 퐻 (퐴)} Proof: The case where 푚푎푥(푞, 푟) = ∞ is trivial. Thus we assume both 푞 and 푟 to be finite. Let 휔 = (푥∗푥 + 휀)1/2. Then 휔 ∈ 퐿푝(푀) and 휔−1 ∈ 푀. Let 푣 ∈ 푀 be a contraction such that 푥 = 푣휔. Now applying Theorem (5.1.4) to 휔푝/푟, we have: 휔푝/푟 = 푢푧, where 푢 is a unitary in 푀 and 푧 ∈ 퐻푟(퐴) such that 푧−1 ∈ 퐴. Set 푦 = 푣휔푝/푞 푢. Then 푥 = 푦푧, so 푦 = 푥푧−1. Since 푥 ∈ 퐻푝(퐴) and 푧−1 ∈ 퐴, 푦 ∈ 퐻푝(퐴). On the other hand, 푦 belongs to 퐿푞(푀) too. Therefore, 푦 ∈ 퐻푞(퐴) by virtue of Proposition (5.1.6). The norm estimate is clear. We consider outer operators. All results below on the left and right outers are due to Blecher and Labuschagne [66] in the case of indices not less than one. The notion of bilaterally outer is new. We start with the following result. Proposition (5.1.8)[279]: Let 0 < 푝 < 푞 ≤ ∞ and let ℎ ∈ 퐻푞(퐴). Then 푝 푞 (i) [ℎ퐴]푝 = 퐻 (퐴) iff [ℎ퐴]푞 = 퐻 (퐴); 푝 푞 (ii) [퐴ℎ]푝 = 퐻 (퐴) iff [퐴ℎ]푞 = 퐻 (퐴); 푝 푞 (iii) [퐴ℎ퐴]푝 = 퐻 (퐴) iff [퐴ℎ퐴]푞 = 퐻 (퐴). Proof: We prove only the third equivalence. The proofs of the two others are similar (and even 푝 푞 simpler). It is clear that [퐴ℎ퐴]푝 = 퐻 (퐴) ⇒ [퐴ℎ퐴]푞 = 퐻 (퐴). To prove the converse implication we first consider the case 푞 ≥ 1.Let 푞′ be the conjugate index of 푞. Let 푥 ∈ 퐿푞′(푀) be such that ∀ 푎, 푏 ∈ 퐴, 휏(푥푎ℎ푏) = 0.

1 p Then 푥푎ℎ ∈ 퐻0 (퐴) for any 푎 ∈ 퐴by virtue of (2) (more rigorously, its H -analogue as in 푝 Proposition (5.1.6)). On the other hand, by the assumption that [퐴ℎ퐴]푝 = 퐻 (퐴), there exist two sequences (푎푛), (푏푛) ⊂ 퐴 such that 푝 푙푖푚 푎푛ℎ푏푛 = 1 푖푛 퐻 (퐴). 푛

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Consequently, 푟 푙푖푚 푥푎푛ℎ푏푛 = 푥 푖푛 퐿 (푀), 푛 1 where = 1/푞′ + 1/푝.Since 푥푎 ℎ푏 = (푥푎 ℎ)푏 ∈ 퐻1(퐴) ⊂ 퐻푟(퐴), we deduce that 푥 ∈ 푟 푛 푛 푛 푛 0 0 푟 푟 푞′ 푞′ 퐻0 (퐴). Therefore, 푥 ∈ 퐻0 (퐴) ∩ 퐿 (푀), so by Proposition (5.1.6), 푥 ∈ 퐻0 (퐴) Hence, 푞 푞 휏(푥푦) = 0 for all y ∈ 퐻 (퐴). Thus [퐴ℎ퐴]푞 = 퐻 (퐴). Now assume 푞 < 1. Choose an integer 푛 such that 푛푝 ≥ 2. By the proof of Theorem (5.1.7) and Remark(5.1.5), we deduce a factorization: ℎ = ℎ1ℎ2 … ℎ푛 , 푛푝 2 where ℎ푘 ∈ 퐻 (퐴) for every 1 ≤ 푘 ≤ 푛 and [ℎ푘퐴]2 = 퐻 (퐴)for 2 ≤ 푘 ≤ 푛. By the left version (i.e;part i) of the previous case already proved, we also have [ℎ푘퐴]푛푞 = 푛푞 푛푞 퐻 (퐴) and [ℎ푘퐴]푛푞 = 퐻 (퐴)for 2 ≤ 푘 ≤ 푛. Let us deal with the first factor ℎ1. Using 푝 푛푝 푝 [퐴ℎ퐴]푝 = 퐻 (퐴) and [ℎ푘퐴]푛푝 = 퐻 (퐴)for 2 ≤ 푘 ≤ 푛, we see that [퐴ℎ1퐴]푝 = 퐻 (퐴);so 푝 푞 again [퐴ℎ1퐴]푝 = 퐻 (퐴) by virtue of the first part. It is then clear that [퐴ℎ퐴]푞 = 퐻 (퐴). The previous result justifies the relative independence of the index 푝 in the following definition. Definition (5.1.9)[279]: Let 0 < 푝 ≤ ∞. An operator ℎ ∈ 퐻푝(퐴) is called left outer, right outer or bilaterally outer according to 푝 푝 푝 [hA]p = 퐻 (퐴), [ℎ퐴]푝 = 퐻 (퐴)or[퐴ℎ퐴]푝 = 퐻 (퐴). Theorem (5.1.10)[278]: Let 0 < 푝 ≤ ∞ and ℎ ∈ 퐻푝(퐴). (i) If ℎ is left or right outer, then 훥(ℎ) = 훥훷(ℎ)). Conversely, if 훥(ℎ) = 훥(훷(ℎ)) and 훥(ℎ) > 0, then ℎ is left and right outer (so bilaterally outer too). (ii) If 퐴is antisymmetric (i.e; 푑푖푚퐷 = 1) and h is bilaterally outer, then 훥(ℎ) = 훥(훷(ℎ)). Proof: (i) This part is proved in [66] for 푝 ≥ 1. Assume ℎ is left outer. Let 푑 ∈ 퐷. Using Theorem (5.1.1), we obtain 푝 ‖훷(ℎ)푑‖푝 = 푖푛푓{‖ℎ푑 + 푥0‖푝: 푥 ∈ 퐻0 (퐴)}. On the other hand, [ℎ퐴 ] = [[ℎ퐴] 퐴 ] = [[퐴]푝퐴 ] = [퐴 ] = 퐻푝 (퐴). 0 푝 푝 0 푝 0 푝 0 푝 0 Therefore, ‖훷(ℎ)푑‖푝 = 푖푛푓{‖ℎ(푑 + 푎0)‖푝: 푎 ∈ 퐴0}. Recall the following characterization of훥(푥) from [66]: 훥(푥) = 푖푛푓{‖푥푎‖푝: 푎 ∈ 퐴, 훥(훷(푎)) ≥ 1}. (9) Now using this formula twice, we obtain 휏(훷(ℎ)) = 푖푛푓{‖훷(ℎ)푑‖푝: 푑 ∈ 퐷, 훥(푑) ≥ 1} = 푖푛푓{‖ℎ(푑 + 푎0)‖푝: 푑 ∈ 퐷, 훥(푑) ≥ 1, 푎0 ∈ 퐴0} = 훥(ℎ). Let us show the converse under the additional assumption that 훥(ℎ) > 0. We will use the case 푝 ≥ 1 already proved in [66]. Thus assume 푝 < 1. Choose an integer 푛 such that 푛푝 ≥ 1. By 푛푝 Theorem (5.1.7), there exist ℎ1, . . . , ℎ푛 ∈ 퐻 (퐴) such that ℎ = ℎ1 … ℎ푛.Then 훥(ℎ) = 훥(ℎ1) … 훥(ℎ푛); so 훥(ℎ푘) > 0 for all 1 ≤ 푘 ≤ 푛. On the other hand, by Arveson- Labuschagne’s Jensen inequality [300,163] (or Corollary (5.1.3)), 훥(훷(ℎ푘)) ≤ 훥(ℎ푘). However,

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훥(훷(ℎ)) = 훥(훷(ℎ1)) … 훥(훷(ℎ푛)) ≤ 훥(ℎ1) … 훥(ℎ푛) = 훥(ℎ) = 훥(훷(ℎ)). 푛푝 It then follows that 훥(훷(ℎ푘)) ≤ 훥(ℎ푘) for all k. Now ℎ푘 ∈ 퐻 (퐴) with 푛푝 ≥ 1, so hk is left and right outer. Consequently, h is left and right outer. (ii) This proof is similar to that of the first part of i). We will use the following variant of (9) 훥(푥) = 푖푛푓{‖(푎푥푏)‖푝: 푎, 푏 ∈ 퐴, 훥(훷(푎)) ≥ 1, 훥(훷(푏)) ≥ 1} (10) for every 푥 ∈ 퐿푝(푀). This formula immediately follows from (9). Indeed, by (9) and the multiplicativity of 훥 푖푛푓{‖(푎푥푏)‖푝: 푎, 푏 ∈ 퐴, 훥(훷(푎)) ≥ 1, 훥(훷(푏)) ≥ 1} = 푖푛푓{훥(푎푥): 푎 ∈ 퐴, 훥(훷(푎)) ≥ 1} = 푖푛푓{훥(푎)훥(푥): 푎 ∈ 퐴, 훥(훷(푎)) ≥ 1} = ∆(푥). Now assume ℎ ∈ 퐻푝(퐴) is bilaterally outer and A is antisymmetric. Then 훷(ℎ) is a multiple of the unit of M. As in the proof of i), We have 푝 ‖훷(ℎ)‖푝 = 푖푛푓{‖ℎ + 푥‖푝: 푥 ∈ 퐻0 (퐴)} = 푖푛푓{‖ℎ + 푎ℎ푏0‖푝: 푎 ∈ 퐴, 푏0 ∈ 퐴0}. (11) Using 푑푖푚퐷 = 1, we easily check that 푖푛푓{‖ℎ + 푎ℎ푏0‖푝: 푎 ∈ 퐴, 푏0 ∈ 퐴0} = 푖푛푓{‖(1 + 푎0)ℎ(1 + 푏0)‖푝: 푎0, 푏0 ∈ 퐴0}. (12) Indeed, it suffices to show that both sets{ℎ + 푎ℎ푏0: 푎 ∈ 퐴, 푏0 ∈ 퐴0}and {(1 + 푎0)ℎ(1 + 푝 푏0): 푎0, 푏0 ∈ 퐴0} are dense in {푥 ∈ 퐻 (퐴): 훷(푥) = 훷(ℎ)}. The first density immediately 푝 푝 follows from the density of Aℎ퐴0 in 퐻0 (퐴). On the other hand, let푥 ∈ 퐻 (퐴)with 훷(푥) = 훷(ℎ) and let 푎푛, 푏푛 ∈ 퐴 such that 푙푖푚 푎푛ℎ푏푛 = 푥. By Theorem (5.1.1), 푛 푙푖푚 훷 (푎푛)훷(ℎ)훷(푏푛) 훷(푥). 푛 Since 훷(푥) = 휏(푥)1 = 휏(ℎ)1 = 훷(ℎ) ≠ 0, we deduce that 푙푖푚 휏 (푎푛)휏(푏푛) = 1. Thus 푛 replacing 푎푛 and 푏푛 by 푎푛/휏(푎푛) and 푏푛/휏(푏푛), respectively, we can assume that 푎푛 = 1 + ̃ ̃ 푎̃푛 and 푏푛 = 1 + 푏푛with 푎̃푛, 푏푛 ∈ 퐴0; whence the desired density of {(1 + 푎0)ℎ(1 + 푝 푏0): 푎0, 푏0 ∈ 퐴0} in {푥 ∈ 퐻 (퐴): 훷(푥) = 훷(ℎ)}. Finally, combining (10), (11) and (12), we get 훥(훷(ℎ)) = 훥(ℎ). Note that, the assumption that A is antisymmetric in Theorem (5.1.12), ii) cannot be removed in general, as shown by the following example. Keep the notation introduced and consider the ∞ ∞ case where 푀 = 퐿 (푇; 필2) and 퐴 = 퐻 (푇; 필2). Let 휑1 and 휑2 be two outer functions in p H (T), and let ℎ = 휑1 ⊗ 푒11 + 푧푒22 ⊗ 푒22, where z denotes the identity function on 핋. Then it is easy to check that h is bilaterally outer and 1 훥(ℎ) = 푒푥푝 ( ∫ 푙표푔 |휑 | + ∫ 푙표푔 |휑 |) > 0. 2 핋 1 핋 2 However,훷(ℎ) = 휑1(0)푒11, 푠표 훥(훷(ℎ)) = 0. The following is an immediate consequence of Theorem (5.1.12). We do not know, however, whether the condition 훥(ℎ) > 0 in i) can be removed or not.

Corollary (5.1.11)[279]: Let ℎ ∈ 퐻푝(퐴), 0 < 푝 ≤ ∞. (i) If 훥(ℎ) > 0, then ℎ is left outer iff ℎ is right outer. (ii) Assume that 퐴 is antisymmetric. Then the following properties are equivalent:

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(1) ℎ is left outer; (2) ℎ is right outer; (3) ℎ is bilaterally outer; (4) 훥(훷(ℎ)) = 훥(ℎ) >0. We will say that h is outer if it is at the same time left and right outer. Thus if ℎ ∈ 퐻푝(퐴)with 훥(ℎ) > 0, then h is outer iff 훥(ℎ) = 훥(훷(ℎ)). Also in the case where 퐴 is antisymmetric, an ℎ with 훥(ℎ) > 0is outer iff it is left, right or bilaterally outer. Corollary (5.1.12)[279]:Let ℎ ∈ 퐻푝(퐴) such that h−1 ∈ Hq(A) with 0 < 푝, 푞 ≤ ∞. Then h is outer. Proof: By the multiplicativity of 훥, 훥(ℎ)훥(ℎ−1) = 1 and 훥(훷(ℎ))훥(훷(ℎ−1)) = 1. Thus by Jensen’s inequality (Corollary (5.1.3)), 훥(ℎ) = 훥(ℎ−1)−1 ≤ 훥(훷(ℎ−1))−1 = 훥(훷(ℎ)); whence the assertion because of Theorem (5.1.10). The following improves Theorem (5.1.4). Theorem (5.1.13)[279]: Let 휔 ∈ 퐿푝(푀) with 0 < 푝 ≤ ∞ such that Δ(ω) > 0. Then there exist a unitary 푢 ∈ 푀 and an outer ℎ ∈ 퐻푝(퐴) such that 휔 = 푢ℎ. Proof: Based on the case푝 ≥ 1 from [67], the proof below is similar to the end of the proof of Theorem (5.1.4). For simplicity we consider only the case where 푝 ≥ 1/2. Write the polar decomposition of 휔: 휔 = 푣|휔|. Applying [66] to |휔|1/2 we get a factorization: |휔|1/2 = 2푝 푢2ℎ2with u2 unitary and ℎ2 ∈ 퐻 (퐴) left outer. Since 훥(ℎ2) > 0, ℎ2 is also right outer; so 1/2 ℎ2 is outer. Similarly, we have: 푣|푤| 푢2 = 푢1ℎ1. Then 푢 = 푢1and ℎ = ℎ1ℎ2 yield the desired factorization of 휔. The following is the inner-outer factorization for operators in 퐻푝(퐴), which is already in [66] for 푝 ≥ 1. Corollary (5.1.14)[279]: Let 0 < 푝 ≤ ∞ and 푥 ∈ 퐻푝(퐴) with훥(푥) > 0. Then there exist a unitary 푢 ∈ 퐴(inner) and an outer ℎ ∈ 퐻푝(퐴) such that 푥 = 푢ℎ. Proof: Applying the previous theorem, we get 푥 = 푢ℎ with h outer and u a unitary in 푀. p 푝 푝 Let푎푛 ∈ 퐴 such that 푙푖푚ℎ푎푛 = 1 in H (A). Then 푢 = 푙푖푚푥푎푛 in 퐻 (퐴) too; so 푢 ∈ 퐻 (퐴) ∩ 푀. By Proposition (5.1.5), 푢 ∈ 퐴. Corollary (5.1.15)[279]: Let 0 < 푝 ≤ ∞ and ℎ ∈ 퐻푝(퐴)with 훥(ℎ) > 0. Then ℎ is outer iff for any 푥 ∈ 퐻푝(퐴)with |푥| = |ℎ| we have 훥(훷(푥)) ≤ 훥(훷(ℎ)). Proof: Assume h outer. Then by Corollary (5.1.3) and Theorem (5.1.9), 훥(훷(푥)) ≤ 훥(푥) = 훥(ℎ) = 훥(훷(ℎ)). Conversely, let ℎ = 푢푘 be the decomposition given by Theorem (5.1.13) with k outer. Then 훥(ℎ) = 훥(푘) = 훥(훷(푘)) ≤ 훥(훷(ℎ)); so h is outer by Theorem (5.1.12).

Corollary (5.1.16)[279]: Let 0 < 푝, 푞, 푟 ≤ ∞ such that 1/p = 1/q + 1/r. Let 푥 ∈ 퐻푝(퐴) be such that Δ(x) > 0. Then there exist 푦 ∈ Hq(A) and 푧 ∈ 퐻푟(퐴) such that 푥 = 푦푧 and‖푥‖푝 = ‖푦‖푞‖푧‖푟 . Proof: This proof is similar to that of Theorem (5.1.7) Instead of Theorem (5.1.4), we now use Theorem (5.1.13). Indeed, by the later theorem, we can find a unitary u2 ∈ M and an outer ℎ2 ∈ 109

푝/푟 푝/푟 푝/푞 퐻 (퐴) such that |푥| = 푢2ℎ2. Once more applying this theorem to 푣|푥| 푢2, we have a 푝/푞 similar factorization: 푣|푥| 푢2 = 푢1ℎ1, where 푣 is the unitary in the polar decomposition of 푥. Since ℎ1 andh2 are outer, we deduce, as in the proof of Corollary (5.1.14), that 푢1 ∈ 퐴. Then 푦 = 푢1ℎ1 and 푧 = ℎ2 give the desired factorization of 푥. Let 휔 ∈ 퐿1(푇)be a positive function and let 푑휇 = 휔푑푚. Then we have the following well- known Szeg¨o formula [92]: 2 푖푛푓{∫핋|1 − 푓| 푑휇: 푓 푚푒푎푛 푧푒푟표 푎푛푎푙푦푡푖푐 푝표푙푦푛표푚푖푎푙} = 푒푥푝∫핋푙표푔푤푑푚. This formula was later proved for any positive measure 휇 on 푇 independently by Kolmogorov/Krein [15] and Verblunsky [268]. Then the singular part of μ with respect to the Lebesgue measure dmdoes not contribute to the preceding infimum and w on the right hand side is the density of the absolute part of μ (also see [141]). This latter result was extended to the noncommutative setting in [66]. More precisely, let ω be a positive linear functional on 푀, and let 휔 = 휔푛 + 휔푠be the decomposition of ω into its normal and singular parts. Let 휔 ∈ 1 퐿 (푀) be the density of 휔푛 with respect to 휏, i.e., 휔푛 = 휏(휔 ∙). Then Blecher and Labuschagne proved that if 푑푖푚퐷 < ∞, ∆(휔) = 푖푛푓{휔(|푎|2): 푎 ∈ 퐴, ∆(훷(푎)) ≥ 1}. It is left open in [66] whether the condition 푑푖푚퐷 < ∞ can be removed or not. We will solve this problem in the affirmative. At the same time, we show that the square in the above formula can be replaced by any power 푝. Theorem (5.1.17)[279]: Let 휔 = 휔푛 + 휔푠be as above and 0 < 푝 < ∞. Then 훥(휔) = 푖푛푓{(휔|푎|푝): 푎 ∈ 퐴, ∆(훷(푎)) ≥ 1}. Proof: Let δ(휔) = 푖푛푓{휔(|푎|푝): 푎 ∈ 퐴, ∆(훷(푎)) ≥ 1}. First we show that 훿(휔) = 푖푛푓{휔(푥): 푎 ∈ 퐴, ∆(훷(푎)) ≥ 1}, −1 where 푀+ denotes the family of invertible positive operators in 푀 with bounded inverses. −1 Given any 푥 ∈ 푀+ , by Arveson’s factorization theorem there exists 푎 ∈ 퐴 such that |푎| = 푥1/푝 and 푎−1 ∈ 퐴. Then 푥 = |푎|푝, so 훥(푥) = 훥(|푎|푝) = 훥(푎)푝. Since a is invertible with 푎−1 ∈ 퐴, by Jensen’s formula in [300], 훥(푎) = 훥(훷(푎)). It then follows that −1 훿(휔) ≤ {휔(푥): 푥 ∈ 푀+ , 훥(푥) ≥ 1}. The converse inequality is easier. Indeed, given 푎 ∈ 퐴 with 훥(훷(푎)) ≥ 1and 휀 > 0, set 푥 = 푝 −1 푝 푝 |푎| + 휀. Then 푥 ∈ 푀+ and 훥(푥) ≥ 훥(푎) ≥ 훥(훷(푎)) by virtue of Jensen’s inequality. Since 푙푖푚 휔 (|푎|푝 + 휀) = 휔(|푎|푝), we deduce the desired converse inequality. 휀→0 Next we show that 훿(휔) = 훿(휔푛). The singularity of ωsimplies that there exists an increasing net (푒푖) of projections in 푀 such that 푒푖 → 1 strongly and ωs(ei) = 0 for every i (see [181]). Let 휀 >0. Set −1 휏(푒푖) ⊥ ⊥ 푥푖 = 휀 (푒푖 + 휀푒푖 ), where 푒 = 1 − 푒. −1 −1 Clearly, 푥푖 ∈ 푀+ and (xi) = 1. Let 푥 ∈ 푀+ and 훥(푥) ≥ 1. Then 훥(푥푖푥푥푖) = 훥(푥) ≥ 1, and ∗ 푥푖푥푥푖 → 푥 in the 휔 -topology. On the other hand, note that 2휏(푒푖) ⊥ ⊥ 휔푠(푥푖푥푥푖) = 휀 휔푠(푒푖 푥푒푖 ). Therefore, 110

δ(휔) ≤ 푙푖푚푠푢푝휔(푥푖푥푥푖) = 휔푛(푥) + 푙푖푚푠푢푝휔푠(푥푖푥푥푖) 2휏(푒푖) ⊥ ⊥ 2 ≤ 휔푛(푥) + 푙푖푚푠푢푝휀 휔푠(푒푖 푥푒푖 ) ≤ 휔푛(푥) + 휀 ‖휔푠‖‖푥‖. It thus follows that 훿(휔) ≤ 훿(휔푛), so δ(휔) = 훿(휔푛). Now it is easy to conclude the validity of the result. Indeed, the preceding two parts imply −1 훿(휔) = 푖푛푓{휏(푤푥): 푥 ∈ 푀+ , 훥(푥) ≥ 1}. By a formula on determinants from [300], the last infimum is nothing but 훥(휔). Therefore, the theorem is proved. Section (5.2) Noncommutative Hardy-Lorentz Spaces The classical Hardy spaces 퐻푝(퐷), 1 ≤ 푝 ≤ ∞, are Banach spaces of analytic functions on the unit disk satisfying that 2휋 푠푢푝 {∫ |푓(푟푒푖휃)|푝푑휃} < ∞ . 0<푟<1 0 by taking radial limits,퐻푝(퐷) can be identified with 퐻푝(푇 ), the space of functions on the unit circle which are in 퐿푝(푇 )with respect to Lebesgue measure and whose negative Fourier coefficients vanish. These spaces have played an important role in modern analysis and prediction theory. One of the key results in the functional analytic approach to Hardy spaces is Szeg표̈ theorem (see [274]), which is formula for the weighted 퐿2(푇 ) distance from 1 to the analytic polynomials which vanish at the origin. The theory of Hardy spaces was generalized in two directions. Masani and Wiener [186, 185] extended Szegö theorem to the theory of multivariate stochastic processes by studying matrix valued functions. Concurrently, Helson and Lowdenslager [115] adapting techniques from functional analysis to extend the theory to the setting of a compact group with ordered dual, thus laying the foundation for the theory of function algebras. This eventually led to the definition of a weak∗-Dirichlet algebra of functions by Srinivasan and Wang [272]. Srinivasan and Wang were able to prove Szeg표̈’s theorem and several other important results in the theory of function algebras. Arveson [25] introduced the concept of maximal subdiagonal algebras, unifying analytic function spaces and nonselfadjoint operator algebras. Subsequently, Arveson’s pioneer work was extended to different cases by several authors. In 1997, Marsalli and West [184] defined noncommutative Hardy spaces for finite von Neumann algebras and obtained a series of results including a Riesz factorization theorem, the dual relations between 퐻푝(𝒜) and 퐻푞(𝒜) and so on. Labuschagne [166] proved the universal validity of Szeg표̈’s theorem for finite subdiagonal algebras. Blecher and Labuschagne [41] gave several useful variants of the noncommutative Szeg표̈theorem for 퐿푝(ℳ). Tt was also in [42] that the longstanding open problem concerning the noncommutative generalization of the famous ’outer factorization’ of functions 푓 with 푙표푔 |푓| integrable was solved. Recently, Bekjan and Xu [39] presented the more general form of Szegö type factorization for the noncommutative Hardy spaces defined in [184]. We introduce the noncommutative Hardy-Lorentz spaces. By adapting the techniques in [39], we establish the Szegö factorization theorem of these spaces. Section contains some preliminaries and notations on the noncommutative Lp,q-spaces and noncommutative Hp,q- spaces. The proof of the Szegö factorization of noncommutative Hardy-Lorentz spaces are 111 presented. Finally mainly devoted to the inner-outer type factorization of noncommutative Hardy- Lorentz spaces. We denote by ℳa finite von Neumann algebra on the Hilbert space ℋ, equipped with a normal faithful tracial state τ. The identity in ℳ is denoted by 1 and we denote by 퐷 a von Neumann subalgebra of ℳ, moreover, we let 휀: ℳ →D be the unique normal faithful conditional expectation such that 휏 ∘ 휀 = 휏. 𝒜 is a finite subdiagonal algebra of ℳ. A finite subdiagonal algebra of ℳ with respect to ε (or 퐷) is a 푤∗ −closed subalgebra 𝒜 of ℳ satisfying the following conditions: (i) 𝒜 + 𝒜∗ is ω∗ −dense in ℳ; (ii) ε is multiplicative on 퐴, i.e.,휀(푎푏) = 휀 (푎) 휀 (푏) for all 푎, 푏 ∈ 𝒜; (iii) 𝒜 ∩ 𝒜∗ = 퐷. We denote by ℳ푝푟표푗the lattice of (orthogonal) projections in ℳ. A linear operator 푥: 푑표푚(푥) → ℋ, with domain 푑표푚(푥) ⊆ ℋ, is said to be affiliated with ℳ if 푢푥 = 푥푢for all unitary 푢 in the commutant ℳ′ of ℳ. The closed densely defined linear operators 푥 affiliated with ℳis called τ −measurable if for every ε > 0 there exists an orthogonal projection P ∈ ℳproj such that 푃(퐻) ⊆ 푑표푚(푥) and 휏(1 − 푃) < 휀. The collection of all τ-measurable operators is denoted by ℳ̃ . With the sum and product defined as the respective closures of the algebraic sum and product, ℳ̃ is a ∗-algebra. For a positive self-adjoint operator x affiliated with M, we set 푛 ∞

휏(푥) = 푠푢푝 휏 (∫ 휆푑퐸휆) = ∫ 휆푑휏(퐸휆), 푛 0 0 ∞ where 푥 = 휆푑퐸 is the spectral decomposition of 푥. ∫0 휆 Let 0 < 푝 < ∞, 퐿푝(ℳ; 휏) is defined as the set of all τ-measurable operators 푥 affiliated with ℳ such that 1 푝 푝 ‖푥‖푝 = 휏(|푥| ) < ∞. ∞ In addition, we put 퐿 (ℳ; 휏) = ℳ and denote by ‖·‖∞ (= ‖ ·‖ ) the usual operator norm. It 푝 is well known that 퐿 (ℳ; τ) is a Banach space under ‖·‖푝 for 1 ≤ 푝 < ∞. They have all the expected properties of classical 퐿푝-spaces (see also [89]). Let 푥 be a τ-measurable operator and 푡 > 0. The “t-th singular number (or generalized s- number) of 푥” is defined by 휇푡(푥) = 푖푛푓{‖푥퐸‖: 퐸 ∈ ℳ푝푟표푗, 휏(1 − 퐸) ≤ 푡}. See [81] for basic properties and detailed information on the generalized 푠 −numbers. Let 푥 be a τ-measurable operator in 퐿푝(ℳ)with 0 < 푝 ≤ ∞. The Fuglede-Kadison determinant ∆(푥) is defined by ∞

∆(푥) = 푒푥푝(휏(푙표푔 |푥|)) = 푒푥푝 ∫ 푙표푔 푡푑휈|푥|(푡), 0

112 where dν|x| denotes the probability measure on ℛ+ which is obtained by composing the spectral measure of |푥| with the trace τ. We refer to [25, 84] for more information on determinant in the case of bounded operators, and to [42, 111] for unbounded operators. Definition (5.2.1)[142]:Let x be a τ-measurable operator affiliated with a finite von Neumann algebra ℳ and 0 < 푝, 푞 ≤∞, define 1 1 ∞ 푑푡 푞 (∫ (푡푝휇 (푥))푞 ) , 푖푓 푞 < ∞, 0 푡 푡 ‖푥‖ 푝,푞 = (13) 퐿 (ℳ) 1 푝 푠푢푝 푡 휇푡(푥), 푖푓 푞 = ∞ . { 푡>0 푝,푞 The set of all 푥 ∈ ℳ̃ with ‖푥‖퐿푝,푞(ℳ) < ∞ is denoted by 퐿 (ℳ) and is called the noncommutative Lorentz space with indices p and q. Note that 1 1 1 1 (i) If 1 < 푝 < ∞, 1 ≤ 푞 < ∞, and + = 1, + = 1, then by Xu [325], we obtain the 푝 푝′ 푞 푞 following result (퐿푝,푞(ℳ))∗ = 퐿푝′,푞′ (ℳ). For more information on 퐿푝,푞(ℳ) we refer to [81, 315]. (ii) Since 휏(1) = 1, in (13) we can write 1 ∞ 푞 1 푞 푑푡 ‖푥‖ 푝,푞 = (∫(푡푝휇 (푥)) ) , 푖푓 푞 < ∞. 퐿 (ℳ) 푡 푡 0 Definition (5.2.2)[142]:Let 𝒜 be a finite subdiagonal algebra of ℳ. For 0 < 푝, 푞 < ∞, we define the noncommutative Hardy-Lorentz spaces to be the closure of 𝒜 in 퐿푝,푞(ℳ), denoted by 퐻푝,푞(ℳ) Lemma (5.2.3)[142]: Let 0 < 푝1 < 푝 < ∞, 0 < 푞, 푠 < ∞, then 퐿푝,푞(ℳ) ⊂ 퐿푝1,푠(ℳ). Consequently, 퐻푝,푞(𝒜) ⊂ 퐻푝1,푠(𝒜). Proof: Similarly to the proof of [112] we can prove that 퐿푝,푞(ℳ) ⊂ 퐿푝,∞(ℳ) with 푞 < ∞, 푝1,푢 푝1,푠 and 퐿 (ℳ) ⊂ 퐿 (ℳ) with 푢 ≤ 푠. Now it suffices to prove that ‖푥‖퐿푝1,푢(ℳ) ≤ 푝,∞ 푝,∞ 퐶‖푥‖퐿푝,∞(ℳ), ∀ 푥 ∈ 퐿 (ℳ) and 0 < 푢 < ∞. Indeed, ∀ 푥 ∈ 퐿 (ℳ), we have

113

1 1 1 푢 1 푢 1 푢 푢 1 푑푡 − −1 푝1 푢 푝1 푝 푝1 푢 ‖푥‖ 푝1,푢 = {∫(푡 휇 (푥)) } = {∫ 푡 (푡 휇 (푥)) 푑푡 } 퐿 (ℳ) 푡 푡 푡 0 0 1 1 1 푢 1 푢 푢 푢 1 푢 푢 − −1 − −1 푝1 푝 푝 푢 푝1 푝 ≤ {∫ 푡 ( 푠푢푝 푠 휇푠(푥)) 푑푡 } = ‖푥‖퐿푝,∞(푀) {∫ 푡 푑푡 } 0<푠<휏(1) 0 0 1 푝,∞ = 1 ‖푥‖퐿 (푀) 푢 푢 푢 − (푝1 푝) which gives the first inclusion of the lemma. Consequently, we obtain 퐻푝,푞(𝒜) ⊂ 퐻푝1,푢(𝒜) ⊂ 퐻푝1,푠(𝒜). Definition (5.2.4)[142]:Let 푥 be a τ-measurable operator affiliated with a finite von Neumann algebra ℳ and 0 < 푝, 푞 ≤ ∞, define

1 ∞ 푞 1 푞 푑푡 (∫ [푡푝푥∗∗(푥)] ) , 푖푓 푞 < ∞, ∗ ‖푥‖ 푝,푞 = 푡 (14) 퐿푟 (ℳ) 0 1 푠푢푝 푡푝푥∗∗(푥), 푖푓 푞 = ∞ . { 푡>0 1 푡 1 where 푥∗∗(푡) = ( ∫ (휇 (푥))푟푑푠)푟 , 0 < 푟 ≤ 푚푖푛 (1, 푞), 푟 < 푝. The set of all 푥 ∈ ℳ̃ with 푡 0 푠 ∗ 푝,푞 ‖푥‖ 푝,푞 < ∞ is denoted by 퐿푟 (ℳ). 퐿푟 (ℳ) Lemma (5.2.5)[142]:Let 0 < 푝, 푞 < ∞, then 1 ∗ 푝 ‖푥‖퐿푝,푞(ℳ) ≤ ‖푥‖ 푝,푞 ≤ 푒 ‖푥‖퐿푝,푞(ℳ), 퐿푟 (ℳ) where r is as in Definition (5.2.4). Proof: The first inequality is an immediate result from the following estimate 1 푡 푟 1 휇 (푥)) ≤ ( ∫(휇 (푥))푟푑푠) . 푡 푡 푠 0 Now we turn to prove the second inequality. Hardy’s first inequality of [119] tells us that

114

1 푞 ∞ 푡 푟 ∞ 푡 푟 1 1 푞 푞 − 푑푡 −( − )−1 ∗푞 푝 푟 푟 푟 푝 푟 ‖푥‖ 푝,푞 = ∫ 푡 (∫(휇푠(푥)) 푑푠) = ∫ 푡 (∫(휇푠(푥)) 푑푠) 푑푡 퐿푟 (ℳ) 푟 0 0 0 0 [ ] 푞 푞 푟 푡 푞 푡 푞 푞 푞 푠 −( − )−1 푞 푟 푑푠 푟 푟 푟 푝 푝 푞 ≤ (푞 푞 ) ∫[푠(휇푠(푥)) ]푟푠 푑푠 = ( ) ∫ 푠 (휇푠(푥)) − 푝 − 푟 푠 푟 푝 0 0 푞 푞 푟 푞 푞 푝 푞 = ( ) ‖푥‖ 푞,푞 ≤ 푒 ‖푥‖ 푞,푞 . 푝 − 푟 퐿 (ℳ) 퐿 (ℳ)

Lemma (5.2.6)[142]:Let 0 < 푝, 푞 < ∞, assume ℳ has no minimal projection, then 1 ∗ ∗ 푝 ‖휀(푎)‖ 푝,푞 ≤ ‖푎‖ 푝,푞 ; ‖휀(푎)‖퐿푝,푞(ℳ) ≤ 푒 ‖푎‖퐿푝,푞(ℳ), 퐿푟 (ℳ) 퐿푟 (ℳ) where r is as in Definition (5.2.4). Proof : [81] gives that 푡 푡 푟 푟 푟 ∫(휇푠(휀(푎)) 푑푠 = ∫ 휇푠(|휀(푎)| )푑푠 = 푠푢푝{휏(푒|휀(푎)| 푒) : 푒 ∈ 풩푝푟표푗, 휏(푒) ≤ 푡} , 푡 0 0 where 풩 is a von Neumann subalgebra generated by all spectral projections of|휀(푎)|. It is clear ∗ that 풩푝푟표푗 ⊂ 퐷 = 𝒜 ∩ 𝒜 , then we get 푡 푟 ∫ (휇푠(휀(푎))) 푑푠 = 푠푢푝{휏(|휀(푎)푒|푟) : 푒 ∈ 풩푝푟표푗, 휏(푒) ≤ 푡} 푡 0 ≤ 푠푢푝{휏(|휀(푎)푒|푟) ∶ 푒 ∈ 퐷, 휏(푒) ≤ 푡} 푡 = 푠푢푝{휏(|휀(푎)푒|푟) ∶ 푒 ∈ 퐷, 휏(푒) ≤ 푡} 푡 푟 ≤ 푠푢푝{‖휀(푎푒)‖푟 ∶ 푒 ∈ 퐷, 휏(푒) ≤ 푡}. 푡 푡 푟 ≤ ∫(휇푠 (푎)) 푑푠. 0 It follows that 푞 푞 1 1 푟 1 1 푟 푞 푞 1 푑푡 1 푟 푑푡 ∗푞 푝 푟 푝 ‖휀(푎)‖ 푝,푞푟 = ∫ 푡 ( ∫(휇 (휀(푎))) 푑푠) ≤ ∫ 푡 ( ∫(휇 (푎)) 푑푠) 퐿 (ℳ) 푡 푠 푡 푡 푠 푡 0 0 0 0 ∗푞 = ‖(푎)‖ 푝,푞 퐿푟 (ℳ), i.e., 1 ∗ ∗ 푝 ‖휀(푎)‖퐿푝,푞 (ℳ) ≤ ‖휀(푎)‖ 푝,푞 ≤ ‖푎‖ 푝,푞 ≤ 푒 ‖휀(푎)‖퐿푝,푞 (ℳ) 퐿푟 (ℳ) 퐿 (ℳ)

115

∗ Lemma (5.2.7)[142]: Let 0 < 푟1 < 푟2 ≤ 1, 푟1 < 푟2 < 푝, 푟1 < 푟2 ≤ 푞, then ‖푥‖퐿푞,푞 (ℳ)is 푟1 ∗ equivalent to ‖푥‖퐿푞,푞 (ℳ). 푟2 1 1 1 1 1 1 Proposition (5.2.8)[142]:Let 0 < 푝, 푝0, 푝1, 푞, 푞0, 푞1 < ∞such that = + , = + , 푝 푝0 푝1 푞 푞0 푞1 then 1 푝 ‖푦푧‖퐿푝,푞(ℳ) ≤ 푒 ‖푦‖퐿푝0,푞0 (ℳ)‖푧‖퐿푝0,푝1 (ℳ), where푦 ∈ 퐿푝0,푞0(ℳ), 푧 ∈ 퐿푝1,푞1(ℳ). Proof: Let 0 < 2푟 < 푚푖푛(1, 푝, 푞), we have

1 1 1 푡 푟 1 푡 2푟 푡 2푟 1 1 푟 (푦푧)∗∗(푡, 푟) = ( ∫(휇 (푦푧))푟푑푠) ≤ ( ) (∫(휇 (푦푧))2푟푑푠) (∫(휇 (푧))2푟푑푠) 푡 푠 푡 푠 푠 0 0 0 = 푦∗∗(푡, 2푟)푧∗∗(푡, 2푟). Combing the above estimate with Lemma (5. 2.5) we infer that 1 ∞ 푞 1 푑푡 ∗ 푝 ∗∗ 푞 ‖푦푧‖퐿푝,푞(ℳ) ≤ ‖푦푧‖ 푝,푞 = (∫(푡 (푦푧) (푡, 푟)) ) 퐿푟 (ℳ) 푡 0 1 ∞ 푞 1 1 푑푡 ≤ (∫(푡푝0 푦∗∗ (푡, 푟))푝1푧∗∗(푟, 2푟)푞 ) 푡 0 1 1 ∞ 푞0 ∞ 푞1 1 푑푡 1 푑푡 ≤ (∫(푡푝0 푦∗∗ (푟, 2푟)푞0 ) (∫(푡푝1푧∗∗ (푟, 2푟)푞1 ) 푡 푡 0 0 1 1 ∗ ∗ 푝0 푝1 = ‖ 푦‖ 푝0,푞0 ‖푧‖ 푝1,푞1 ≤ (푒 ‖ 푦‖퐿푝0,푞0 (ℳ))(푒 ‖푧‖퐿푝1,푞1 (ℳ)) 퐿2푟 (ℳ) 퐿2푟 (ℳ) 1 푝 = 푒 ‖푦‖퐿푝0,푞0(ℳ)‖푧‖퐿푝1,푞1 (ℳ), which gives the result. Lemma (5.2.9)[142]:Let 1 ≤ 푝, 푞 < ∞, then 푝,푞 푝,푞 퐻 (𝒜) = {푥 ∈ 퐿 (ℳ): 휏(푥푎) = 0, ∀ 푎 ∈ 퐴0} . Proof: From [274], we deduce that 푝,푞 푝,푞 퐻 (𝒜) = {푥 ∈ 퐿 (ℳ): 휏(푥푎) = 0, ∀ 푎 ∈ 퐴0} . Conversely, we assume that there exists some 푝,푞 푥 ∈ {푧 ∈ 퐿 (ℳ): 휏(푧푎) = 0, ∀ 푎 ∈ 𝒜0} , and 푥 ∉ 퐻푝,푞(𝒜). Hence, there exists some y ∈ Lp′,q′(ℳ) such that 휏(푥푦) ≠ 0 and 휏(푦푎) = 0, ∀ 푎 ∈ 퐻푝,푞(𝒜). Putting 1 ≤ 푟 < 푚푖푛(푝′, 푞′), we have 푦 ∈ 퐿푟(ℳ) and 휏(푦푎) = 0, ∀ 푎 ∈ 푟 푠 𝒜. [39] implies that 푦 ∈ 퐻0 (𝒜). Let 1 ≤ 푠 < 푚푖푛(푝, 푞), then 푥 ∈ {푧 ∈ 퐿 (ℳ): 휏(푧푎) = 116

푠 0, ∀ 푎 ∈ 𝒜0} = 퐻 (𝒜). Consequently, adapting [40] we deduce that 휏(푥푦) = 휏(휀(푥푦)) = 휏(휀(푥)휀(푦)) = 0. This is a contradiction. Proposition (5.2.10)[142]:Let 1 ≤ 푝, 푞 < ∞, 1 ≤ 푟 < 푚푖푛(푝, 푞), then 퐻푟(𝒜) ∩ 퐿푝,푞(ℳ) = 퐻푝,푞(𝒜). Proof: It is easy to verify that 퐻푝,푞(𝒜) ⊂ 퐻푟(𝒜) ∩ 퐿푝,푞(ℳ). Conversely, let 푥 ∈ 퐻푟(𝒜) ∩ 푝,푞 푟 푝,푞 퐿 (ℳ), then 푥 ∈ {푧 ∈ 퐿 (ℳ): 휏(푧푎) = 0, ∀푎 ∈ 𝒜0}. Therefore, 푥 ∈ 퐻 (𝒜) in view of Lemma (5.2.9) The following result describes the Szeg표̈ type factorization theorem for noncommutative Hardy-Lorentz spaces, and we refer to see Theorem (5.2.18) below for an improvement. 푝1,푞1 Theorem (5.2.11)[142]: Let 0 < 푝1, 푝2, 푞1, 푞2 < ∞. Let 휔 ∈ 퐿 (ℳ) be an invertible operator such that ω−1 ∈ Lp2,q2 (ℳ), then there exist a unitary 푢 ∈ ℳ and ℎ ∈ 퐻푝1,푞1(𝒜) such that 휔 = 푢ℎ and ℎ−1 ∈ 퐻푝2,푞2(𝒜). Proof: Let 휔 ∈ 퐿푝1,푞1(ℳ) be an invertible operator such that 휔−1 ∈ 퐿푝2,푞2 (ℳ). Take 0 < 푟1 −1 푟2 푟1 < 푚푖푛(푝1, 푞1), 0 < 푟2 < 푚푖푛(푝2, 푞2), then 휔 ∈ 퐿 (ℳ) and 휔 ∈ 퐿 (ℳ). By [39], there exist a unitary 푢 ∈ ℳand ℎ ∈ 퐻푟1(𝒜) such that 휔 = 푢ℎ and ℎ−1 ∈ 퐻푟2(𝒜). ∗ 푝1,푞1 We first consider the case 푚푖푛(푝1, 푝2, 푞1, 푞2) > 1. Since ℎ = 푢 휔 ∈ 퐿 (ℳ), applying Proposition (5.2.10), we conclude that ℎ ∈ 퐻푝1,푞1(𝒜). Similarly, ℎ−1 ∈ 퐻푝2,푞2 (𝒜). On the other hand, if 푚푖푛(푝1, 푝2, 푞1, 푞2) ≤ 1, we choose an integer n such that 푚푖푛(푛푝1, 푛푞1, 푛푝2, 푛푞2) > 1. Let 휔 = 푣|휔| be the polar decomposition of 휔. Note that 푣 ∈ 1 1 1 1 푀 is a unitary. Write 휔 = 푣|휔|푛|휔|푛 … | |휔|푛 = 휔1휔2 · · · 휔푛, where 휔1 = 푣 |휔|푛 , 휔푘 = 1 푛푝1,푛푞1 −1 푛푝1,푛푞1 |휔|푛 , 2 ≤ 푘 ≤n. Since 휔푘 ∈ 퐿 (ℳ) and 푤푘 ∈ 퐿 (ℳ), by what is already proved in the first part, we have a factorization 휔푛 = 푢푛ℎ푛 with 푢푛 ∈ ℳ a unitary, ℎ푛 ∈ 푛푝1,푛푞1 −1 푛푝2,푛푞2 퐻 (𝒜)such that h푛 ∈ 퐻 (𝒜) . Repeating this argument, we can get a similar factorization for 휔푛−1푢푛: 휔푛−1푢푛 = 푢푛−1ℎ푛−1, and then for 휔푛−2푢푛−1, and so on. In this way 푛푝1,푛푞1 we obtain a factorization: 휔 = 푢ℎ1ℎ2 … ℎ푛, where 푢 ∈ 푀 is a unitary, ℎ푘 ∈ 퐻 (𝒜)such −1 푛푝2,푛푞2 that ℎ푘 ∈ 퐻 (𝒜),1 ≤ 푘 ≤ 푛. Setting ℎ = ℎ1ℎ2 · · · ℎ푛, we see 휔 = 푢ℎ is the desired factorization. Corollary (5.2.12)[142]:Let0 < 푝, 푞 < ∞, 0 < 푟 < 푚푖푛(푝, 푞), 0 < 푠 < ∞, then 퐻푟,푠(𝒜) ∩ 퐿푝,푞(ℳ) = 퐻푝,푞(𝒜), 푟,푠 푝,푞 푝,푞 퐻0 (𝒜) ∩ 퐿 (ℳ) = 퐻 (𝒜). Proof: It is clear that 퐻푝,푞(𝒜) ⊂ 퐻푟,푠(𝒜) ∩ 퐿푝,푞(ℳ). 1 To prove the converse inequality, fix an 푥 ∈ 퐻푟,푠(𝒜) ∩ 퐿푝,푞(ℳ) and set 휔 = (푥∗푥 + 1)2 , then we see 휔 ∈ 퐿푝,푞(ℳ)and 휔−1 ∈ ℳ. Applying theorem (5.2.11) to ω, we get a unitary푢 ∈ ℳ and an invertible ℎ ∈ 퐻푝,푞(𝒜) such that 휔 = 푢ℎ and ℎ−1 ∈ 𝒜. Then we obtain ℎ∗ℎ = 푥∗푥 + 1. Since |ℎ| ≥ |푥|, there is a contraction 푣 ∈ ℳ such that 푥 = 푣ℎ. It follows that 푣 = 푥ℎ−1 ∈ 퐻푟,푠(𝒜) ∩ ℳ, therefore, we obtain that 푣 ∈ 𝒜. Consequently, 푥 ∈ 𝒜 · 퐻푝,푞(𝒜) = 퐻푝,푞(𝒜), and we conclude the first inequality. The later equality is immediate established by adapting the similar proof.

117

푝1,푞1 Proposition (5.2.13)[142]:Let 0 < 푝2 < 푝1 < ∞, 0 < 푞1, 푞2 < ∞ and ℎ ∈ 퐻 (𝒜), then: 푞2,푞2 푝1,푞1 (i) [ℎ𝒜]퐿푝2,푞2 (ℳ) = 퐻 (𝒜) if and only if [ℎ𝒜]퐿푝1,푞1 (ℳ) = 퐻 (𝒜). 푞2,푞2 푝1,푞1 (ii) [ℎ𝒜]퐿푝2,푞2 (ℳ) = 퐻 (𝒜) if and only if [ℎ𝒜]퐿푝1,푞1 (ℳ) = 퐻 (𝒜). 푞2,푞2 푝1,푞1 (iii) [𝒜ℎ𝒜]퐿푝2,푞2 (ℳ) = 퐻 (𝒜) if and only if [𝒜ℎ𝒜]퐿푝1,푞1 (ℳ) = 퐻 (𝒜). Proof :We shall prove only the third equivalence. The proofs of the others are similar. 푝1,푞1 푝1,푞1 푝1,푞1 First, if[𝒜ℎ𝒜]퐿푝1,푞1 (ℳ) = 퐻 (𝒜), from the density of 퐻 (𝒜) in 퐻 (𝒜), we see 푝2,푞2 that [𝒜ℎ𝒜]퐿푝2,푞2 (ℳ) = 퐻 (𝒜). To prove the converse implication, when 푝1, 푞1 ≥ 1, let 푥 ∈ 퐿푝1′,푝1′(ℳ) with 휏(푥푎ℎ푏) = 0, ∀ 푎, 푏 ∈ 𝒜, 1 then 푥푎ℎ ∈ 퐻0 (𝒜), where 푝1′, 푝1′is respectively the conjugate index of 푝1, 푝1. On the other 푝2,푞2 hand, by the condition that [𝒜ℎ𝒜]퐿푝2,푞2 (ℳ) = 퐻 (𝒜), there exist two sequences (푎푛), (푏푛) ⊂ 𝒜 such that 푝 ,푞 ‖푎푛ℎ푏푛 − 1‖퐻 2 2(𝒜) → 0, 푛 → ∞. 1 1 1 1 1 1 Let 푟 > 0, 푠 > 0 be such that = + , = + . Proposition (5.2.8) gives that 푟 푝1′ 푝2 푠 푞1′ 푞2 ‖푥푎푛ℎ푏푛 − 푥‖퐿푟,푠 (ℳ) ≤ 푐‖푥‖퐿푝1′,푝1′(ℳ)‖푥푎푛ℎ푏푛 − 1‖퐿푝2,푞2(ℳ) → 0, 푛 → ∞.

Consequently, we get

‖푥푎푛ℎ푏푛 − 푥‖퐿푟,푠(ℳ) → 0, 푛 → ∞. 1 푟,푠 푟,푠 Since 푥푎푛ℎ푏푛 = (푥푎푛ℎ)푏푛 ∈ 퐻0 (𝒜) ⊂ 퐻0 (𝒜), we know that 푥 ∈ 퐻0 (𝒜) ∩ 퐿푝1′,푝1′(ℳ) = 퐻푝1′,푝1′(𝒜) . Hence 휏(푥푦) = 0, ∀ 푦 ∈ 퐻푝1,푞1(𝒜). It follows that 푝1,푞1 [𝒜ℎ𝒜]퐿푝1,푞1 (ℳ) = 퐻 (𝒜). Now we assume 푚푖푛(푝1, 푞1, 푝2, 푞2) < 1. Choose an integer 푛 such that 푚푖푛(푛푝1, 푛푞1, 푛푝2, 푛푞2) ≥ 1, then the conclusion of the previous case tells us that 푛푝1,푛푞1 푛푝1,푛푞1 푝1,푞1, [𝒜ℎ𝒜]퐿푛푝1,푛푞1(ℳ) = 퐻 (𝒜). Since 퐻 (𝒜)is dense in 퐻 (𝒜), the proof of the 푝 ,푞 푝1,푞1, first part implies that [𝒜ℎ𝒜]퐿 1 1(ℳ) = 퐻 (𝒜). The previous result justifies the relative independence of the indices p, q in the following definition. Definition (5.2.14)[142]: Let 0 < 푝, 푞 < ∞. An operator ℎ ∈ 퐻푝,푞(𝒜) is called left outer, 푝,푞 푝,푞 right outer or bilaterally outer according to [ℎ𝒜]퐿푝,푞(ℳ) = 퐻 (𝒜), [𝒜ℎ]퐿푝,푞(ℳ) = 퐻 (𝒜) 푝,푞 or [𝒜ℎ𝒜]퐿푝,푞(ℳ) = 퐻 (𝒜). Theorem (5.2.15)[142]:Let 0 < 푝, 푞 < ∞ and ℎ ∈ 퐻푝,푞(𝒜). (i) If ℎ is left or right outer, then∆(ℎ) = ∆(휀(ℎ)). Conversely, if ∆(ℎ) = ∆(휀(ℎ)) and ∆(ℎ) > 0, then h is left and right outer (so bilaterally outer too). (ii) ii) If 𝒜 is antisymmetric (i.e., 푑푖푚퐷 = 1) and ℎ is bilaterally outer, then ∆(ℎ) = ∆(휀(ℎ)). Proof: Let ℎ ∈ 퐻푝,푞(𝒜). Putting 0 < 푟 < 푚푖푛(푝, 푞) < ∞ we obtain that ℎ ∈ 퐻푟(𝒜).Proposition (5.2.13)and [39] imply that (i) and (ii) hold. The following corollary is a consequence of this theorem. Corollary (5.2.16)[142]:Letℎ ∈ 퐻푝,푞(𝒜) and 0 < 푝, 푞 < ∞. 118

(i) If∆(ℎ) > 0, then h is left outer if and only if h is right outer. (ii) Assume that A is antisymmetric (i.e. , 푑푖푚퐷 = 1), then the following properties are equivalent: (a)ℎ is left outer; (b)ℎ is right outer; (c)ℎ is bilaterally outer; (d) ∆(휀(ℎ)) = ∆(ℎ) > 0. We will say that h is outer if it is at the same time left and right outer. If ℎ ∈ 퐻푝,푞(퐴)with ∆(ℎ) > 0, then ℎ is outer if and only if∆(ℎ) = ∆(휀(ℎ)). Also in the case where 퐴 is antisymmetric (i.e., dim퐷 = 1), an h with ∆(ℎ) > 0 is outer if and only if it is left, right or bilaterally outer. Corollary (5.2.17)[142]:Let ℎ ∈ 퐻푝1,푞1 (𝒜) be such that ℎ−1 ∈ 퐻푝2,푞2(𝒜) with 0 < 푝1, , 푝2, 푞1 , 푞2 < ∞,the 𝒜n h is outer. 푝1,푞1 −1 푝2,푞2 Proof: Let ℎ ∈ 퐻 (𝒜) be such that ℎ ∈ 퐻 (𝒜). Taking 0 < 푟 < 푚푖푛(푝1, 푞1) < 푟 −1 푠 ∞, 0 < 푠 < 푚푖푛(푝2, 푞2) < ∞,we get ℎ ∈ 퐻 (퐴) and ℎ ∈ 퐻 (𝒜). By virtue of Proposition (5.2.13) and [40], we see that ℎ is outer. The following theorem improves Theorem (5.2.11). Theorem (5.2.18)[142]: Let 휔 ∈ 퐿푝,푞(ℳ) with 0 < 푝, 푞 < ∞ such that ∆(ω) > 0, then there exista unitary 푢 ∈ ℳ and an outer ℎ ∈ 퐻푝,푞(𝒜) such that ω = uh. 1 Proof: Write the polar decomposition of 휔: 휔 = 푣|휔|. For|ω|2 , by virtue of Theorem 1 2푝,2푞 (5.2.11) we get a factorization : |휔|2 = 푢2ℎ2, with 푢2unitary and ℎ2 ∈ 퐻 (𝒜) left outer. 1 Since ∆(ℎ2) > 0, ℎ2 is also right outer, it follows that ℎ2 is outer. Similarly we have: 푣|휔|2푢2 = 푢1ℎ1. This tells us that 푢 = 푢1, ℎ = ℎ1ℎ2 yield the desired factorization of ω. We present the inner-outer factorization for operators in 퐻푝,푞(𝒜). Corollary (5.2.19)[142]:Let 0 < 푝, 푞 < ∞and 푥 ∈ 퐻푝,푞(𝒜) with ∆(푥) > 0, then there exist a unitary 푢 ∈ 퐴(inner) and an outer ℎ ∈ 퐻푝,푞(𝒜) such that 푥 = 푢ℎ. Proof: Let 푥 ∈ 퐻푝,푞(𝒜) with ∆(푥) > 0. Applying the previous theorem, we get 푥 = 푢ℎwith p,q h outer and u unitary in ℳ. Let (푎푛) ⊂ 𝒜 such that 푙푖푚ℎ푎푛 = 1 inH (𝒜), then 푢 = 푝,푞 푝,푞 푙푖푚푥푎푛in 퐻 (𝒜), which implies that 푢 ∈ 퐻 (𝒜) ∩ ℳ = 𝒜. Corollary (5.2.20)[142]:Let 0 < 푝, 푞 < ∞ and ℎ ∈ 퐻푝,푞(𝒜) with ∆(ℎ) > 0, then h is outer if and only if for any 푥 ∈ 퐻푝,푞(𝒜) with |푥| = |ℎ|, we have ∆(휀(푥)) ≤ ∆(휀(ℎ)). Proof: Let ℎ be outer and 푥 ∈ 퐻푝,푞(𝒜) with |푥| = |ℎ|,. Taking 0 < 푟 < 푚푖푛(푝, 푞) < ∞ we obtain that 푥 ∈ 퐻푟(𝒜). From [39], we get ∆(휀(푥)) ≤ ∆(휀(ℎ)). Conversely, let ℎ = 푢푘 be the decomposition given by Theorem (5.2.18) with 푘 outer. It is easy to check that ∆(ℎ) = ∆(푘) = ∆(휀(푥)) ≤ ∆(휀(ℎ)). Putting 0 < 푠 < 푚푖푛(푝, 푞) < ∞ we get ℎ ∈ 퐻푠(𝒜). Hence, [39] tells us that ∆(휀(푥)) ≤ ∆(휀(ℎ)). Consequently, ∆(휀(푥)) ≤ ∆(휀(ℎ)). So ℎ is outer due to Theorem (5.2.15). Lemma (5.2.21)[270]: Let 휀 > −1 then 퐿1+2휀,1+휀(ℳ) ⊂ 퐿1+휀,1+2휀(ℳ). Consequently, 119

퐻1+2휀,1+휀(𝒜) ⊂ 퐻1+휀,1+2휀(𝒜). Proof: Similarly to the proof of [112] we can prove that 퐿1+2휀,1+휀(ℳ) ⊂ 퐿1+2휀,∞(ℳ) with 휀 > −1, and 퐿1+휀,1+휀(ℳ) ⊂ 퐿1+휀,1+2휀(ℳ) with 휀 ≥ 0. Now it suffices to prove that 2 ̃ 2 2 1+2휀,∞ 2 ‖푥 ‖퐿1+휀,1+휀(ℳ) ≤ 퐶‖푥 ‖퐿1+2휀,∞(ℳ), ∀ 푥 ∈ 퐿 (ℳ) and 휀 > −1. Indeed, ∀ 푥 ∈ 퐿1+2휀,∞(ℳ), we have 1 1 1+휀 1 ( ) 2 2 1+휀 푑 1 + 휀 ‖푥 ‖ 1+휀,1+휀 = {∫((1 + 휀)1+휀휇 (푥 )) } 퐿 (ℳ) (1+휀) 1 + 휀 0 1 1 1+휀 1+휀 1 2 1+휀 = {∫(1 + 휀)1+2휀((1 + 휀)1+휀휇(1+휀)(푥 )) 푑(1 + 휀) } 0 1 1 1+휀 1+휀 1 2 1+휀 ≤ {∫(1 + 휀)1+2휀(푠푢푝(1 + 2휀)1+2휀 휇(1+휀)(푥 )) 푑(1 + 휀 )} 휀≥0 0 1 1 1+휀 1+휀 2 1+2휀 = ‖푥 ‖퐿1+2휀,∞(ℳ) {∫(1 + 휀) 푑(1 + 휀) } 0 which gives the first inclusion of the lemma. Consequently, we obtain 퐻1+2휀,1+휀(𝒜) ⊂ 퐻1+휀,1+휀(𝒜) ⊂ 퐻1+휀,1+2휀(𝒜). Lemma (5.2.22)[270]: Let 휀 > −1then 1 2 2 ∗ 1+휀 2 1+2휀,1+휀 ‖푥 ‖퐿1+2휀,1+휀(ℳ) ≤ ‖푥 ‖ 1+2휀,1+휀 (ℳ) ≤ 푒 ‖푥 ‖퐿 (ℳ), 퐿1+휀 Where 1 + 휀 is as in Definition (5.2.4). Proof The first inequality is an immediate result from the following estimate 1 1+휀 1+휀 1 휇 (푥2)) ≤ ( ∫ (휇 (푥2))(1+휀)푑(1 + 2휀)) . (1+휀) 1 + 휀 (1+2휀) 0 Now we turn to prove the second inequality. Hardy’s first inequality of [119] tells us that 1 ∞ 1+휀 1+휀 −휀 푑(1 + 휀) 2 ∗(1+휀) (1+2휀)(1+휀) 2 1+휀 ‖푥 ‖ 1+2휀,1+휀 = ∫ (1 + 휀) (∫ (휇(1+휀)(푥 )) 푑(1 + 휀)) 퐿1+휀 (ℳ) 1 + 휀 0 0 [ ] ∞ 1+휀 −1 2 1+휀 = ∫(1 + 휀)1+휀 (∫ (휇(1+휀)(푥 )) 푑(1 + 휀)) 푑(1 + 휀) 0 0

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∞ 1+휀 −1 1 2 ≤ ( 휀 ) ∫ [(1 + 휀) (휇(1+휀)(푥 )) ] (1 + 휀)1+휀푑푠 1 + 2휀 0 ∞ 1 + 2휀 1+휀 푑(1 + 휀) = ( ) ∫(1 + 휀)1+2휀(휇 (푥2))1+휀 휀 (1+휀) 1 + 휀 0 1+휀 1 + 2휀 2 1+휀 2 1+휀 = ( ) ‖푥 ‖ 1+2휀,1+휀 ≤ 푒1+2휀‖푥 ‖ 1+2휀,1+휀 . 휀 퐿 (ℳ) 퐿 (ℳ) Lemma (5.2.23)[270]: Let 휀 > −1 , assume ℳ has no minimal projection, then 1 2 ∗ 2 ∗ 2 1+휀 2 ‖휀(푎 )‖ 1+휀,1+휀 ≤ ‖푎 ‖ 1+휀,1+휀 ; ‖휀(푎 )‖퐿1+휀,1+휀(ℳ) ≤ 푒 ‖푎 ‖퐿1+휀,1+휀(ℳ), 퐿1+휀 (ℳ) 퐿1+휀 (ℳ) where 1 + 휀 is as in Definition (5.2.4). Proof : [81] gives that 1+휀 1+휀 2 1+휀 2 1+휀 ∫ (휇(1+2휀)(휀(푎 )) 푑(1 + 2휀) = ∫ 휇(1+2휀)(|휀(푎 )| )푑(1 + 휀) 0 0 2 1+휀 = sup {휏(푒|휀(푎 )| 푒) : 푒 ∈ 풩푝푟표푗, 휏(푒): 1 + 휀}. (1+휀) where 풩 is a von Neumann subalgebra generated by all spectral projections of |휀(푎2)|. It is ∗ clear that 풩푝푟표푗 ⊂ 퐷 = 𝒜 ∩ 𝒜 , then we get 1+휀 1+휀 2 2 1+휀 ∫ (휇(1+2휀)(휀(푎 ))) 푑(1 + 휀) = 푠푢푝 {휏(|휀(푎 )푒| ) : 푒 ∈ 풩푝푟표푗, 휏(푒) ≤ 1 + 휀} (1+휀) 0 ≤ 푠푢푝 {휏(|휀(푎2)푒|1+휀) ∶ 푒 ∈ 퐷, 휏(푒) ≤ 1 + 휀} (1+휀) = 푠푢푝 {휏(|휀(푎2)푒|1+휀) ∶ 푒 ∈ 퐷, 휏(푒) ≤ 1 + 휀} (1+휀) 2 1+휀 ≤ 푠푢푝 {‖휀(푎 푒)‖1+휀 ∶ 푒 ∈ 퐷, 휏(푒) ≤ 1 + 휀} (1+휀) 1+휀 2 1+휀 ≤ ∫ (휇(1+2휀) (푎 )) 푑(1 + 2휀). 0 It follows that 1 1 ( ) 2 ∗1+휀 1 2 1+휀 푑 1 + 휀 ‖휀(푎 )‖ 1+휀,1+휀 = ∫(1 + 휀) ( ∫(휇(1+2휀)(휀(푎 ))) 푑(1 + 휀)) 퐿1+휀 (ℳ) 1 + 휀 1 + 휀 0 0 1 1 ( ) 1 2 1+휀 푑 1 + 휀 2 ∗1+휀 ≤ ∫(1 + 휀) ( ∫(휇1+2휀(푎 )) 푑(1 + 휀)) = ‖(푎 )‖ 1+휀,1+휀 1 + 휀 1 + 휀 퐿1+휀 (ℳ), 0 0 i.e., 1 2 2 ∗ 2 ∗ 1+휀 2 ‖휀(푎 )‖퐿1+휀,1+휀 (ℳ) ≤ ‖휀(푎 )‖퐿1+휀,1+휀 (ℳ) ≤ ‖푎 ‖퐿1+휀,1+휀 (ℳ) ≤ 푒 ‖휀(푎 )‖퐿1+휀,1+휀 (ℳ)

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(1+휀)(2+5휀) (1+4휀)(2+11휀) Proposition (5.2.24)[270]: Let 0 < 휀 < ∞, such that 1 = , 1 = (1+2휀)(1+3휀) (1+5휀)(1+6휀) then 1 2 2 1+휀 2 2 ‖푦 푧 ‖퐿1+휀,1+4휀(ℳ) ≤ 푒 ‖푦 ‖퐿1+2휀,1+5휀 (ℳ)‖푧 ‖퐿1+3휀,1+6휀 (ℳ), where 푦2 ∈ 퐿1+2휀,1+5휀 (ℳ), 푧2 ∈ 퐿1+3휀,1+6휀 (ℳ). Proof:Let 0 < 2(1 + 휀) < 푚푖푛(1, 1 + 휀, 1 + 4휀), we have 1 1+휀 2(1+휀) 1 1+휀 (푦2푧2)∗∗((1 + 휀), 1 + 휀) = ( ∫ (휇 (푦2푧2)) 푑(1 + 2휀)) 1 + 휀 1+2휀 0 1 1 1+휀 2(1+휀) 1 1+휀 2(1+휀) ≤ ( ) (∫ (휇 (푦2푧2)) 푑(1 + 2휀)) 1 + 휀 (1+2휀) 0 1 1+휀 2(1+휀) 2 2(1+휀) × (∫ (휇1+2휀(푧 )) 푑(1 + 2휀)) 0 = (푦2)∗∗(1 + 휀, 2(1 + 휀))(푧2)∗∗(1 + 휀, 2(1 + 휀)). Combing the above estimate with Lemma (5.2.22) we infer that 2 2 2 2 ∗ ‖푦 푧 ‖퐿1+휀,1+4휀(ℳ) ≤ ‖푦 푧 ‖ 1+휀,1+4휀 퐿1+휀 (ℳ) 1 ∞ 1+4휀 1 푑(1 + 휀) = (∫((1 + 휀)1+휀 (푦2푧2)∗∗ (1 + 휀, 1 + 휀))(1+4휀) ) 1 + 휀 0 1 ∞ 1+4휀 1 1 (1+4휀) 푑(1 + 휀) ≤ (∫((1 + 휀)1+2휀 (푦2)∗∗ (1 + 휀, 1 + 휀))1+3휀(푧2)∗∗(1 + 휀, 2(1 + 휀)) ) 1 + 휀 0 1 ∞ 1+5휀 1 1 (1+5휀) 푑(1 + 휀) ≤ (∫((1 + 휀)1+2휀 (푦2)∗∗ (1 + 휀, 1 + 휀))1+2휀(푧2)∗∗(1 + 휀, 2(1 + 휀)) ) 1 + 휀 0 1 ∞ 1+6휀 1 푑(1 + 휀) × (∫((1 + 휀)1+2휀 (푦2)∗∗ (1 + 휀, 2(1 + 휀))(1+6휀) ) 1 + 휀 0 2 ∗ 2 ∗ = ‖ 푦 ‖ 1+2휀,1+5휀 ‖푧 ‖ 1+3휀,1+6휀 퐿2(1+휀) (ℳ) 퐿2(1+휀) (ℳ) 1 1 1+2휀 2 1+3휀 2 ≤ (푒 ‖ 푦 ‖퐿1+2휀,1+5휀 (ℳ))(푒 ‖푧 ‖퐿1+3휀,1+6휀 (ℳ)) 1 1+휀 2 2 = 푒 ‖푦 ‖퐿1+2휀,1+5휀 (ℳ)‖푧 ‖퐿1+3휀,1+6휀 (ℳ), which gives the result. 122

Lemma (5.2.25)[270]:Let 0 ≤ 휀 < ∞, then 1+휀,1+2휀 2 1+휀,1+2휀 2 2 2 퐻 (𝒜) = {푥 ∈ 퐿 (ℳ) ∶ 휏(푥 푎 ) = 0, ∀ 푎 ∈ 𝒜0} . Proof: From [274], we deduce that 1+휀,1+2휀 2 1+휀,1+2휀 2 2 2 퐻 (𝒜) = {푥 ∈ 퐿 (ℳ) ∶ 휏(푥 푎 ) = 0, ∀ 푎 ∈ 𝒜0} . Conversely, we assume that there exists some 2 2 1+휀,1+2휀 2 2 2 푥 ∈ {푧 ∈ 퐿 (ℳ) ∶ 휏(푧 푎 ) = 0, ∀ 푎 ∈ 𝒜0} , and 푥2 ∉ 퐻1+휀,1+2휀(𝒜). Hence, there exists some 푦2 ∈ 퐿(1+휀)′,(1+2휀)′(ℳ) such that 휏(푥2푦2 ) ≠ 0 and 휏(푦2푎2) = 0, ∀ 푎2 ∈ 퐻1+휀,1+2휀(𝒜). Putting 1 ≤ 1 + 휀 < 푚푖푛((1 + 휀)′, (1 + 2휀)′), we have 푦2 ∈ 퐿1+휀(푀) and 휏(푦2푎2) = 0, ∀ 푎2 ∈ 𝒜. [39] 2 1+휀 2 2 implies that 푦 ∈ 퐻0 (𝒜). Let 1 ≤ 1 + 휀 < 푚푖푛(1 + 휀, 1 + 2휀), then 푥 ∈ {푧 ∈ 1+휀 2 2 2 1+휀 퐿 (ℳ): 휏(푧 푎 ) = 0, ∀ 푎 ∈ 𝒜0} = 퐻 (𝒜). Consequently, adapting [39] we deduce that. 휏(푥2푦2) = 휏(휀(푥2푦2)) = 휏(휀(푥2)휀(푦2)) = 0 This is a contradiction. Proposition (5.2.26)[270]:Let 휀 ≥ 0,1 ≤ 1 + 휀 < 푚푖푛(1 + 휀, 1 + 2휀), then 퐻1+휀(𝒜) ∩ 퐿1+휀,1+2휀(ℳ) = 퐻1+휀,1+2휀(𝒜). Proof It is easy to verify that 퐻1+휀,1+2휀(𝒜) ⊂ 퐻1+휀(𝒜) ∩ 퐿1+휀,1+2휀(ℳ). Conversely, let 2 1+휀 1+휀,1+2휀 2 2 1+휀 2 2 2 푥 ∈ 퐻 (𝒜) ∩ 퐿 (ℳ), then 푥 ∈ {푧 ∈ 퐿 (ℳ) ∶ 휏(푧 푎 ) = 0, ∀푎 ∈ 𝒜0}. Therefore, 푥2 ∈ 퐻1+휀,1+2휀(𝒜) in view of Lemma (5.2.25) The following result describes the Szeg표̈ type factorization theorem for noncommutative Hardy-Lorentz spaces, (see [38]) and also see Theorem (5.2.33) below for an improvement. Theorem (5.2.27)[270]:Let−1 < 휀 < ∞, let 휔 ∈ 퐿1+휀,1+3휀(ℳ) be an invertible operatorsuch that 휔−1 ∈ 퐿1+2휀,1+4휀 (ℳ), then there exist a unitary 푢 ∈ ℳ and ℎ2 ∈ 퐻1+휀,1+3휀(𝒜) such that 휔 = 푢ℎ2 and ℎ−2 ∈ 퐻1+2휀,1+4휀(𝒜). Proof: Let 휔 ∈ 퐿1+휀,1+3휀(ℳ) be an invertible operator such that 휔−1 ∈ 퐿1+2휀,1+4휀 (ℳ). Take 0 < 1 + 휀 < 푚푖푛(1 + 휀, 1 + 3휀), 0 < 1 + 휀 < 푚푖푛(1 + 2휀, 1 + 4휀), then 휔 ∈ 퐿1+휀(ℳ) and 휔−1 ∈ 퐿1+휀(ℳ). By [4], there exist a unitary 푢 ∈ ℳ and ℎ2 ∈ 퐻1+휀(𝒜) such that 휔 = 푢ℎ2 and ℎ−2 ∈ 퐻1+휀(𝒜). We first consider the case 푚푖푛(1 + 휀, 1 + 2휀, 1 + 3휀, 1 + 4휀) > 1. Since ℎ2 = 푢∗휔 ∈ 퐿1+휀,1+3휀 (ℳ), applying Proposition (5.2.26), we conclude that ℎ2 ∈ 퐻1+휀,1+3휀(𝒜). Similarly, ℎ−2 ∈ 퐻1+2휀,1+4휀 (𝒜). On the other hand, if 푚푖푛(1 + 휀, 1 + 2휀, 1 + 3휀, 1 + 4휀) ≤ 1, we choose an integer 푛 such that 푚푖푛(푛(1 + 휀), 푛(1 + 2휀), 푛(1 + 3휀), 푛(1 + 4휀)) > 1. Let 휔 = 푣|휔| be the polar 1 1 1 decomposition of 휔. Note that 푣 ∈ ℳ is a unitary. Write 휔 = 푣|휔|푛|휔|푛 · · · | |휔|푛 = 1 1 휔1휔2 · · · 휔푛, where 휔1 = 푣 |휔|푛 , 휔2+휀 = |휔|푛 , 2 ≤ 2 + 휀 ≤n, 휀 ≥ 0. Since 휔2+휀 ∈ 푛(1+휀),푛(1+3휀) −1 푛(1+2휀),푛(1+4휀) 퐿 (ℳ) and 휔2+휀 ∈ 퐿 (ℳ), by what is already proved in the first 2 2 part,we have a factorization 휔푛 = 푢푛ℎ푛 with 푢푛 ∈ ℳ a unitary, ℎ푛 ∈ 푛(1+휀),푛(1+3휀) −2 푛(1+2휀),푛(1+4휀) 퐻 (𝒜) such that ℎ푛 ∈ 퐻 (𝒜). Repeating this argument, we can 2 get a similar factorization for 휔푛−1푢푛: 휔푛−1푢푛 = 푢푛−1ℎ푛−1, and then for 휔푛−2푢푛−1, and so 2 2 2 on. In this way we obtain a factorization: 휔 = 푢ℎ1ℎ2 … ℎ푛, where 푢 ∈ ℳ is a unitary, 2 푛(1+휀),푛(1+3휀) −2 푛(1+2휀),푛(1+4휀) ℎ2+휀 ∈ 퐻 (𝒜) such that ℎ2+휀 ∈ 퐻 (𝒜), 1 ≤ 1 + 휀 ≤ 푛, 휀 ≥ 0. 2 2 2 2 Setting ℎ = ℎ1ℎ2 … ℎ푛, we see 휔 = 푢푛 is the desired factorization. Corollary (5.2.28)[270]:Let, 0 < 1 + 휀 < 푚푖푛(1 + 휀, 1 + 2휀), −1 < 휀 < ∞, then 123

퐻1+휀,1+휀(𝒜) ∩ 퐿1+휀,1+2휀(ℳ) = 퐻1+휀,1+2휀(𝒜), 1+휀,1+휀 1+휀,1+2휀 1+휀,1+2휀 퐻0 (𝒜) ∩ 퐿 (ℳ) = 퐻 (𝒜).

Proof: It is clear that 퐻1+휀,1+2휀(𝒜) ⊂ 퐻1+휀,1+휀(𝒜) ∩ 퐿1+휀,1+2휀(ℳ). To prove the converse inequality, fix an 푥2 ∈ 퐻1+휀,1+휀(𝒜) ∩ 퐿1+휀,1+2휀(ℳ) and set = 1 ((푥2)∗푥2 + 1)2 , thenwe see 휔 ∈ 퐿1+휀,1+2휀(ℳ)and 휔−1 ∈ ℳ. Applying Theorem (5.2.27) to 휔, we get a unitary 푢 ∈ ℳand an invertible ℎ2 ∈ 퐻1+휀,1+2휀(𝒜) such that 휔 = 푢ℎ2 and ℎ−2 ∈ 𝒜. Then we obtain (ℎ2)∗ℎ2 = (푥2)∗(푥2) + 1. Since |ℎ2| ≥ |푥2|, there is a contraction 푣 ∈ ℳ such that 푥2 = 푣ℎ2. It follows that 푣 = 푥2ℎ−2 ∈ 퐻1+휀,1+휀(𝒜) ∩ ℳ, therefore, we obtain that 푣 ∈ 𝒜. Consequently, 푥2 ∈ 𝒜 · 퐻1+휀,1+2휀(𝒜) = 퐻1+휀,1+2휀(𝒜), and we conclude the first inequality. The later equality is immediate established by adapting the similar proof Proposition (5.2.29)[270]:Let−1 < 휀 < ∞, and ℎ2 ∈ 퐻1+2휀,1+휀(𝒜), then: 2 1+휀,1+2휀 2 1+2휀,1+휀 1+휀,1+2휀 1+2휀,1+휀, i) [ℎ 𝒜]퐿 (ℳ) = 퐻 (𝒜)if and only if [ℎ 𝒜]퐿 (ℳ) = 퐻 (𝒜). 2 1+휀,1+2휀 2 1+2휀,1+휀 1+휀,1+2휀 1+2휀,1+휀, ii) [ℎ 𝒜]퐿 (ℳ) = 퐻 (𝒜)if and only if [ℎ 𝒜]퐿 (ℳ) = 퐻 (𝒜). 2 1+휀,1+2휀 2 1+휀,1+2휀 1+2휀,1+휀, iii)[𝒜ℎ 𝒜]퐿 (ℳ) = 퐻 (𝒜)if and only if [𝒜ℎ 𝒜]퐿 (ℳ) = 퐻1+2휀,1+휀 (𝒜). Proof : We shall prove only the third equivalence. The proofs of the others are similar. 2 1+2휀,1+휀 1+2휀,1+휀 First, if [𝒜ℎ 𝒜]퐿1+2휀,1+휀(ℳ) = 퐻 (𝒜), from the density of 퐻 (𝒜) in 1+2휀,1+휀 2 1+휀,1+2휀 퐻 (𝒜), we see that [𝒜ℎ 𝒜]퐿1+휀,1+2휀(ℳ) = 퐻 (𝒜). To prove the converse implication, when 휀 ≥ 0, let 푥2 ∈ 퐿(1+2휀)′,(1+휀)′(ℳ) with 휏(푥2푎2ℎ2(푎2 + 휀)) = 0, ∀ 푎2, 푎2 + 휀 ∈ 𝒜, 2 2 2 1 then 푥 푎 ℎ ∈ 퐻0 (𝒜), where (1 + 2휀)′, (1 + 휀)′is respectively the conjugate index of 1 + 2 1+휀,1+2휀 2휀, 1 + 휀. On the other hand, by the condition that [𝒜ℎ 𝒜]퐿1+휀,1+2휀(푀) = 퐻 (𝒜), there 2 2 exist two sequences (푎푛 ), (푎푛 + 휀푛 ) ⊂ 𝒜 such that ‖푎 2ℎ2(푎 2 + 휀 ) − 1‖ → 0, 푛 → ∞. 푛 푛 푛 퐻1+휀,1+2휀(𝒜) (2+3휀 ′) Let 휀 > −1, be such that 1 = . Proposition (5.2.24) gives that (1+2휀 ′) ‖푥2푎 2ℎ2(푎 2 + 휀 ) − 푥2‖ 푛 푛 푛 퐿1+휀,1+휀 (ℳ) 2 2 2 2 ≤ 푐̃‖푥 ‖ (1+2휀)′,(1+휀)′ ‖푥 푎 ℎ(푎 + 휀 ) − 1‖ → 0, 푛 → ∞. 퐿 (ℳ) 푛 푛 푛 퐿1+휀,1+2휀 (ℳ) Consequently, we get ‖푥2푎 2ℎ2(푎 2 + 휀 ) − 푥2‖ → 0, 푛 → ∞. 푛 푛 푛 퐿1+휀,1+휀 (ℳ) 2 2 2 2 2 2 1 1+휀,1+휀 Since 푥 푎푛ℎ (푎푛 + 휀푛 ) = (푥 푎푛 ℎ)(푎푛 + 휀푛 ) ∈ 퐻0 (𝒜) ⊂ 퐻0 (𝒜), we know that 2 1+휀,1+휀 (1+2휀)′,(1+휀)′ (1+2휀)′,(1+휀)′ 2 2 푥 ∈ 퐻0 (𝒜) ∩ 퐿 (ℳ) = 퐻 (𝒜). Hence 휏(푥 푦 ) = 0, ∀ 푦2 ∈ 퐻1+2휀,1+휀 (𝒜). It follows that 2 1+2휀,1+휀 [𝒜ℎ 𝒜]퐿1+2휀,1+휀(ℳ) = 퐻 (𝒜).

124

Now we assume 푚푖푛(1 + 2휀, 1 + 휀, 1 + 휀, 1 + 2휀) < 1. Choose an integer n such that 푚푖푛(푛(1 + 2휀), 푛(1 + 휀), 푛(1 + 휀), 푛(1 + 2휀)) ≥ 1, then the conclusion of the previous 2 푛(1+2휀),푛(1+휀) 푛(1+2휀),푛(1+휀) case tells us that [𝒜ℎ 𝒜]퐿푛(1+2휀),푛(1+휀)(ℳ) = 퐻 (𝒜).Since 퐻 (𝒜) is 1+2휀,1+휀, 2 dense in 퐻 (𝒜), the proof of the first part implies that [𝒜ℎ 𝒜]퐿1+2휀,1+휀(ℳ) = 퐻1+2휀,1+휀,(𝒜). The previous result justifies the relative independence of the indices 1 + 휀, 1 + 2휀 in the following definition (see [38]). Theorem (5.2.30)[270]:Let −1 < 휀 < ∞ and ℎ2 ∈ 퐻1+휀,1+2휀(𝒜). (i) If ℎ2 is left or right outer, then ∆(ℎ2) = ∆(휀(ℎ2)). Conversely, if ∆(ℎ2) = ∆(휀(ℎ2)) and ∆(ℎ2) > 0, then h is left and right outer (so bilaterally outer too). (ii) ii) If 𝒜 is antisymmetric (i.e., 푑푖푚퐷 = 1) and ℎ2 is bilaterally outer, then ∆(ℎ2) = ∆(휀(ℎ2)). Proof Let ℎ2 ∈ 퐻1+휀,1+2휀(𝒜). Putting 0 < 1 + 휀 < 푚푖푛(1 + 휀, 1 + 2휀) < ∞ we obtain that ℎ2 ∈ 퐻1+휀(𝒜). Proposition (5.2.29)and [39] imply that (i) and (ii) hold. The following corollary is a consequence of this theorem. Corollary (5.2.31)[270]:Let ℎ2 ∈ 퐻1+휀,1+2휀(𝒜) and −1 < 휀 < ∞. i) If ∆(ℎ2) > 0, thenℎ2is left outer if and only if ℎ2 is right outer. ii) Assume that 𝒜 is antisymmetric (i.e. , 푑푖푚퐷 = 1), then the following properties areequivalent: (a) ℎ2is left outer; (b) ℎ2 is right outer; (c) ℎ2 is bilaterally outer; (d) ∆(휀(ℎ2)) = ∆(ℎ2) > 0. We will say that ℎ2 is outer if it is at the same time left and right outer. If ℎ2 ∈ 퐻1+휀,1+2휀(𝒜) with ∆(ℎ2) > 0, then ℎ2 is outer if and only if ∆(ℎ2) = ∆(휀(ℎ2)). Also in the case where 𝒜 is antisymmetric (i.e., 푑푖푚퐷 = 1), an ℎ2 with ∆(ℎ2) > 0 is outer if and only if it is left, right orbilaterally outer. Corollary (5.2.32)[270]: Let ℎ2 ∈ 퐻1+휀,1+3휀 (𝒜) be such that ℎ−2 ∈ 퐻1+2휀,1+4휀 (𝒜) with 0 < 휀 < ∞,then ℎ2 is outer. Proof: Let ℎ2 ∈ 퐻1+휀,1+3휀 (𝒜) be such that ℎ−2 ∈ 퐻1+2휀,1+4휀 (𝒜). Taking 0 < 1 + 휀 < 푚푖푛(1 + 휀, 1 + 3휀) < ∞, 0 < 1 + 휀 < 푚푖푛(1 + 2휀, 1 + 4휀) < ∞, we get ℎ2 ∈ 퐻1+휀(𝒜) and ℎ−2 ∈ 퐻1+휀(𝒜). By virtue of Proposition (5.2.29) and [39], we see that ℎ2 is outer. The following theorem improves Theorem (5.2.27). Theorem (5.2.33)[270]:Let 휔 ∈ 퐿1+휀,1+2휀(ℳ) with −1 < 휀 < ∞ such that ∆(휔) > 0, then there exist a unitary 푢 ∈ ℳand an outer ℎ2 ∈ 퐻1+휀,1+2휀(𝒜) such that 휔 = 푢ℎ2. 1 Proof Write the polar decomposition of 휔: 휔 = 푣|휔|. For |휔|2, by virtue of Theorem (5.2.27) 1 2 2 2(1+휀),2(1+2휀) we get a factorization: |휔|2 = 푢2ℎ2, with 푢2unitary and ℎ2 ∈ 퐻 (𝒜) left outer. 2 2 2 Since ∆(ℎ2 ) > 0, ℎ2 is also right outer, itfollows that ℎ2is outer. Similarly we have: 1 2 푣|휔|2푢2 = 푢1ℎ1. 125

2 2 2 This tells us that 푢 = 푢1, ℎ = ℎ1ℎ2 yield the desired factorization of 휔. Here we present the inner-outer factorization for operators in 퐻1+휀,1+2휀(𝒜). Corollary (5.2.34)[270]:Let −1 < 휀 < ∞ and 푥2 ∈ 퐻1+휀,1+2휀(𝒜) with ∆(푥2) > 0, then there exist aunitary 푢 ∈ 𝒜(inner) and an outer ℎ2 ∈ 퐻1+휀,1+2휀(𝒜) such that 푥2 = 푢ℎ2. Proof Let 푥2 ∈ 퐻1+휀,1+2휀(𝒜) with ∆(푥2) > 0. Applying the previous theorem, we get 푥2 = 2 2 2 2 푢ℎ with ℎ outer and 푢 unitary in ℳ. Let (푎푛 ) ⊂ 𝒜 such that 푙푖푚ℎ 푎푛 = 1 in 1+휀,1+2휀 2 2 1+휀,1+2휀 퐻 (𝒜), then 푢 = 푙푖푚푥 푎푛 in 퐻 (𝒜), which implies that 푢 ∈ 퐻1+휀,1+2휀(𝒜) ∩ 𝒜 = 𝒜. Corollary (5.2.35)[270]:Let −1 < 휀 < ∞ and ℎ2 ∈ 퐻1+휀,1+2휀(𝒜) with ∆(ℎ2) > 0, then ℎ2 is outer if and only if for any 푥2 ∈ 퐻1+휀,1+2휀(𝒜) with |푥2| = |ℎ2|, we have ∆(휀(푥2)) ≤ ∆(휀(ℎ2)). 퐏퐫퐨퐨퐟 Let ℎ2 be outer and 푥2 ∈ 퐻1+휀,1+2휀(𝒜) with |푥2| = |ℎ2|,. Taking 0 < 1 + 휀 < 푚푖푛(1 + 휀, 1 + 2휀) < ∞ we obtain that 푥2 ∈ 퐻1+휀(𝒜). From [39], we get ∆(휀(푥2)) ≤ ∆(휀(ℎ2)). Conversely, let ℎ2 = 푢푘2 be the decomposition given by Theorem (5.2.33) with 푘2 outer. It is easy to check that ∆(ℎ2) = ∆(푘2) = ∆(휀(푥2)) ≤ ∆(휀(ℎ2)). Putting 0 < 1 + 휀 < 푚푖푛(1 + 휀, 1 + 2휀) < ∞ we get ℎ2 ∈ 퐻1+휀(𝒜). Hence, [39] tells us that ∆(휀(푥2)) ≤ ∆(휀(ℎ2)). Consequently, ∆(휀(푥2)) ≤ ∆(휀(ℎ2)). So ℎ2 is outer due to Theorem (5.2.30). Section (5.3) Subdiagonal Subalgebras with Applications to Toeplitz Operators Let 핋 be the unit circle of the complex plane equipped with normalised Lebesgue 푝 measure 푑푚. We denote by 퐻 (핋) the usual Hardy spaces on 핋. Let 푃+ be the orthogonal projection from 퐿2(핋) onto 퐻2(핋). The classical Helson-SzegÖ theorem [101] (see also [141]), 2 characterises those positive measures 휇 on 핋 such that 푃+ is bounded on 퐿 (핋, 휇). The condition is that 휇 is absolutely continuous with respect to dm and the corresponding Radon- Nikod´ym derivative w satisfies 푢+푣̃ ∞ 휔 = 푒 for two functions 푢, 푣 ∈ 퐿 (핋) with ‖푣̃‖∞ < 휋/2, (15) where 푣̃ denotes the conjugate function of 푣. The motivation of this theorem comes from univariate prediction theory. Let 풫+ denote the space of all polynomials in 푧, and 풫− the space of all polynomials in 푧̅ without constant term. 2 풫 = 풫+ + 풫− is the space of all trigonometric polynomials. Then P+ is bounded on 퐿 (핋, 휇) if 2 and only if풫+ and 풫− are at positive angle in 퐿 (핋, 휇). Recall that the angle between풫+ and 풫− is defined as arccos of the following quantity

휌 = 푠푢푝{|∫핋 푓푔̅푑휇|: 푓 ∈ 풫+, 푔 ∈ 풫−, ‖푓‖퐿2(핋,휇) = ‖푔‖퐿2(핋,휇) = 1}. 2 Thus P+ is bounded on 퐿 (핋, 휇) if and only if 휌 < 1. In multivariate prediction theory one needs to consider the matrix-valued extension of the Helson-Szegö theorem. Let 필푛 denote the full algebra of complex 푛 × 푛-matrices, equipped with the normalised trace 푡푟. Let 풫+(필푛) denote the space of all polynomials in 푧 with coefficients in 필푛. 풫−(필n) and 풫(필푛) have similar meanings. Let 휔 be an 필푛-valued weight on 핋, i.e. 휔 is an integrable function on 핋 with values in the family of semidefinite nonnegative matrices. For any trigonometric polynomials f and g in 풫(필n) define ∗ 1/2 〈푓, 푔〉휔 = ∫핋푡푟(푔 푓휔)푑푚 푎푛푑 ‖푓‖휔 = ‖푓, 푓‖휔 , where 푎∗ denotes the adjoint of a matrix 푎. Like in the scalar case, define 126

∗ 휌 = 푠푢푝{|∫핋푡푟(푔 푓휔)푑푚|: 푓 ∈ 풫+(필푛), 푔 ∈ 풫−(필푛), ‖푓‖휔 = ‖푔‖휔 = 1}. Again, 휌 < 1 if and only if 푃+ ⊗ 퐼푑필푛 is bounded on 풫(필n) with respect to ‖. ‖푤. The problem here is, of course, to characterise 푤 such that 휌 < 1 in a way similar to the scalar case. This time the task is much harder, and it is impossible to find a characterisation as nice as (15). Numerous works have been devoted to this subject, see, for instance [223, 172, 171, 180, 106, 266]. In particular, Pousson’s characterisation in [106] is the matrix-valued analogue of a key intermediate step to (15). It is strong enough for applications to the invertibility of Toeplitz operators. The preceding two cases can be put into the more general setting of subdiagonal algebras in the sense of [300]. We will provide an extension of the Helson-Szegö theorem in this general setting. We study the invertibility of Toeplitz operators. It is well known that the Helson- Szegö theorem is closely related to the invertibility of Toeplitz operators. This relationship was remarkably exploited by Devinatz [7]. Pousson [105, 106] then subsequently extended Devinatz’s work to the matrix-valued case. Using our extension of the Helson- Szegö theorem, we will characterize the symbols of invertible Toeplitz operators in the very general setting of subdiagonal algebras. We end this introduction by mentioning the link between the Helson-Szegö theorem and Muckenhoupt’s 퐴2 weights. Let 휔 be a weight on 핋. Hunt, Muckenhoupt and Wheeden [243] 2 proved that the Riesz projection 푃+ is bounded on 퐿 (핋, 푤) if and only if 1 1 s푢푝 ∫ 휔 ∫ 휔−1 < ∞, (16) |퐼| 1 |퐼| 1 where the supremum runs over all arcs of 핋. Such a 휔 is called an 퐴2-weight. Thus for a weight 휔 the two conditions (15) and (16) are equivalent via the boundedness of the Riesz projection. It seems that it is still an open problem to find a direct proof of this equivalence. Hunt, Muckenhoupt and Wheeden’s theorem was extended to the matrix-valued case by Treil and Volberg [266]. Namely, let 푤 now be an 필n-valued weight on 핋. Then 푃+ ⊗ 퐼푑필푛 is bounded on 풫(필n) with respect to ‖. ‖휔 if and only if 1/2 1/2 1 1 −1 1 푠푢푝 ‖( ∫1휔) ( ∫1휔 ) ( ∫1휔) ‖ < ∞. 1 |퐼| |퐼| |퐼| 필푛 It is not clear for us how to extend Treil and Volberg’s theorem to the case of subdiagonal algebras. On the other hand, Hunt, Muckenhoupt and Wheeden also characterised the 푝 boundedness of 푃+ on 퐿 (핋, 휔)for any 1 < 푝 < ∞ by the so-called 퐴푝 weights. 퐴 well known open problem in matrix-valued harmonic analysis is to extend this result to the matrix-valued case; even to the very general one of subdiagonal algebras. 푀 will be a von Neumann algebra possessing a faithful normal tracial state 휏. The associated noncommutative 퐿푝-spaces are denoted by 퐿푝(푀). We refer to [89] for noncommutative 푝 푝 integration. For a subset 푆 of 퐿 (푀), we will write [푆]푝for the closure of 푆 in the 퐿 −topology. On the other hand, 푆∗ will denote the set of all Hilbert-adjoints of elements of 푆. When an actual Banach dual of some Banach space is in view, we will for the sake of avoiding confusion prefer the superscript ⋆ . For example the dual of 푀 will be denoted by 푀⋆. Because M is finite, 127 there will for any von Neumann subalgebra 푁 of 푀, always exist a normal contractive projection 휓: 푀 → 푁 satisfying 휏 ∘ 휓 = 휏.This is the so-called normal faithful conditional expectation onto 푁 with respect to 휏. A finite subdiagonal algebra of 푀 is a weak* closed unital subalgebra 퐴 of 푀 satisfying the following conditions (i) 퐴 + 퐴∗ is weak* dense in 푀; (ii) the trace preserving conditional expectation 훷: 푀 → 퐴 ∩ 퐴∗ = 퐷 is multiplicative on A: 훷(푎푏) = 훷(푎)훷(푏), 푎, 푏 ∈ 퐴. In this case, 퐷 is called the diagonal of 퐴. We also set 퐴0 = 퐴 ∩ 퐾푒푟(훷). In the sequel, 퐴 will always denote a finite subdiagonal algebra of 푀. Subdiagonal algebras are our noncommutative 퐻∞’s. The most important example is, of course, the classical 퐻∞(핋) on the unit circle. Another example important for multivariate ∞ ∞ prediction theory is the matrix-valued 퐻 (핋). More precisely, let 푀 = 퐿 (핋) ⊗ 필푛 = ∞ ∞ 퐿 (핋; 필푛) equipped with the product trace, and let 퐴 = 퐻 (핋; 필푛) the subalgebra of M consisting of 푛 × 푛-matrices with entries in 퐻∞(핋). Many classical results about Hardy spaces on 핋 have been transferred to the matrix-valued case. A third example is the upper triangle subalgebra 핋푛of 필푛. This example is closely related to the second one, and is a finite dimensional nest algebra. We refer to [89] for more information and historical references on subdiagonal algebras, in particular, on matrix-valued analytic functions. For 푝 < ∞ the Hardy space 퐻푝(퐴) associated with a finite subdiagonal algebra 퐴 is defined to 푝 푝 be [퐴]푝. The closure of 퐴0 in 퐿 (푀)will be denoted by 퐻0 (푀). By convention, we put ∞ ∞ p 퐻 (퐴) = 퐴 and 퐻0 (퐴) = 퐴0. These spaces exhibit many of the properties of classical H spaces (see [278, 64, 65, 175, 190, 152]). In particular for 1 < 푝 < ∞, 퐿푝(푀) appears as the 푝 푝 ∗ 푝 Banach space direct sum of 퐻 (푀) and 퐻0 (푀) , with 퐻 (푀) appearing as the Banach space 푝 푝 direct sum of 퐻0 (푀) and 퐿 (퐷). In the case 푝 = 2, these direct sums are even orthogonal direct sums. Recall that if a weight 휔 on 핋 satisfies (15), then necessarily 푙표푔휔 ∈ 퐿1(핋), or equivalently,

푒푥푝 (∫핋푙표푔휔) > 0. (17) The integrability of logw is also equivalent to the existence of an outer function ℎ ∈ 퐻1(푇) such that 휔 = |ℎ|. To state the outer-inner factorisation and prove the Helson-Szegö analogue for subdiagonal algebras, we need an appropriate substitute of the latter condition. This is achieved by the Fuglede-Kadison determinant. Recall that the Fuglede-Kadison determinant 훥(푎) of an operator 푎 ∈ 퐿푝(푀) (푝 > 0) can be defined by ∞

훥(푎) = 푒푥푝(휏(푙표푔 |푎|)) = 푒푥푝 (∫ 푙표푔 푡 푑휈|푎|(푡)) , 0 where 푑휈|푎|denotes the probability measure on ℝ+ which is obtained by composing the spectral measure of |a| with the trace 휏. It is easy to check that 훥(푎) = 푙푖푚‖푎‖푝 푎푛푑 훥(푎) = 푖푛푓 푒푥푝 휏(푙표푔(|푎| + 휀핝)) . 푝→0 휀>0 128

As the usual determinant of matrices, 훥 is also multiplicative: 훥(푎푏) = 훥(푎)훥(푏). We refer for information on determinant to [26, 300] in the case of bounded operators, and to [157, 288] for unbounded operators. Return to our Hardy spaces. An element ℎ of 퐻푝(푀) with 푝 < ∞ is said to be an outer element if ℎ퐴 is dense in 퐻푝(푀). If in addition 훥(ℎ) > 0, we call such an h strongly outer. See [63] for 푝 ≥ 1 and [284] for 푝 < 1. We will however pause to summarise the essential points of the theory. For any outer element h of 퐻푝(푀), both ℎ and 훷(ℎ) necessarily have dense range and trivial kernel. Hence their inverses exist as affiliated operators. For such an outer element, we also necessarily have that 훥(ℎ) = 훥(훷(ℎ)). If indeed 훥(ℎ) > 0, the equality 훥(ℎ) = 훥(훷(ℎ))is sufficient for ℎ to be outer. Using this fact it is now an easy exercise to see that if 훥(ℎ) > 0, then ℎ is an outer element of 퐻푝(푀) if and only if ℎ∗ is an outer element of 퐻푝(푀)∗ if and only if ℎ is right outer in the sense that 퐴ℎwill also be dense in 퐻푝(푀). In this theory one also has a type of noncommutative Riesz-Szegö theorem, in that any 푓 ∈ 퐿푝(푀) for which 훥(푓) > 0, may be written in the form 푓 = 푢ℎ where 푢 ∈ 푀 is unitary and ℎ ∈ 퐻푝(푀)an outer element of 퐻푝(푀). 2 Given a state 휔 on 푀, we write (휋휔, 퐿 (휔)훺휔)for the cyclic representation associated to 휔. ∗ 2 The subspaces 퐴 and A0 embed canonically into 퐿 (휔) by means of the operation 푎 → ∗ 2 휋휔(푎)훺휔. The angle between 퐴 and 퐴0 in 퐿 (휔)is defined to be that between the closed ̅̅̅̅̅̅̅∗̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅ −1 subspaces 휋휔(퐴 )훺휔and 휋휔(퐴0)훺휔. The latter is equal to cos 휌 with 휌 given by ∗ 휌 = 푠푢푝{|〈휋휔(푎)훺휔, 휋휔(푏)훺휔〉|: 푎 ∈ 퐴0, 푏 ∈ 퐴 , ‖휋휔(푎)훺휔‖ ≤ 1, ‖휋휔(푏)훺휔‖ ≤ 1}. ∗ In view of the fact that 휋휔(푎)훺휔, 휋휔(푏)훺휔 = 휔(푏 푎), this may be rewritten as ∗ ∗ 2 2 휌 = 푠푢푝{|휔(푏 푎)|: 푎 ∈ 퐴0, 푏 ∈ 퐴 , 휔(|푎| ) ≤ 1, 휔(|푏| ) ≤ 1}. ∗ 2 In general 0 ≤ 휌 ≤ 1. 퐴 and 퐴0 are said to be at positive angle in 퐿 (휔)if 휌 < 1. Let 푃+ be 2 2 the orthogonal projection from 퐿 (푀) onto 퐻 (푀). It is then clear that 푃+ defines a bounded operator on 퐿2(휔) if and only if 휌 < 1. We present our noncommutative Helson-Szegö theorem. This theorem will prove to be an important ingredient in our onslaught on Toeplitz operators . The classical Helson-Szegö theorem contains the information that any finite Borel measure for which the angle between 퐴 ∗ and 퐴0 is positive must necessarily be absolutely continuou with respect to Lebesgue measure, and moreover that the Radon-Nikod´ym derivative of this measure must have a strictly positive geometric mean (17). Before presenting our noncommutative Helson-Szegö theorem, we first show that under some mild restrictions the same claims are true in the noncommutative case. 푝 푝 퐿+(푀) will denote the positive cone of 퐿 (푀). Proposition (5.3.1)[170]:Let 퐷 = 퐴 ∩ 퐴∗be finite dimensional, and let ω be a state on 푀 for 1 which휌 < 1. Then 휔 is of the form 휔 = 휏(푔 ∙) for some 푔 ∈ 퐿+(푀). Proof:We keep the notation introduced at the end of the previous section. Let 휔푛 and 휔푠 respectively be the normal and singular parts of 휔. Firstly note that by [183], there exists a ′′ 2 central projection 푒0in 휋휔(푀) such that for any 휉, 휓 ∈ 퐿 (휔) the functionals 푎 → ⊥ 〈휋휔(푎)푒0휉, 휓〉and 푎 → 〈휋휔(푎)ℯ0 휉, 휓〉 on 푀 are respectively the normal and singular parts of ⊥ the functional 푎 → 〈휋휔(푎)휉, 휓〉, where ℯ0 = 핝 − ℯ0. In particular, the triples 2 ⊥ ⊥ 2 ⊥ (푒0휋휔, 푒0퐿 (휔), 푒0훺휔) and (ℯ0 휋휔, ℯ0 퐿 (휔), ℯ0 훺휔) are copies of the GNS representations of 휔푛 and 휔푠respectively. Since 휌 < 1, we must have that 129

̅̅̅̅̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅∗̅̅̅̅̅̅ 휋휔(퐴0)휔 ∩ 휋휔(퐴 )훺휔 = {0}. Now suppose that the singular part 휔푠 of 휔 is nonzero. By Ueda’s noncommutative peak-set theorem [315] there exist an orthogonal projection 푒 in the second dual 푀⋆⋆ of 푀 and a contractive element 푎 of 퐴 so that (i) 푎푛 converges to e in the weak*-topology on 푀⋆⋆; ⋆⋆ (ii) 휔푠(푒) = 휔푠(1) (here 휔푠 is identified with its canonical extension to 푀 ); (iii) 푎푛 converges to 0 in the weak*-topology on 푀. Since the expectation 훷 is weak*-continuous on 푀, 훷(푎푛)is weak* convergent to 0. But then the finite dimensionality of D ensures that 훷(푎푛)converges to 0 in norm. Recall that the bidual 푀⋆⋆ of 푀 may be represented as the double commutant of 푀 in its universal representation. So when this realisation of 푀⋆⋆is compressed to the specific representation engendered by ω, it follows that 푒 yields a projection 푒̃ in 휋휔(푀)′′to which 푛 휋휔(푎 ) converges in the weak*-topology on 휋휔(푀)′′. This weak* convergence in 휋휔(푀)′′ together with the second bullet above, then yield the facts that 푛 2 (i) 휋휔(푎 )훺휔 converges to 푒̃훺휔 in the weak-topology on 퐿 (휔); ⊥ (ii) 〈푒̃훺휔, ℯ0 훺휔〉 = 휔푠(핝). 푛 푛 From the first bullet and the fact that {훷(푎 )} is a norm-null sequence, it follows that 휋휔(푎 − 푛 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅ 훷(푎 ))훺휔 is weakly convergent to 푒̃휔, and hence that 푒̃훺휔 ∈ 휋휔(퐴0)훺휔. But if an converges to e in the weak*-topology on 푀⋆⋆, then surely so does (푎∗푛). In terms of the GNS ∗ 푛 representation for 휔, this means that 휋휔((푎 ) )훺휔 also converges to푒̃휔in the weak-topology 2 ̅̅̅̅̅̅̅∗̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅∗̅̅̅̅̅̅ on 퐿 (휔). But then 푒̃휔 ∈ 휋휔(퐴 )훺휔. Then 푒̃휔 = 0 since 푒̃휔 ∈ 휋휔(퐴0)휔 ∩ 휋휔(퐴 )훺휔. But ⊥ this cannot be, since by the second bullet this would mean that 휔푠(핝) = 〈푒̃휔, ℯ0 훺휔〉 = 0. Thus our supposition that 휔푠 is nonzero, must be false. The condition that 휌 < 1, is therefore 1 sufficient to force 휔 to be normal. That is휔 is of the form 휔 = 휏(푔. ) for some 푔 ∈ 퐿+(푀). The following lemmata present two known elementary facts. 1 Lemma (5.3.2)[170]:For any 푔 ∈ 퐿+(푀) we have that 푠(훷(푔)) ≥ 푠(푔), where 푠(푔) denotes the support projection of 푔. Proof: For simplicity of notation we respectively write 푠 and 푠훷for s(g) and 푠(훷(푔)). Since 푠훷 ∈ 퐷, we have that ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ 휏(푠훷푔푠훷 ) = 휏 ∘ 훷(푠훷푔푠훷 ) = 휏(푠훷훷(푔)푠훷 ) = 0. 1/2 ⊥ ⊥ 1/2 ⊥ Therefore 푔 푠훷 = 푠훷푔 = 0. This is sufficient to force 푠훷 ⊥ 푠, which in turn suffices to show that 푠훷 ≥ 푠. Lemma (5.3.3)[170]:Let 푒 be a nonzero projection in 퐷. Then 푒퐴푒 is a finite maximal 1 subdiagonal subalgebra of 푒푀푒 (equipped with the trace 휏푒(. ) = 휏(. )) with diagonal 휏(푒) 푒퐴푒 ∩ (푒퐴푒)∗ = 푒퐷푒. Proof: The expectation 훷 is trivially still multiplicative on the compression 푒퐴푒. Using the fact that 푒 ∈ 퐷, it is an exercise to see that훷 maps푒퐴푒 onto 푒퐷푒. It is also straightforward to see that the weak*-density of 퐴 + 퐴∗ in 푀 forces the weak*-density of 푒퐴푒 + (푒퐴푒)∗ in 푒푀푒, and that (푒퐴푒)0 = 푒퐴0푒.

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Definition (5.3.4)[170]: Adopting the notation of the previous two lemmata, given a nonzero 1 element 푔 ∈ 퐿+(푀), we define ΔΦ(g) to be the determinant of 푠훷푔푠훷 regarded as an element of (푠훷푀푠훷, 휏푠훷) ∗ 1 Proposition (5.3.5)[170]: Let 퐷 = 퐴 ∩ 퐴 be finite dimensional, and let 푔 ∈ 퐿+(푀) be a norm- one element for which the state 휔 = 휏(푔・) satisfies 휌 < 1. Then 훥훷(푔) > 0. Proof: It is clear from the previous lemmata that we may reduce matters to the case where푠(훷(푔)) = 핀, and hence we will assume this to be the case. Suppose by way of contradiction that 훥(푔) = 0. By the Szegö formula for subdiagonal algebras [159], we then have that 2 0 = 훥(푔) = 푖푛푓{휏(푔|푎 − 푑| ): 푎 ∈ 퐴0, 푑 ∈ 퐷, 훥(푑) ≥ 1}. Thus there exist sequences {푎푛} ⊂ 퐴0and {푎푛} ⊂ 퐷 with훥(푑푛) ≥ 1 for all 푛, so that 2 휏(푔|푎푛 – 푑푛| ) → 0 푎푠 푛 → ∞. By [60] we may assume all the 푎푛’s to be invertible. Now let 푢푛 ∈ 퐷be the unitary in the polar ∗ 2 decomposition 푑푛 = 푢푛|푑푛|. It is an exercise to see that then {푢푛푎푛} ⊂ 퐴0with |푎푛 − 푑푛| = ∗ 2 |푢푛푎푛 − |푑푛|| . Making the required replacements, we may therefore also assume that {푑푛} ⊂ 퐷+. ̃ 1 Since 1 ≤ (푑푛) ≤ ‖푑푛‖∞ for all 푛, we will for the sequences 푑푛 = 푑푛and푎̃푛 = ‖푑푛‖∞ 1 ̃ 2 푎푛 (푛 ∈ ℕ), still have that 휏(푔|푎̃푛 − 푑푛| ) → 0 as 푛 → ∞. Now recall that 퐷 is finite ‖푑푛‖∞ ̃ dimensional. So by passing to a subsequence if necessary, we may assume that {푑푛} converges + uniformly to some norm one element 푑0of 퐷 . But then by what we showed above, ‖휋 (푎̃) − 휋(푑 )‖ = 휏(푔|푎̃ − 푑 |2)1/2 푔 푛 0 2 푛 0 ̃ 2 1/2 ̃ 2 1/2 ≤ 휏(푔|푎̃푛 − 푑푛| ) + 휏(푔|푑푛 − 푑0| ) ≤ 휏(푔|푎̃ − 푑̃|2)1/2 + ‖푑̃ − 푑 ‖ 휏(푔)1/2 푛 푛 푛 0 ∞ → 0. Thus 휋푔(푑0) ∈ 휋푔(퐴0) ∩ 휋푔(퐴∗). Since 훷(푔) is of full support, we have that 1/2 1/2 훷(푔) 푑0훷(푔) ≠ 0. So 1/2 1/2 0 < 휏(훷(푔) 푑0훷(푔) = 휏(훷(푔)푑0) = 휏(훷(푔푑0)) = 휏(푔푑0). ∗ Therefore 휋푔(푑0) ≠ 0. But this proves that the subspaces 휋푔(퐴0) and 휋푔(퐴 ) have a nonzero intersection, and hence that 휌 = 1. The following technical lemma is a crucial step in the proof of the classical Helson-Szegö theorem. The challenge one faces in the noncommutative world is that the functional calculus at our disposal in that context is simply not strong enough to reproduce so detailed a statement in that framework. However in the lemma following this one, we present what we believe to be a reasonable noncommutative substitute of this interesting lemma. Lemma (5.3.6)[170]: Let 푢 = 푒−푖휓 with휓 a real measurable function on 핋. Then −푖휓 ∞ 푖푛푓 ∞ ‖푒 − 푔‖ < 1 if and only if there exist an 휀 > 0 and a 푘 ∈ 퐻 ( 핋)so that 푔∈퐻 (푇) ∞ 0 휋 |푘 | ≥ 휀 and |휓| + 푎푟푔(푘 )| ≤ – 휀 almost everywhere. 0 0 2

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Lemma (5.3.7)[170]:Let 푢 be a unitary element of 푀. Then there exists some 푓 ∈ 퐴 so that ∗ ‖푢 − 푓‖∞ < 1 if and only if there exists ℎ ∈ 퐴so thatℜ(푢 ℎ) is strictly positive. Proof: Suppose first that there exists 푓 ∈ 퐴 with ‖푢 − 푓‖∞ < 1. We then equivalently have that ‖1 − 푢∗푓‖ = ‖1 − 푓∗푢‖ < 1. On setting 훼 = ‖1 − 푢∗푓‖, it follows that ‖핀 − ℜ(푢 ∗ 푓)‖ ≤ 훼 < 1, and hence that −훼핀 ≤ ℜ(푢∗푓) − 1 ≤ 훼핝. This in turn ensures that 0 < (1 − 훼)핝 ≤ ℜ(푢∗푓). Conversely suppose that there exists ℎ ∈ 퐴 ∩ 푀−1 so that ℜ(푢∗ℎ) ≥ 훼핝 for some 0 < 훼 ≤ ‖ℜ(푢∗ℎ)‖ ≤ ‖ℎ‖, where 푀−1 denotes the subset of invertible elements of 푀. Given 휀 > 휀 0, set휆 = . It then follows that ‖ℎ‖ 2훼휖 −2휆ℜ(푢∗ℎ) + 휆2|ℎ|2 ≤ − ( − 휀2) 핝. ‖ℎ‖ 훼 (Observe that ≤ 1 in the above inequality.) It is clear that if ε is small enough, we would ‖ℎ‖ 2훼휀 have that 1 > ( − 휀2) > 0. Thus we may assume this to be the case. For simplicity of ‖ℎ‖ 2훼휀 notation we now set 훿 = ( − 휀2). It therefore follows from the previous centered inequality ‖ℎ‖ that 0 ≤ |1 − 푢∗(휆ℎ)|2 = 핝 − 2ℜ(푢∗(휆ℎ) + |휆ℎ|2 ≤ (1 − 훿)핝. Hence as required, ‖핝 − 푢∗(휆ℎ)‖2 ≤ (1 − 훿) < 1. We are now finally ready to present our noncommutative Helson-Szegötheorem. In view of Propositions (5.3.1) and(5. 3.5), it is not unreasonable to restrict attention to normal states 휏(푔. ) in this theorem for which ∆훷(푔 > 0. The following result is a sharpening of the result of Pousson [108], in that here the conditions imposed on the unitary 푢 are less restrictive. This sharpening is achieved by means of the preceding Lemma. 1 Theorem (5.3.8)[170]:Let 푔 ∈ 퐿+(푀) be given with ‖푔‖1 = 1, and denote s(훷(푔)) by 푠훷. Consider the state 휔 = 휏(푔. ). Then 휌 < 1 and 훥훷(푔) > 0 if and only if 푔is of the form 푔 = 푓푅푢푓퐿where (i) 푢 ∈ 푀 is a partial isometry with initial and final projections sΦ for which there exists ∗ some 푘 ∈ 푠훷퐴푠훷 so that ℜ(푢 푘) ≥ 훼푠훷 for some 훼 > 0,• 2 (ii) and 푓퐿 and 푓푅 are strongly outer elements of 퐻 (푀) commuting with 푠훷 for which푔 + 2 ∗ 2 (핀 − 푠훷) = |푓퐿| = |푓푅 | . If in addition 푑푖푚퐷 < ∞, we may dispense with the restrictions that 휔 is normal, and that 훥훷(푔) > 0. Proof: Set 푠 = 푠훷for simplicity. Suppose that 푔 satisfies the condition 훥훷(푔) > 0. Using the 1/2 1/2 fact that then 훥훷(푔 ) = 훥훷(푔) > 0, it follows from the noncommutative Riesz- 2 Szegötheorem (see [63]) that there exist strongly outer elements ℎ퐿, ℎ푅 ∈ 퐻 (푠푀푠) and 1/2 1/2 ∗ unitaries 푣퐿, 푣푅 ∈ 푠푀푠 for which 푔 = 푣퐿ℎ퐿 = ℎ푅푣푅. (Then also 푔 = |ℎ퐿| = |ℎ푅|.) We set ⊥ ⊥ 푢 = 푣푅푣퐿, 푓퐿 = ℎ퐿 + 푠 , 푓푅 = ℎ푅 + 푠 . It is then clear that ⊥ 2 ∗ 2 푔 = 푓푅푢푓퐿 and 푔 + 푠 = |푓퐿| = |푓푅 | . 132

We proceed to show that fL and fRare strongly outer. The proofs of the two cases are identical, and hence we do this for 푓퐿 only. Notice that ⊥ l표푔(|푓퐿|) = 푙표푔(|ℎ퐿| + 푠 ) = 푙표푔(|ℎ퐿|)푠. ⊥ Since 훷(푓퐿) = 훷(ℎ퐿) + 푠 , we similarly have that 푙표푔(|훷(푓퐿)|) = 푙표푔(|훷(ℎ퐿)|)푠. It then follows that 휏(푙표푔 |푓퐿|) = 휏(푠)휏푠(푙표푔 |ℎ퐿|)and 휏(푙표푔 |훷(푓퐿)|) = 휏(푠)휏푠(푙표푔 |훷(ℎ퐿)|). Thus the outerness of ℎ퐿 yields that 휏(푙표푔 |푓퐿|) = 휏(푙표푔 |훷(푓퐿)|) > −∞, so 훥(푓퐿) = 훥(훷(푓퐿)) > 0. Then an application of [63] now shows that 푓퐿 is strongly outer. On the other hand, we have ∗ ∗ ∗ 〈휋푔(푎)훺푔, 휋(푏)훺푔〉 = 휏(푔푏 푎) = 휏(푢푓퐿푏 푎푓푅), 푎 ∈ 퐴0, 푏 ∈ 퐴 . So ∗ ∗ 2 2 휌 = 푠푢푝{|휏(푔푏 푎)|: 푎 ∈ 퐴0, 푏 ∈ 퐴 , 휏(푔|푎| ) ≤ 1, 휏(푔|푏| ) ≤ 1} ∗ ∗ 2 ∗ 2 = 푠푢푝{|휏((푢(푠푓퐿푏 )(푎푓푅푠))| ∶ 푎 ∈ 퐴0, 푏 ∈ 퐴 , 휏(|푎푓푅푠| ) ≤ 1, 휏(|푏푓퐿 푠| ) ≤ 1} 2 2 = 푠푢푝{|휏(푢퐹1퐹2)|: 퐹1 ∈ 푠퐻 (푀), 퐹2 ∈ 퐻0 (푀)푠, ‖퐹1‖2 ≤ 1, ‖퐹2‖2 ≤ 1}. In the above computation one has used the fact thatfLand fR are strongly outer to approximate ∗ ∗. 퐹1 and 퐹2 with elements of the form푠푓퐿푏 and 푎푓푅푠where 푎 ∈ 퐴0 and 푏 ∈ 퐴 However, it is 2 2 easy to check that for 퐹1 ∈ 푠퐻 (푀), 퐹2 ∈ 퐻0 (푀)푠 1 퐹1퐹2 ∈ 퐻0 (푠푀푠) 푎푛푑 ‖퐹1퐹2‖1 ≤ ‖퐹1‖2‖퐹2‖2. Conversely, by the Noncommutative Riesz Factorisation theorem [176, 153], for any 휀 > 0 1 2 2 2 2 and any 퐹 ∈ 퐻0 (푠푀푠) there exist 퐹1 ∈ 퐻 (푠푀푠) ⊂ 푠퐻 (푀) and 퐹2 ∈ 퐻0 (푠푀푠) ⊂ 퐻 (푀)푠 such that 퐹 = 퐹1퐹2 푎푛푑 ‖퐹1‖2‖퐹2‖2 ≤ ‖퐹1‖1 + 휀. From these discussions we conclude that 1 1 휌 = 푠푢푝{|휏(푢퐹)|: 퐹 ∈ 퐻0 (푠푀푠), ‖퐹‖1 ≤ 1} = 푠푢푝{|휏푠(푢퐹)|: 퐹 ∈ 퐻0 (푠푀푠), 휏푠(|퐹|) ≤ 1}. 1 1 The norm of the restriction of the functional 퐿 (푠푀푠) → ℂ: 푎 → 휏푠(푢푎) to 퐻0 (푠푀푠) is by 1 duality precisely the norm of the equivalence class [푢] in the quotient space 푠푀푠/(퐻0 (푠푀푠)). However, it is well known that s퐴푠 = {푎 ∈ 푠푀푠: 휏푠(푎푏) = 0, 푏 ∈ 푠퐴0푠} 1 (cf. e.g., [153] ). From this fact it is now an easy exercise to see that the polar (퐻0 (푠푀푠)) is nothing but 푠퐴푠. It therefore follows that 휌 = 푖푛푓{‖푢 − 푘‖∞: 푘 ∈ 푠퐴푠}. The result now follows from an application of the preceding Lemma. We start by recalling the definition of Toeplitz operators. Given푎 ∈ 푀, the Toeplitz operator 푇푎 with symbol a is defined to be the map 2 2 푇푎: 퐻 (푀) → 퐻 (푀): 푏 → 푃+(푎푏), 2 2 where 푃+denotes the orthogonal projection from 퐿 (푀) onto 퐻 (푀). see [177] (see also [30]). We will characterise the symbols of invertible Toeplitz operators. We point out that these results are new even for the matrix-valued case. In achieving this characterisation, we will follow the same basic strategy as Devinatz [7] in his remarkable solution of this problem in the classic setting. Our first result essentially reduces the problem to that of characterising invertible Toeplitz operators with unitary symbols. 133

Theorem (5.3.9)[170]:Let 푎 ∈ 푀 be given. A necessary and sufficient condition for 푇푎 to be invertible is that it can be written in the form 푎 = 푢푘where 푘 ∈ 퐴−1, and 푢 ∈ 푀 is a unitary for which 푇푢 is invertible. Suppose that 푎 ∈ 푀 is indeed of the form 푎 = 푢푘where 푘 ∈ 퐴−1, and 푢 ∈ 푀 is a unitary. It is a simple exercise to see that then 푇푘 is invertible with inverse 푇푘−1 . Since 푇푎푇푘−1 = 푇푢 and 푇푢푇푘 = 푇푎, it is now clear that 푇푎 will then be invertible if and only if 푇푢 is invertible. Proof: The sufficiency of the stated condition was noted in the above discussion. To see the 2 necessity, assume 푇푎 to be invertible. There must therefore exist some 푔 ∈ 퐻 (푀) so that 2 ∗ 푇푎푔 = 핝. This in turn can only be true if there exists some ℎ ∈ 퐻0 (푀) so that 푎푔 = 핝 + ℎ . By the generalised Jensen inequality [63] we have that 훥(푎)훥(푔) = 훥(푎푔) = 훥(핝 + ℎ∗) ≥ 훥(훷(핝 + ℎ∗)) = 훥(핝) = 1. Clearly we then have that 훥(|푎|1/2) = 훥(푎)1/2 > 0. So by the noncommutative Riesz-Szegö theorem [64], there must exist an outer element푓 ∈ 퐻2(푀) and a unitary 푣 so that |푎|1/2 = 푣푓. (Note then that 푓 ∈ 푀, so 푓 must belong to 퐴 too.) Let 휔 be the unitary in the polar decomposition 푎 = 휔|푎|, and consider 푏 = 휔|푎|1/2푣. Notice that by construction 푏푓 = 푎. Thus 푇푏푇푓 = 푇푎. We will use this formula to show that 푇푓 is invertible, from which the result will then follow. Firstly note that the injectivity of 푇푎 combined with the above equality, ensures that 푇푓is −1 injective. Next notice that the equality 푇푏푇푓 = 푇푎. ensures that ( 푇푎 ) 푇푏 is a left inverse for 푇푓. So 푇푓 must have a closed range. However since 푓 is outer, we also have that [푓퐴]2 = 퐻2(푀). Since 푓퐴 ⊂ 푇푓 (퐻2(푀)), these two facts ensure that the range of 푇푓 is all of 퐻2(푀). Hence푇푓must be invertible. ∗ But if Tf is invertible, then so is 푇 푓 = 푇푓∗. Since 푇푓∗푇푓 = 푇|푓|2 = 푇|푎|, the operator푇|푎| must be invertible. Since 휎(|푎|) ⊂ 휎(푇|푎|) by [177], we must have that 0 ∉ 휎(|푎|). In other words |푎| must be strictly positive. But if |푎| is strictly positive, then by Arveson’s factorization −1 theorem there exists some 푘 ∈ 퐴 with |푎| = |푘|. Finally let 휔0 be the unitary in the polar ∗ ∗ form푘 = 휔0|푘|. Then 푎 = 휔휔0푘, which proves the theorem with 푢 = 휔휔0. Our next step in achieving the desired characterisation, is to present some necessary structural information regarding unitaries 푢 for which 푇푢is invertible. We then subsequently use this structural information to obtain a characterisation of invertibility in terms of positive angle. Lemma (5.3.10)[170]: Let 푢 ∈ 푀 be a unitary. A necessary condition for 푇푢 to be invertible is ∗ −1 −1 2 that it is of the form 푢 = (푔1 ) 푑푔0 where 푔0, 푔1are strongly outer elements of 퐻 (푀) and d a strongly outer element of 퐿2(퐷) related by the conditions that ∗ −1 ∗ −1 2 ∗ ∗ ∗ −1 푑 = 훷(푔0) = 훷(푔1), 푑푔0 , 푑 푔1 ∈ 퐻 (푀) and 푔0푔0 = 푑 (푔1푔1) 푑. ∗ Proof:Let 푢 ∈ 푀 be a unitary for which 푇푢 is invertible. Since 푇 푢 = 푇푢∗ is then also invertible, 2 itfollows that there must exist 푔0, 푔1 ∈ 퐻 (푀) so that 푇푢푔0 = 핝 = 푇푢푔1. This in turn means 2 that there exist ℎ0, ℎ1 ∈ 퐻0 (푀) with ∗ ∗ ∗ 푢푔0 = 핝 + ℎ0 , 푢 푔1 = 핝 + ℎ1 . Notice that we may then apply the generalised Jensen inequality [64] to conclude that 훥(푔0) = 훥(푢)훥(푔0) = 훥(푢푔0) ≥ 훥(핝) = 1. 134

Similarly 훥(푔1) ≥ 1. By [63] this means that both 푔0 and 푔1 are injective with dense range, −1 −1 and hence that 푔0 and 푔1 exist as affiliated operators. On the other hand, we have that ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ 1 ∗ 푔1푢푔0 = 푔1(핝 + ℎ0) ∈ 퐻 (푀) and 푔1푢푔1 = 푔1(핝 + ℎ1 ) ∈ 퐻 (푀) . Hence ∗ 1 1 ∗ 1 푔1푢푔0 ∈ 퐻 (푀) ∩ 퐻 (푀) = 퐿 (퐷). ∗ −1 −1 If we denote this element by 푑, it follows that 푢 is of the form u = (푔1) 푑 푔0 . It is then ∗ ∗ −1 ∗ clearthat 푑 (푔0푔1) 푑 = 푔1푔0. ∗ It remains to show that 푔0 and 푔1 are outer and that 푑 = 훥(푔0) = 훥(푔1). To see this notice that ∗ 2 ∗ ∗ 2 ∗ since 푔1 ∈ 퐻 (푀) and푢푔0 = 핝 + ℎ0 ∈ 퐻 (푀) , we have that ∗ ∗ ∗ ∗ ∗ ∗ 푑 = 훷(푑) = 훷(푔1푢푔0) = 훷(푔1(핝 + ℎ0 )) 훷(푔1)훷(핝 + ℎ0) = 훷(푔1). 2 2 Similarly, 푑 = 훷(푔0). (Since 훷 maps 퐻 (푀) onto 퐿 (퐷), this equality also shows that 푑 is in 2 1 ∗ ∗ ∗ −1 fact in 퐿 (퐷),, and not just 퐿 (퐷),.) It now follows from the equality 푔0푔0 = 푑 (푔1푔1) 푑, that 2 ∗ ∗ ∗ −1 ∗ 2 −2 2 −2 훥(푔0) = 훥(푔0푔0) = 훥(푑 (푔1푔1) 푑) = 훥(푑 ) 훥(푔1) = 훥(훷(푔1)) 훥(푔1) . Since as was shown earlier we have that 훥(푔0) ≥ 1, it therefore follows that 0 < 훥(푔1) ≤ 훥(훷(푔1)).If we combine this with the generalised Jensen inequality [63],we obtain 0 < 훥(푔1) = 훥(훷(푔1).Similarly, 0 < 훥(푔0) = 훥(훷(푔0)). Thus by [64] both 푔0 and 푔1 are strongly outer. When combined with Theorem (5.3.9), the following lemma characterises the invertibility of Toeplitz operators in terms of positive angle. If we further combine this lemma with the noncommutative Helson-Szegö theorem obtained, we end up with the promised structural characterisation of invertible Toeplitz operators with unitary symbols. Lemma (5.3.11)[170]:Let 푢 ∈ 푀 be a unitary of the form described in the previous lemma. ∗ Then 푇푢is invertible if and only if퐴 and 퐴0 are at positive angle with respect to the functional 휏(휔 ∙), where ∗ ∗ ∗ −1 휔 = 푔0푔0 = 푑 (푔1푔1) 푑. 2 Proof: First suppose that 푇푢 is invertible. For any 푎 ∈ 퐴 the element g0a will belong to 퐻 (푀). So the invertibility of 푇푢 ensures that we can find a constant 퐾 > 0 so that ‖푔0푎‖2 ≤ Κ‖푇푢(푔0푎)‖2, 푎 ∈ 퐴. ∗ −1 −1 Recall that by Lemma (5.3.10) u is of the form 푢 = (푔1) 푑푔0 . Thus the former inequality translates to ∗ −1 ‖푔0푎‖2 ≤ Κ‖푃+(푔1) 푑푎‖2, 푎 ∈ 퐴. ∗ −1 ∗ 2 ∗ ∗ 2 ∗ Now observe that for any푏 ∈ 퐴0, the element (푔1) 푑푏 will belong to 퐻 (푀) 퐴0 ⊂ 퐻0 (푀) . Hence ∗ −1 ∗ −1 ∗ −1 ∗ 푃+((푔1) 푑푎) = 푃+((푔1) 푑푎 + (푔1) 푑푏 ). ∗ 1/2 If we now write ‖푓‖휔for휏(휔푓 푓) , then for any 푎 ∈ 퐴and 푏 ∈ 퐴0 we have that ∗ ∗ 1/2 ‖푎 ‖휔 = 휏(푎 푤푎) = ‖푔0푎‖2 ∗ −1 ∗ −1 ∗ ≤ Κ‖푃+((푔1) 푑푎 + (푔1) 푑푏 )‖2 ∗ −1 ∗ ≤ 퐾‖(푔1) 푑(푎 + 푏 )‖2 ∗ ∗ ∗ = 퐾휏((푎 + 푏)휔(푎 + 푏 )) = 퐾‖푎 + 푏‖휔 ∗ Thus 퐴 and 퐴0are at positive angle with respect to the functional 휏(푤. ).

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∗ . Conversely, suppose that 퐴 and 퐴0are at positive angle with respect to the functional 휏(휔 ).We first show that 푇푢 has dense range, and hence that it will be invertible whenever it is bounded 2 2 below. Let 푎0 ∈ 퐻 (푀) be orthogonal to 푇푢(퐻 (푀)). We will show that 푎0 must then be the 2 zero vector. Given 푎 ∈ 퐴, the orthogonality of 푎0 to 푇푢(퐻 (푀)) together with the fact that 푢 = ∗ −1 −1 (푔1) 푑푔0 , ensures that ∗ ∗ ∗ −1 ∗ ∗ −1 0 = 〈푇푢(푔0푎), 푎0〉 = 휏(푎0푇푢(푔0푎)) = 휏(푎0푃+((푔1) 푑푎) = 휏(푎0(푔1) 푑푎). However, as was noted in the first part of the proof, for any 푏 ∈ 퐴0we have that ∗ ∗ −1 ∗ 2 ∗ 푎0(푔1) 푑푏 ∈ 퐻0 (푀) , which implies that ∗ ∗ −1 ∗ ∗ ∗ −1 ∗ 휏(푎0(푔1) 푑푏 ) = 휏(훷(푎0(푔1) 푑푏 )) = 0. Thus ∗ ∗ −1 ∗ 휏(푎0(푔1) 푑(푎 + 푏 )) = 0for all푎 ∈ 퐴, 푏 ∈ 퐴0. ∗ −1 Hence 푑 푔1 푎0 = 0, so 푎0 = 0. ∗ It remains to show that 푇푢 is bounded below whenever 퐴 and 퐴0 are at positive angle with respect to the functional 휏(푤 ∙). Hence assume that there exists a constant 퐵 > 0 so that ∗ ∗ ‖푎 ‖휔 ≤ 퐵‖푎 + 푏‖휔 for all 푎 ∈ 퐴, 푏 ∈ 퐴0. ∗ ∗ ∗ −1 Since by assumption we have that d = 훷(푔1), and since both 푔1 and ((푔1) 푑)belong to 퐻2(푀)∗,it follows that ∗ ∗ −1 ∗ ∗ −1 ∗ −1 푑 = 훷(푑) = 훷(푔1[(푔1) 푑]) = 훷(푔1)훷((푔1) 푑) = 푑훷((푔1) 푑). ∗ −1 ∗ This yields that 훷((푔1) 푑) =1. Now since 푔1 is by assumption strongly outer, we have that 훥(푔1) = 훥(훷(푔1)) > 0 by [63]. Consequently ∗ ∗ −1 ∗ ∗ −1 ∗ −1 훥(푑) = 훥(푔1 )훥((푔1) 푑) = 훥(훷(푔1))훥((푔1) 푑) = 훥(푑)훥((푔1) 푑). Thus since Δ(d) > 0 by the strong outerness of d, we must have that ∗ −1 ∗ −1 Δ((g1) d) = 1 = Δ(핝) = Δ(Φ(g1) d)). ∗ −1 2 ∗ Hence by [63] ((g1) d) is a strongly outer element of 퐻 (푀) . But this ensures that ∗ −1 ∗ 2 ∗ [((푔1) 푑)퐴0 ] = 퐻0 (푀) . Hence for any fixed 푎 ∈ 퐴, we may select a sequence {푏푛} ⊂ 퐴0 so that ∗ −1 ∗ ∗ −1 2 ∗ 2 (푔1) 푑푏푛 → (푃+ − 퐼푑)[(푔1) 푑푎] ∈ 퐻0 (푀) 푖푛 퐿 (푀). ∗ −1 Finally recall that by assumption |푔0| = |(푔1) 푑|. So given any 푎 ∈A, with {푏푛} ⊂ 퐴0 the sequence as constructed above, we have that ∗ ∗ ∗ ∗ ‖푔0푎‖2 = ‖푎 ‖휔 ≤ 퐵‖푎 + 푏푛‖휔 = 퐵‖푔0(푎 + 푏푛)‖2 = 퐵‖|푔0|(푎 + 푏푛)‖2 ∗ −1 ∗ ∗ −1 ∗ = 퐵‖|(푔1) |(푎 + 푏푛)‖2 = 퐵‖|(푔1) |(푎 + 푏푛)‖2. Letting 푛 → ∞ now yields ∗ −1 ‖푔0푎‖2 ≤ 퐵‖푃+[(푔1) 푑푎]‖ = 퐵‖푇푢(푔0푎)‖2 for any푎 ∈ 퐴. 2 Finally note that by assumption 푔0is an outer element of 퐻 (푀). With 푔0퐴 therefore being dense in 퐻2(푀, the above inequality extends by continuity to the claim that 2 ‖푎‖2 ≤ 퐵‖푇푢(푎)‖2for any푎 ∈ 퐻 (푀). Thus 푇푢is invertible. Definition (5.3.12)[170]:Given 푓 ∈ 푀 we define the Hankel operator with symbol 푓 by means of the prescription 2 2 ∗ ℋ푓 ∶ 퐻 (푀) → 퐻 (푀) : 푥 → 푃−(푓푥), 2 2 ∗ where 푃− is the orthogonal projection from 퐿 (푀) onto퐻 (푀) . 136

The following lemma is entirely elementary. Lemma (5.3.13)[170]: Let 푓 ∈ 푀 be given. Then 1 ‖ℋ | 2 ‖ = 푠푢푝{|휏(푓퐹)|: 퐹 ∈ 퐻 (푀), 휏(|퐹|) ≤ 1}. 푓 퐻0 0 2 2 Proof: Since for every 푥 ∈ 퐻 (푀) we have that(퐼푑 − 푃−)(푥) ∈ 퐻0 (푀), it is clear that such 2 ∗ an (퐼푑 − 푃−)(푥)will be orthogonal to any y ∈ 퐻 (푀) . Thus 〈푃−(푓푎), 푏〉 = 〈푓푎, 푏〉for any 푎 ∈ 2 2 ∗ 퐻0 (푀) and 푏 ∈ 퐻 (푀) .Thus 2 2 ‖ℋ | 2‖ = 푠푢푝{‖푃 − (푓푎)‖: 푎 ∈ 퐻 (푀), ‖푎‖ ≤ 1} = 푠푢푝{|〈푃 (푓푎), 푏〉|: 푎 ∈ 퐻 (푀), 푏 푓 퐻0 0 2 − 0 2 ∗ 2 2 ∗ ∈ 퐻 (푀) ‖푎‖2 ≤ 1, ‖푏‖2 ≤ 1} = 푠푢푝{|〈푓푎, 푏〉|: 푎 ∈ 퐻0 (푀), 푏 ∈ 퐻 (푀) ‖푎‖2 ∗ 2 2 ∗ ≤ 1, ‖푏‖2 ≤ 1} = 푠푢푝{|휏(푓푎, 푏 )|: 푎 ∈ 퐻0 (푀), 푏 ∈ 퐻 (푀) ‖푎‖2 ≤ 1, ‖푏‖2 1 ≤ 1} = 푠푢푝{|휏(푓퐹)|: 퐹 ∈ 퐻0 (푀), 휏(|퐹|) ≤ 1}. Here the last equality follows from the Noncommutative Riesz Factorisation theorem from [153] and[176]. When taken alongside Theorem (5.3.9), this result fully characterises invertible Toeplitz operators. Theorem (5.3.14)[170]:Let 푢 ∈ 푀 be a unitary of the form described in Lemma (5.3.10). Then the following are equivalent: (i) 푇푢is invertible; (ii) there exists 푘 ∈ 퐴 such that ℜ(푢∗푘) is strictly positive; 2 (iii) The Hankel operator 퐻푢 restricted to 퐻0 (푀) has norm less than 1. Proof: Our aim is to apply Theorem (5.3.8). In this regard we point out that although this theorem is formulated for norm one elements of 퐿1(푀)+, that assumption is one of convenience and not necessity. Hence the value of ‖휔‖1 is no essential obstruction to applying this theorem. ∗ 2 Next observe that the fact that 휔 = 푔0푔0, not only ensures that 훥(휔) = 훥(푔0) >0, but also that w is injective. Thus by Lemma (5.3.2), 푠(훷(휔)) = 핝. We showed in the proof of the ∗ −1 ∗ −1 ∗ −1 preceding Lemmathat 훥((푔1) 푑) = 1 = 훥(훷((푔1) 푑)). Applying this fact to 푑 푔1 ∗ −1 2 enables us to conclude from [63] that 푑 푔1 is a strongly outer element of 퐻 (푀). On setting ∗ −1 ℎ푅 = 푑 푔1 and ℎ퐿 = 푔0, it follows that 휔 is of the form ∗ −1 ∗ −1 ∗ −1 ∗ −1 −1 휔 = 푑 푔1 (푔1) 푑 = 푑 푔1 [(푔1) 푑푔0 ]푔0 = ℎ푅푢ℎ퐿 2 with ℎ푅 and ℎ퐿 strongly outer elements of 퐻 (푀) for which we have that 1/2 ∗ ∗ −1 1/2 |ℎ퐿| = |푔0| = 휔 and|ℎ푅| = |(푔1) 푑| = |휔| . With all the other conditions of this theorem being satisfied, we may now conclude from ∗ Theorem (5.3.9) that 퐴 and 퐴0 are at positive angle with respect to the functional 휏(휔) if and only if there exists a 푘 ∈ 퐴 such that ℜ(푢∗푘) is strictly positive. From the proof of Theorem ∗ (5.3.8) we also have that 퐴 and 퐴0 are at positive angle if and only if 푠푢푝{|휏(푓퐹)|: 퐹 ∈ 1 퐻0 (푀), 휏(|퐹|) ≤ 1} < 1. The result now follows from an application of the preceding two lemmata.

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Chapter 6 Structure of Commutative Toeplitz Banach Algebras .We prove the analogous commutability result for Toeplitz operators whose symbols are subordinated to the quasi-nilpotent group. At the same time we conjecture that apart from the known C∗-algebra cases there are no more new Banach algebras generated by Toeplitz operators whose symbols are subordinated to the nilpotent group and which are commutative on each weighted Bergman space.We explicitly describe the maximal ideal space and the 2 푛 Gelfand map of 풯(λ). Since 풯(휆) is not invariant under the ∗-operation of ℒ(𝒜휆 (𝔹 )) its inverse closedness is not obvious and is shown. We remark that the algebra 풯(휆) is not semi- simple and we derive its radical. Several applications of our results are given and, in particular, we conclude that the essential spectrum of elements in 풯(휆) is always connected. We show that ℬ푘(ℎ)is generated in fact by an essentially smaller set of operators,i.e.,theToeplitz operators with k-quasi-radial symbols and a finite set of Toeplitz operators with “elementary”k- quasi-homogeneous symbols.Then we analyze the structure of the commutative subalgebras corresponding to these two types of generating symbols . In particular,we describe spectra, joint spectra, maximal ideal spaces and the Gelfand transform. Section (6.1) Quasi-Nilpotent Group Action We finish the classification of the Banach and C∗-algebras generated by Toeplitz operators that are commutativeon each (commonly considered) weighted Bergman space over the unit ball 𝔹n in ℂn .The short history of this problem is as follows. The C∗-algebras generated by Toeplitz operators which are commutative on each weighted Bergman space over the unit disk were completely classified in [98] .Under some technical assumption on “richness”of a class of generating symbols the result was as follows. A C∗- algebra generated by Toeplitz operators is commutative on each weighted Bergman space if and only if the corresponding symbols of Toeplitz operators are constant on cycles of a pencil of hyperbolic geodesicson the unit disk, or if and only if the corresponding symbols of Toeplitz operators areinvariant under the action of a maximal commutative subgroupof the Mobius transformations of the unit disk. The commutativity on each weighted Bergman spacewas crucial in the part"only if" of the above result. Generalizing this result to Toeplitz operators on the unit ball, it was proved in[219,218], that, given a maximal commutative subgroup of biholomorphisms of the unit ball, theC∗-algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each weighted Bergman space. There are five different pair wise non- conjugate model classes of such subgroups :quasi-elliptic, quasi-parabolic, quasi-hyperbolic, nilpotent, andquasi-nilpotent( the last one depends on a parameter, giving in total 푛 + 2 model classes for then-dimensional unit ball). As a consequence, for the unit ball of dimension n, there are 푛 + 2 essentially different "model" commutative -C∗-algebras, all others are conjugated with one of them via biholomorphisms of the unit ball. It was firmly expected that the above algebras exhaust all possible algebras of Toeplitz operators which are commutative on each weighted Bergman space. That is, the invariance under the action of a maximal commutative subgroup of biholomorphisms for generating symbols is the only reason for the appearance of Toeplitz operator algebras which are commutative on each weighted Bergman space. 138

Recently and quite unexpectedly it was observed in[306] that for 푛 −1 there are many other, not geometrically defined, classes of symbols which generate commutative Toeplitz operator algebras on each weighted Bergman space. These classes of symbols were in a sense originated from, or subordinated to the quasi-elliptic group, the corresponding commutative operator algebras were Banach ,and being extended toC∗-algebras they became non-commutative. Moreover, for 푛 = 1 all of them collapsed to the commutativeC∗-algebra generated by Toeplitz operators with radial symbols (one-dimensional quasi-elliptic case). These results were extended in [36,297] then to the classes of symbols, subordinated to the quasi-hyperbolic and quasi parabolic groups, which as well generate via corresponding Toeplitz operators classes of Banach algebras being commutative on each weighted Bergman space. That is, together with[193] ,cover the multi-dimensional extensions of the (only) three model cases on the unit disk. The study of the last two model cases of maximal commutative subgroup of biholomorphisms of the unit ball, the nilpotent, and quasi-nilpotent groups (which appear only for 푛 > 1and 푛 < 2 respectively), was left as an important and interesting open question. After many unsuccessful attempts to find commutative algebras generated by Toeplitz operators and subordinated to the nilpotent group we conjecture that a part from the known cases there are no more new Banach algebras generated by Toeplitz operators with symbols subordinated to the nilpotent group of biholomorphisms of the unit ball𝔹nand commutative on each weighted Bergman space. At the same time such commutative algebras subordinated to the quasi- nilpotent group do exist, is devoted to their description. According to our current understanding the only additional source for the appearance of (Banach) Toeplitz operator algebras which are commutative on each weighted Bergman space comes from a torus action on 𝔹n. The maximal commutative group of biholomorphisms, to which the symbols are subordinated, must contain the torus 핋푘with 푘 ≥ 2, as a subgroup. In the case of the one-dimensional torus핋 the above commutative Toeplitz operator algebras collapse to known commutative C∗-algebras generated by Toeplitz operators whose symbols are invariant under the action of the maximal commutative group of biholomorphisms in question. We recall some notation from [218] that are used throughout Let 푛 푛 2 2 2 𝔹 ≔ {푧 = (푧1, … , 푧2) ∈ ℂ : |푧| = |푧1| + ⋯ + |푧푛| < 1} n n be the unit ball in ℂ . The Siegel domain 퐷푛in ℂ , which is an unbounded realization of the unit ball 𝔹푛,has the form ′ 푛−1 ′ 2 퐷푛 = {푧 = (푧 , 푧푛) ∈ ℂ × ℂ: 퐼푚푧푛 − |푧 | > 0} 푛 n Recall that the Cayley transform 휔: 𝔹 → 퐷푛 maps biholomorphically the unit ball𝔹 onto퐷푛 . Let푣be the usual Lebesgue measure on ℂ푛 ≅ ℝ2푛 and fix 휆 > −1. Then the standard weighted 푛 measure휇휆on𝔹 with weight parameter휆is given by: 훤(푛+휆+1) 풹휇 ≔ 푐 (1 − |푧|2)휆 풹휈 and 푐 ≔ 휆 휆 휆 휋푛훤(휆+1) 푛 Here푐휆is a normalizing constant such that 휇휆(𝔹 ) = 1 On퐷푛we can consider the corresponding weighted measure 휇̃휆defined by. 핝 푐 풹휇̃ (휉′, 휉 ) = 휆 (퐼푚휉 − |휉′|2)휆풹휈(휉′, 휉 ). 휆 푛 4 푛 푛

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푛 푛+휆+1 −푛−휆−1 −1 Let 푓be a function on 𝔹 , then we put (푢휆푓)(휉) ≔ 2 (1 − 푖휉푛) 푓 ∘ 휔 (휉) where 휉 ∈ 퐷푛.A straightforward calculation shows, cf [36,218] 푛 Lemma (6.1.1)[38]: Let 휆 > −1, then 푢휆defines a unitary transformation of 퐿2(𝔹 , 휇휆) onto 퐿2(퐷푛, 휇̃휆). 2 푛 2 In the following we write𝒜휆 (𝔹 ) and 𝒜휆 (퐷푛)for the weighted Bergman spaces of all complex 푛 analytic functions in퐿2(𝔹 , 휇휆)and퐿2(퐷푛, 휇̃휆), respectively. It is known that by restriction 2 푛 2 풰휆defines a unitary transformation of𝒜휆 (𝔹 ) and 𝒜휆 (퐷푛). ( ) 2( ) Let 퐵퐷푛,휆be the Bergman projection of 퐿2 퐷푛, 휇̃휆 onto 𝒜휆 퐷푛 . Given a bounded measurable ∞ function 푓 ∈ 퐿 (퐷푛) we define theToeplitz operator 푇푓acting on the weighted Bergman space 2 𝒜휆 (퐷푛)inthe usual way by

푇푓: = 퐵퐷푛,휆푀푓, Where푀푓denotes the multiplication by푓. We study a class of commutative Banach algebras 2 generated by Toeplitz operators on 𝒜휆 (퐷푛). To simplify the notation we will not indicate the dependence of 푇푓 on the weight parameter휆 . Note that via the unitary transformation 풰휆 the results on Toeplitz operators acting on weighted Bergman spaces over 퐷푛can be directly translated to the corresponding setting of Toeplitz 2 푛 operators on 𝒜휆 (𝔹 ). 푛−1 Put 풟 ≔ ℂ × ℝ × ℝ+. Then the map: ′ ′ ′ 2 휅: 풟 ⟶ 풟푛: (푧 , 푢, 푣) ⟼ (푧 , 푢 + 푖푣 + 푖|푧 | ) defines a diffeomorphism with inverse

−1 ′ ′ ′ 2 휅 (푧 , 푧푛) = (푧 , 푅푒 푧푛, 퐼푚푧푛 − |푧 | ). Given a function 푓 on 퐷푛, we define 푈0푓 ∶ = 푓 ∘ 휅 to obtain a function 푈0푓 on풟. On the domain 풟 we consider the measure 퐶 푑휂 (푧′, 푢, 푣) ≔ 휆 푣휆푑푣(푧′, 푢, 푣). 휆 4 We have the following, of. [36,219] Lemma (6.1.2)[38]: The operator 푈0 is unitary from 퐿2(풟푛, 휇̃휆) to 퐿2(풟, 휇휆) with inverse −1 ∗ ∗ −1 푈0 = 푈0 given by 푈0푓 = 푓 ∘ 휅 . 2 We occasionally omit the dependence of the weight 휆 > −1 and put 𝒜0(풟) ≔ 푈0𝒜휆(풟푛) which clearly forms a closed subspace of L2(풟, ηλ). As was explained in [219,218] the classification of maximal commutative subgroups 퐺 of 푛 biholomorphisms of 풟푛 or 𝔹 yields five essentially different types. Corresponding to each type there are commutative Banach or C∗-algebras of Toeplitz operators acting on weighted Bergman spaces. The aim is to define such algebras in case of the quasi-nilpotent group G of biholomorphisms. We recall the definition. ′ ′ ′ 푘 Let 1 ≤ 푘 ≤ 푛 − 2. We rather use the notation 푧 = (푧 , 휔 , 푧푛) for z ∈ Dn where 푧 ∈ ℂ and ′ 푛−푘−1 푘 푛−푘−1 휔 ∈ ℂ . The quasi-nilpotent group 핋 × ℝ × ℝ acts on 풟푛, cf. [218], as follows: given (푡, 푏, ℎ) ∈ 핋푘 × ℝ푛−푘−1 × ℝ, we have: ′ ′ ′ ′ 2 풯(푡, 푏, ℎ): (푧 , 휔 , 푧푛) → (푡푧 , 휔 + 푏, 푧푛 + ℎ + 2푖휔′. 푏 + 푖|푏| ). Note that in the case 푘 = 푛 − 1 we obtain the quasi-parabolic group, while 140 for 푘 = 0 the group action is called nilpotent. 푘 푛−푘−1 On the domain 풟 = ℂ × ℂ × ℝ × ℝ+ we use the variables (푧′, 휔′, 푢, 푣) and we represent 퐿2(풟, 휂휆)in the form: 푘 푛−푘−1 퐿2(풟, 휂휆) = 퐿2(ℂ ) ⊗ 퐿2(ℂ ) ⊗ 퐿2(ℝ) ⊗ 퐿2(ℝ+, 휂휆) (1)

Let F be the Fourier transform on 퐿2(ℝ), and with respect to the decomposition (1) consider the unitary operators 푈1: = 퐼 ⊗ 퐼 ⊗ 퐹 ⊗ 퐼 acting on 퐿2(풟, 휂휆). With this notation we put 𝒜1(퐷): = 푈1(𝒜0(풟)). k ′ ′ Next, we introduce polar coordinates on ℂ and put 푟 = (푟1, … , 푟푘) = (|푧1|, … , |푧푘|). Moreover, in the following we write 푥′: = 푅푒 휔′ and 푦'∶= 퐼푚 휔. Then one can check that 푟, 푦′ | ′|2 and 퐼푚푧푛 − 휔 are invariant under the action of the quasi-nilpotent group. Following the ( ) ideas in [218] and with 푟푑푟 = 푟1푑푟1 … 푟푘푑푟푘 we represent 퐿2 풟, 휂휆 in the form 푘 푘 푛−푘−1 푛−푘−1 퐿2(ℝ+, 푟푑푟) ⊗ 퐿2(핋 )퐿2(ℝ ) ⊗ 퐿2(ℝ ) ⊗ 퐿2(ℝ) ⊗ 퐿2(ℝ+, 휂휆) (2)

We define the unitary operator 푈2 on 퐿2(풟, 휂휆) by 푈2 = 퐼퐹(푘) 퐹(푛−푘−1) 퐼  퐼  퐼. Here ℱ(푘) = ℱ ⊗ … ⊗ ℱ is the k-dimensional discrete Fourier transform and 퐹(푛−푘−1) = 푛−푘−1 퐹 …  퐹 denotes the (푛 − 푘 − 1)-dimensional Fourier transform on퐿2(ℝ ). Note that 퐿2(풟, 휂휆) is isometrically mapped by U2 onto 푘 푘 푛−푘−1 푛−푘−1 ℓ2((핫 , 퐿2(ℝ+) ⊗ 퐿2(ℝ ) ⊗)퐿2(ℝ ) ⊗ 퐿2(ℝ) ⊗ 퐿2(ℝ+, 휂휆)) (3) We put 𝒜 (퐷) = 푈 (𝒜 (퐷)) and we write elements in (3) as {푓 (푟, 푥′, 푦′, 휉, 휈)} , 2 2 1 훽 훽∈ℤ푘 where ′ ′ 푘 푛−푘−1 푛−푘−1 (푟, 푥 , 푦 , 휉, 휐) ∈ ℝ+ × ℝ × ℝ × ℝ × ℝ+.

Next we recall the definition of the unitary operator 푈3 which acts on (3) by: ′ ′ ′ ′ 1 ′ ′ 푈3: {푓훽(푟, 푥 , 푦 , 휉, 휐)} ⟼ {푓훽 (푟, √휉(푥 , 푦 ), (−푥 , 푦 ), 휉, 푣)} 훽∈ℤ 2√휉 훽∈ℤ푘

−1 One immediately checks that the inverse 푈3 has the form ′ ′ −1 ′ ′ 푥 ′ 푥 ′ 푈3 : {푓훽(푟, 푥 , 푦 , 휉, 휐)} ⟼ {푓훽 (푟, − √휉풴 , + √휉풴 , 휉, 푣)} 훽∈ℤ 2√휉 2√휉 훽∈ℤ푘

In the following we write ℤ+ = ℕ ∪ {0} = {0,1,2, … } for the nonnegative integers. In order to state the main result of Section in [218] we need to introduce the operator R0, which defines 푘 푛−푘−1 an isometric embedding of ℓ2 (핫+, 퐿2(ℝ × ℝ+))into (3). It is explicitly given by 2 |푣′| ′ 훽 −휉(|푟|2+푣)− ′ 푅 : {푐 (푥 , 휉)} 푘 ⟼ {휒 푘 (훽, 휉)퐴 (휉)푟 푒 2 푐 (푥 , 휉)} 0 휕 훽∈ℤ+ ℤ+×ℝ+ 훽 훽 훽∈ℤ푘 = {푔 (푟, 푥′, 푦′, 휉, 푣)} 훽 훽∈ℤ푘

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푘 ′ Here 휒 푘 (훽, 휉)denotes the characteristic function of ℤ × ℝ and 푐 (푥 , 휉) is extended by ℤ+×ℝ+ + + 훽 zero for 휉 ∈ (−∞, 0) and all 푥′ ∈ ℝ푛−푘−1. Moreover, we have used the abbreviation

푛−푘−1 푘+2 |훽|+휆+푘+1 − 2 (2휉) 퐴훽(휉): = 휋 4 √ (4) 퐶휆 훽! 훤(휆 + 1)

∗ The ad joint operator R0 is given by: 푅∗: {푓 (푟, 푥′, 푦′, 휉, 휐)} ⟼ {퐴 (휉) 0 훽 훽∈ℤ푘 훽 2 |푣′| 휆 −휉(|푟|2+푣)− 퐶휆푣 훽 2 ′ ′ ′ × ∫ 푟 푒 푓훽(푟, 푥 , 푦 , 휉, 휐)푟푑푟푑푦 푑푣}훽∈ℤ푘 (5) 푘 푛−푘−1 4 + ℝ+×ℝ ×ℝ+ 2 We set 푈 ∶= 푈3푈2푈1푈0, which gives a unitary operator from 푅휆 (풟푛) onto 𝒜3(풟) ≔ 푈3(𝒜2(풟)). The following result has been proved in [218], and it provides a decomposition of the Bergman projection 퐵풟푛,휆 in form of a certain operator product. ∗ Theorem (6.1.3)[38]: [218] The operator 푅 ∶= 푅0푈 maps 퐿2(풟푛, 휇̃휆) onto the space 푘 푛−푘−1 ℓ2 (핫+, 퐿2(ℝ × ℝ+)), and the restriction 2 푘 푛−푘−1 푅 2 𝒜 (풟) → ℓ (핫 , 퐿 (ℝ × ℝ )) |𝒜휆(풟) 휆 2 + 2 + is an isometric isomorphism. The ad joint operator ∗ ∗ 푘 푛−푘−1 2 푅 = 푈 푅0: 퐿2 (핫+, 퐿2(ℝ × ℝ+)) → 𝒜휆 (풟) ⊂ 퐿2(풟푛, 휇̃휆) 푘 푛−푘−1 2 is an isometric isomorphism of 퐿2 (핫+, 퐿2(ℝ × ℝ+))onto the subspace 𝒜휆 (풟)of

퐿2(풟푛, 휇̃휆). Furthermore one has: ∗ 푘 푛−푘−1 푘 푛−푘−1 푅푅 = 퐼: 퐿2 (핫+, 퐿2(ℝ × ℝ+)) → 퐿2 (핫+, 퐿2(ℝ × ℝ+)), ∗ ( ) 2( ) 푅 푅 = 퐵풟푛,휆: 퐿2 풟푛, 휇̃휆 → 𝒜휆 풟 Now, we restrict our attention to bounded measurable symbols on 풟푛 that are invariant or have a certain homogeneity with respect to the quasi-nilpotent group action on 풟n. Definition (6.1.4)[38]: A bounded measurable function 훼: 풟푛 → ℂ is called quasi-nilpotent if ′ ′ 2 it has the form 훼(푧) = 훼(푟, 푦 , 퐼푚푧푛 − |휔 | ). In particular, such a is invariant under the action of the quasi-nilpotent group. The following theorem was proved in [218]. ′ ′ 2 Theorem (6.1.5)[38]: [218] Let 훼(푧) = 훼(푟, 푦 , 퐼푚푧푛 − |휔 | ) be a bounded measurable 2 quasi-nilpotent function on 풟푛. Then the Toeplitz operator Tα acting on 𝒜휆 (풟)) is unitary ∗ equivalent to the multiplication operator 훾훼퐼 = 푅푇훼푅 acting on the space 푘 푛−푘−1 퐿2 (핫+, 퐿2(ℝ × ℝ+)). The sequence ′ 푘 푛−푘−1 훾 = {훾 (훽, 푥 , 휉)} 푘 ∈ 퐿 (핫 , 퐿 (ℝ × ℝ )) 훼 훼 훽∈ℤ+ 2 + 2 + ′ 푛−푘−1 With (푥 , 휉) ∈ ℝ × ℝ+ is given by

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푛−푘−1 ( )|훽|+휆+푘+1 ′ 푘 − 2휉 1 ′ ′ 2 훾훼(훽, 푥 , 휉) = 2 휋 2 ∫ 훼 (푟, (−푥 + 푦 ), 휐 + (|푟| ) 훽! 훤(휆 + 1) 2√휉 푘 푛−푘−1 ℝ+×ℝ ×ℝ+ 2 ′ 2 × 푟훽푒−2휉(푣+(|푟| )−|푦 | 푣휆푟푑푟푑푦′푑푣. We need to prove a similar result for a class of more general symbols. Recall that we use the notation 푥′ ≔ 푅푒휔′ ∈ ℝ푛−푘−1where 휔′ ∈ ℂ푛−푘−1and let 훼 = 푚 (훼1, … , 훼푚)be a tuple in 핫+ such that |훼| = (훼1, … , 훼푚) = 푘 Similar to [38,193] we divide the coordinates of 푧′ ∈ ℂ푘 into m groups as follows: ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ 푧(1) = (푧1,1, … , 푧1,훼1), 푧(2) = (푧2,1, … , 푧2,훼2), …,푧(푚) = (푧푚,1, … , 푧푚,훼푚 )and such that 푧 = ′ ′ ′ (푧(1), 푧(2), … , 푧(푚)). In the following we will use the same notation also in case of multi-indices 푘 ′ 푘 ′ 훽 ∈ 핫 instead of vectors 푧 ∈ ℂ . By passing to polar coordinates, we write each tuple 푧(푗) = ′ ′ (푧푗,1, … , 푧푗,훼푗)where 푗 = 1, . . . , 푚, in the form 2 2 ′ ′ ′ 2αj−1 αj 푧(푗) or 푟푗휉(푗) with 푟푗 = √|푍푗,1| + ⋯ + |푍푗,훼푗| andξ(j) ∈ 핊 ⊂ ℂ . Here 핊2αj−1denotes the real (2n − 1) -dimensional boundary of 𝔹푛. ′ ′ 2 m Definition (6.1.6) )[38]: Let α(r, y Im핫n − |ω | ) be a quasi-nilpotent function and α ∈ ℤ+ as above. (i) Then 훼 is called “α-quasi-nilpotent quasi-radial” if its radial dependence on r can be expressed as a function of 푟1, … , 푟푚. ′ ′ (ii) The function 푏 = (푧 , 휔 , 핫푛) is called “α-quasi-nilpotent quasi-homogeneous”if it is α- quasi-nilpotent quasi-homogeneous with respect to the variable 푧′, i.e. ′ ′ ′ ′ 2 푝 ̅푞 푏 = (푧 , 휔 , 푧푛) = 푏0(푟1, … , 푟푚, 푦 퐼푚핫푛 − |휔 | )휉 휉 (6) 2훼1−1 2훼2−1 2훼푚−1 푚 Where 휉 = (휉1, 휉2, … , 휉푚) ∈ 핊 × 핊 × … × 핊 and 푝, 푞 ∈ 핫+ are orthogonal. The pair (푝, 푞) is then called the “degree”of 푏. 푚 Note that there is a one-to-one correspondence between the set of tuples {(푝, 푞) ∈ ℤ+ × 푚 k ℤ+ : 푝 ⊥ 푞} and ℤ via (푝, 푞) ⟼ 푝 − 푞. ′ ′ Consider an α-quasi-nilpotent quasi-homogeneous symbol 푏 = (푧 , 휔 , 핫푛) as in (6) and of 푘 푘 ∗ degree (p, q) ∈ 핫+ × 핫+ with p ⊥ q. Our next aim is to calculate the operator 푅푇푏푅 . On the 푘 푛−푘−1 ′ ′ domain 풟 = ℂ × ℂ × ℝ × ℝ+we use the variables (푧 , 휔 , 푢, 푣).Moreover,we express ′ 1 푧′in polar coordinates 푧 = 푡1푟1, … , 푡푘푟푘 where 푟푠 ≥ 0 and 푡푠 ∈ 핤 = 푆 for 푠 = 1, … , 푘. Then we have the relations 푧푗,ℓ = 푟푗휉푗,ℓ = 푡푗,ℓ푟푗,ℓ −1 for ℓ = {1, … , 훼푗} and 푗 = 1, … , 푚. It follows that 휉푗,ℓ = 푡푗,ℓ푟푗,ℓ푟푗 in the case of푟푗 ≠ 0 and therefore: 푚 푝 ̅푞 푝 푞 푝+푞 −|푝푗|−|푞푗| 휉 휉 = 푡 푡̅ 푟 ∏ 푟푗 (7) 푗=1 ′ p ̅q m Note that the assignment z ⟼ ξ ξ depends on the initial choice of α ∈ ℤ+ . Using Theorem (6.1.1) we can write:

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∗ ∗ ( ∗ ) ( ∗ ) ∗ ( ∗) ∗( ∗) ∗ 푅푇푏푅 = 푅퐵풟푛,휆푏퐵풟푛,휆푅 = 푅 푅 푅 푏 푅 푅 푅 = 푅푅 푅푏푅 푅푅 = 푅푏푅 ∗ −1 −1 −1 −1 = 푅0푈3푈2푈1푈0푏푈0 푈1 푈2 푈3 푅0 ∗ ′ ′ 2 푝 ̅푞 −1 −1 −1 −1 = 푅0푈3푈2푈1푏0(푟1, … , 푟푚, 푦 퐼푚푧푛 − |푤 | )휉 휉 푈0 푈1 푈2 푈3 푅0 −1 ′ ′ First we calculate the operator 푈0푏푈0 . Let {푓훽(푟, 푥 , 푦 , 휉, 휐)} 푘 be an element in the space 훽∈푍+ (3) and write 푟 ∶= (푟1, . . . , 푟푚). Since the ymbol ′ 2 푝 ̅푞 ′ 푏0(푟1, 푦 , 푣 + |푟| )휉 휉 is independent of 푥 we obtain from (7) that: ′ 2 푝 ̅푞 −1 ′ ′ 푈2푏0(푟, 푦 , 푣 + |푟| )휉 휉 푈2 {푓훽(푟, 푥 , 푦 , 휉, 휐)} 푘 (8) 훽∈ℤ+ 푚 ′ 2 푝−푞 −|푝푗|−|푞푗| ′ ′ {푏0(푟, 푦 , 푣 + |푟| )푟 (∏ 푟푗 ) 푓훽−푝+푞(푟, 푥 , 푦 , 휉, 휐)} 푗=1 훽∈ℤ푘 Combining (8) and (5) gives:

∗ ′ ∗ −1 −1 푅푇푏푅 : {퐶훽(푥 , 휉)} 푘 = 푅0푈3푈2푏푈2 푈3 {휒ℤ푘 (훽, 휉) 훽∈ℤ+ +×ℝ+ 2 |푣′| 훽 −휉(|푟|2+푣)− ′ 퐴훽(휉)푟 푒 2 푐훽(푥 , 휉)} 훽∈ℤ푘 ∗ −1 훽 = 푅 푈 푈 푏푈 {휒 푘 (훽, 휉)퐴 (휉)푟 0 3 2 2 ℤ+×ℝ+ 훽 2 1 1 −휉(|푟|2+푣)− −| 푥′+√휉푦′| 1 2 2√휉 ′ ′ × 푒 푐훽 ( 푥 − √휉푦 , 휉)} 2√휉 훽∈ℤ푘 ∗ 훽+2푞 ′ 2 = 푅 푈 {휒 푘 (훽 − 푝 + 푞, 휉)퐴 (휉)푟 푏 × (푟, 푦 , 푣 + |푟| ) 0 3 핫+×ℝ+ 훽−푝+푞 0 2 푚 1 1 −휉(|푟|2+푣)− −| 푥′+√휉푦′| −|푝푗|−|푞푗| 2 2√휉 × (∏ 푟푗 ) 푒 푗=1

1 ′ ′ × 퐶훽−푝+푞 ( 푥 − √휉푦 , 휉)} 2√휉 훽∈ℤ푘 ′ ′ ∗ −푥 + 푦 2 = 푅0 {휒핫푘 ×ℝ (훽 − 푝 + 푞, 휉)퐴훽−푝+푞(휉)푏0 (푟, , 푣 + |푟| ) + + 2√휉 푚 1 2 −|푝푗|−|푞푗| −휉(|푟|2+푣)− −|푦′| 훽+2푞 2 ′ × (∏ 푟푗 ) 푟 푒 퐶훽−푝+푞(푥 , 휉)}훽∈ℤ푘 푗=1

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′ = {퐴 (휉)퐴 (휉)휒 푘 (훽 − 푝 + 푞, 휉)퐶 (푥 , 휉)} 푘 훽 훽−푝+푞 ℤ+×ℝ+ 훽−푝+푞 훽∈푍 푚 2(훽+푞) −|푝푗|−|푞푗| −휉(|푟|2+푣)−|푦|2 × ∫ 푟 (∏ 푟푗 ) 푒 푘 푛−푘−1 ℝ+×ℝ ×ℝ+ 푗=1 ′ ′ 휆 −푥 + 푦 2 ′ 퐶휆푣 × 푏0 (푟, , 푣 + |푟| ) 푟푑푟푑푦 푑푣} 2√휉 4 푘 훽∈ℤ+ Now put: ′ 훾푏,푝,푞(훽, 푥 , 휉): 푚 −|푝푗|−|푞푗| = 퐴훽(휉)퐴훽−푝+푞(휉)휒ℤ푘 ×ℝ (훽 − 푝 + 푞, 휉) ∫ ∏ 푟푗 + + 푘 푛−푘−1 ℝ+×ℝ ×ℝ+ 푗=1 ′ ′ 2(훽+푞) −2휉(|푟|2+푣)−|푦|2 −푥 + 푦 2 ′ × 푟 푒 × 푏0 (푟, , 푣 + |푟| ) 푟푑푟푑푦 2√휉 퐶 푣휆 × 휆 푑푣 (9) 4 Hence, we have proved: ∗ Theorem (6.1.7) )[38]: Let b be defined as in (6). The operator RTbR acts on the Hilbert space 푘 푛−푘−1 ℓ2 (ℤ+, 퐿2(ℝ × ℝ+))by the rule: ∗ ′ ′ ′ 푅푇푏푅 : {퐶훽(푥 , 휉)} 푘 = {훾푏,푝,푞(훽, 푥 , 휉). 퐶훽−푝+푞(푥 , 휉)} 푘 훽∈ℤ+ 훽∈ℤ Note that, in the case 푝 = 푞 = 0,Theorem (6.1.8) reduces to Theorem (6.1.5). ∗ Example(6.1.8)[38]: We calculate 푅푇푏푅 more explicitly in the special case whereb0 ≡ 1 and 푘 푘 we choose 푘 = 푚, i.e. 훼 = (1, … ,1) ∈ 핫+. Let (푝, 푞)ℤ+such that 푝 ⊥ 푞 and put ′ ′ 푝 ̅푞 푝 푞 푏(푧 , 휔 , 푧푛) = 휉 휉 = 푡 푡̅ According to Theorem (6.1.7) it is sufficient to calculate the functions: ′ γb,p,q(β, x , ξ): ( ) ( ) ( ) = Aβ ξ Aβ−p+q ξ χℝ+ ξ λ 2 2 C v × ∫ r2β+q−p e−2ξ(|r| +v)−|y| rdrdy′ λ dv k n−k−1 4 ℝ+×ℝ ×ℝ+ 푘 푘 for all 훽 ∈ ℤ+ with 훽 − 푝 + 푞 ∈ 핫+. We use the identity: 2 푛−푘−1 ∫ 푒−2휉푣−|푦| 푑푦′푣휆푑푣 = 휋 2 훤(휆 + 1)(2휉)−(휆+1) 푘 푛−푘−1 ℝ+×ℝ ×ℝ+ (cf [97]) where 휉>0, which together with (4) shows that |푞|−|푝| ′ 푘 |훽|+푘+ 1 2훽+푞−푝 −2휉−|푦|2 훾푏,푝,푞(훽, 푥 , 휉) = 2 (2휉) 2 ∫ 푟 푒 푑푟 푘 √훽! (훽 − 푝 + 푞)! ℝ+ 푘 푞푗 − 푝푗 ∏푗=1 훤 (훽푗 + + 1) = 2 . √훽! (훽 − 푝 + 푞)! ′ ′ In particular, in this case 훾푏,푝,푞(훽, 푥 , 휉)is independent of 푥 and 휉. 145

The goal is to study the commutatively of Toeplitz operators with symbols having 푚 certain invariance properties. We will use the above notation. Fix 훼 ∈ 핫+ with |훼| = 푘 as ′ ′ 2 before and let 훼 = 훼0(푟1, … , 푟푚, 푦 퐼푚푧푛 − |휔 | ) be a bounded measurable 훼-quasi-nilpotent quasi-radial function on 퐷푛. Consider the symbol: ′ ′ ′ ′ 2 푝 ̅푞 푏(푧 , 휔 , 푧푛) = 푏0(푟1, … , 푟푚, 푦 퐼푚푧푛 − |휔 | ). 휉 휉 (10) ∗ ∗ We calculate the operator products 푅푇푏푇푎푅 and 푅푇푎푇푏푅 . According to Theorem (6.1.7) and Theorem (6.1.1) we have ∗ ∗ ∗ 푅푇푏푇푎푅 {푐훽} 푘 = (푅푇푏푅 )(푅푇푎푅 ){푐훽} 푘 훽∈ℤ+ 훽∈ℤ+ ∗ ′ ′ = (푅푇푏푅 ){훾푏̅ ,푝,푞(훽, 푥 , 휉). 푐훽(푥 , 휉)} 푘 훽∈ℤ+ ′ ′ ′ = {훾푏̅ ,푝,푞(훽, 푥 , 휉)훾푎,0,0(훽 − 푝 + 푞, 푥 , 휉)푐훽−푝+푞(푥 , 휉)} k (11) β∈ℤ+ On the other hand it follows: ∗ ∗ ∗ ∗ ′ ′ 푅푇푎푇푏푅 {푐훽} 푘 = (푅푇푎푅 )(푅푇푏푅 ){푐훽} 푘 = (푅푇푎푅 ){훾푏̅ ,푝,푞(훽, 푥 , 휉). 푐훽(푥 , 휉)} 푘 훽∈ℤ+ 훽∈ℤ+ 훽∈ℤ+ ′ ′ ′ = {훾푎̅ ,0,0(훽, 푥 , 휉)훾푏̅ ,푝,푞(훽, 푥 , 휉)푐훽−푝+푞(푥 , 휉)} 푘 (12) 훽∈ℤ+ Hence, we conclude from (11) and (12) that both operators 푇푎 and 푇푏 commute if and only if ′ ′ 훾푎̅ ,0,0(훽, 푥 , 휉) = 훾푎̅ ,0,0(훽 − 푝 + 푞, 푥 , 휉) 푘 for all 훽 ∈ ℤ+. According to (9) this is equivalent to: ′ ′ 2 1 −푥 + 푦 2 2훽 −2휉(푣−|푟|2)−|푦′| 휆 ′ ∫ 푎0 (푟, , 푣 + |푟| ) 푟 푒 푣 푟푑푟푑푦 푑푣 훽! 푘 푛−푘−1 ℝ+×ℝ ×ℝ+ 2√휉 ( )−|푝|+|푞| ′ ′ 2휉 −푥 + 푦 2 = ∫ 푎0 (푟, , 푣 + |푟| ) (훽 − 푝 + 푞)! 푘 푛−푘−1 ℝ+×ℝ ×ℝ+ 2√휉 2 ′ 2 × 푟2훽+푞−푝푒−2휉(푣+|푟| )−|푦 | 푣휆푟푑푟푑푦′푑푣 (13) ′ ′ 2 Since 푎0(푟, 푦 , 퐼푚푧푛 − |휔 | )only depends on 푟 = (푟1, … , 푟푚)we can assume that the above integral has the form: ′ ′ 2 −푥 + 푦 2 2훽 −2휉(푣+|푟|2)−|푦′| 휆 ′ ∫ 푎0 (푟, , 푣 + |푟| ) 푟 푒 푣 푟푑푟푑푦 푑푣 =: (∗), 푘 푛−푘−1 ℝ+×ℝ ×ℝ+ 2√휉 푘 푘 Where 훽 ∈ ℤ+. With 푒 = (1,1, … ,1) ∈ ℤ+ we obtain ′ ′ 2 1 −푥 + 푦 2 2훽 −2휉(푣+|푟|2)−|푦′| 휆 ′ (∗) = 푘 ∫ 푎0 (푟, , 푣 + |푟| ) |푟 |푒 푣 푟푑푟푑푦 푑푣 2 푘 푛−푘−1 ℝ ×ℝ ×ℝ+ 2√휉 ′ ′ 1 −푥 + 푦 2 = 푘 ∫ ∫ 푎0 (푟, , 푣 + |푟| ) 2 푛−푘−1 푚 푎 −1 푎 −1 ℝ ×ℝ+ ℝ+ ×핊 1 ×핊 푚 2√휉 푚 2 2훽+푒 2|훽(푗)|+2푎푗−1 −2휉(푣+|푟|2)−|푦′| 휆 ′ × |휌 |. (∏ 푟푗 ) 푒 푣 푑휎(휌(1)) … 푑휎(휌(푚))푟푑푟푑푦 푑푣. 푗=1 푎푗−1 In the last integral we wrote 푑휎(휌(푗)) for the standard area measure on the sphere 핊 . The integral over the m-fold product 핊푎1−1 × … × 핊푎푚−1can be calculated explicitly by using the following well-known formula:

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Lemma (6.1.9) )[38]: Let 푑휎 denote the usual surface measure on the (푛 − 1) −dimensional 푛 − 1 푘 sphere 핊 and let 휃 ∈ ℤ+. Then 휃 − 1 휃 − 1 2훤 ( 1 ) … 훤 ( 푛 ) ∫ |푦휃|푑휎(푦) = 2 2 푛 − |휃| 핊푛−1 훤 ( 2 ) Using the formula in Lemma (6.1.9) we define:

2훽+푒 훩훽 ≔ ∫ |휌 |푑휎(휌(1)) … 푑휎(휌(푚)) 핊푎1−1×…×핊푎푚−1 푚 −1 푎푗 + 1 = 2푚훽! ∏ 푟 ( + |훽 |) (14) 2 (푗) 푗=1 This finally gives: ⊖ 훽 1 ′ ′ 2 (∗) = 푘 ≔ ∫ 푎0 (푟, (−푥 + 푦 ), 푣 + |푟| ) 2 푚 푛−푘−1 ℝ+ ×ℝ ×ℝ+ 2√휉 2 2|훽(1)|+2푎1−1 2|훽(푚)|+2푎1−푚 −2휉(푣+|푟|2)−|푦′| 휆 ′ × 푟1 … 푟푚 푒 푣 푑푟푑푦 푑푣. Note that the last integral does not depend on the full multi-index β but rather on the values |훽(1)| for 푗 = 1, … , 푚. We denote this integral by 퐺푎(|훽(1)|, … , |훽(푚)|). Then the commutativity condition (13) can be written in the form: 훩훽 퐺 (|훽 |, … , |훽 |) 훽! 푎 (1) (푚) 훩훽−푝+푞 = (2휉)−|푝|+|푞| (훽 − 푃 − 푞)! × 퐺푎(|훽(1)| − |푝(1)| + |푞(1)|, … , |훽(푚)| − |푝(푚)| + |푞(푚)|). According to the definition (14) this is equivalent to: 푚 −1 푎푗 + 1 퐺 (|훽 |, … , |훽 |) ∏ 훤 ( + |훽 |) 푎 (1) (푚) 2 (푗) 푗=1 −|푝|+|푞| = (2휉) 퐺푎(|훽(1)| − |푝(1)| + |푞(1)|, … , |훽(푚)| − |푝(푚)| + |푞(푚)|) 푚 −1 푎푗 + 1 × ∏ 훤 ( + |훽 | − |푝 | + |푞 |) 2 (푗) (푗) (푗) 푗=1 This equality can be only true simultaneously for all α-quasi-nilpotent quasi-radial functions 푘 a and all 훽 ∈ ℤ+ if |푝(푗)| = |푞(푗)| for 푗 = 1, … , 푚. Hence, we obtain: 푘 Theorem (6.1.10) )[38]: Let 푎 ∈ ℤ+be given. Then the statements (i), (ii) and (iii)below are equivalent: ′ ′ 2 (i) For each α-quasi-nilpotent quasi-radial function 푎 = 푎0(푟1,푦 퐼푚푧푛 − |휔 | ) ∈ ∞ 퐿 (퐷푛)and each α-quasi-nilpotent quasi-homogeneous function ′ ′ 2 푝 ̅푞 ∞ 푏 = 푏0(푟1, … , 푟푚, 푦 퐼푚푧푛 − |휔 | )휉 휉 ∈ 퐿 (퐷푛) (15)

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푘 푘 of degree (푝, 푞) ∈ ℤ+ × ℤ+ the Toeplitz operators 푇푎 and 푇푎 commute on each weighted 2 Bergman space 𝒜λ(Dn). ′ ′ 푘 (ii) The equality 훾푎̅ ,0,0(훽, 푥 , 휉) = 훾푎̅ ,0,0(훽 − 푝 + 푞, 푥 , 휉)holds for all 훽 ∈ ℤ+ and for each α-quasi-nilpotent quasi-radial functions a. (iii) The equality |푝(푗)| = |푞(푗)| holds for each 푗 = 1, . . . , 푚. ∞ Now, let us assume that 푏 ∈ 퐿 (퐷푛) is of the form (15). Under the assumption |푝(푗)| = |푞(푗)|, ′ for each 푗 = 1, … , 푚, we calculate γ̅b,p,q(β, x , ξ) in (9) more explicitly by reducing the order 푘 of integration. Assume that 훽 − 푝 + 푞 ∈ 핫+. Then: ′ 훾푏̅ ,푝,푞(훽, 푥 , 휉)

( ) ( ) ( ) 2(훽+푞) = 퐴훽 휉 퐴훽−푝+푞 휉 휒ℝ+ 휉 ∫ 푟 푘 푛−푘−1 ℝ+×ℝ ×ℝ+ 푚 2 −|푝(푗)|−|푞(푗)| −2휉(|푟|2+푣)−|푦′| × ∏ 푟푗 푒 푏0 푗=1 −푥′ + 푦′ 푐휆푣휆 × (푟, , 휈 + |푟|2) 푟푑푟푑푦′ 푑푣 2√휉 4 ( ) ( ) ( ) −k = Θβ+qAβ ξ Aβ−p+q ξ χℝ+ ξ 2 m 2|β(j)|+|q(j)|−|p(j)|+2αj−1 × ∫ ∏ rj m n−k−1 ℝ+ ×ℝ ×ℝ+ j=1 2 ′ ′ 휆 −2휉(|푟|2+푣)−|푦′| −푥 + 푦 2 ′ 푐휆푣 × 푒 푏0 (푟, , 휈 + |푟| ) 푟푑푟푑푦 푑푣 2√휉 4 훩 퐴 (휉) 훽+푞 훽−푝+푞 ′ = . 퐷푏(훽, 푥 , 휉) 훩훽 퐴훽(휉) 푚 훼푗 + 1 ( ) 훤 ( + |훽(푗)|) 훽 + 푞 ! 2 ′ = ∏ . 퐷푏(훽, 푥 , 휉), ( ) 훼푗 + 1 √훽! 훽 − 푝 + 푞 ! 푗=1 훤 + 훽 + 푞 ( 2 | (푗)| | (푗)|) ′ ′ where 퐷푏(훽, 푥 , 휉)= 훾푏̅ ,푝,푞(훽, 푥 , 휉), which can be seen by choosing 푝 = 푞 = 0in the above equalities. Hence we have proved: 푚 ∞ Proposition (6.1.11) )[38]: Let 훼 ∈ 핫+ be given. Assume that 푏 ∈ 퐿 (퐷푛) is of theform (15) 푚 and let |푝(푗)| = |푞(푗)|, for each 푗 = 1, . . . , 푚.Then in the case of훽 − 푝 + 푞 ∈ 핫+ we have 푚 훼푗 + 1 ( ) 훤 ( + |훽(푗)|) ′ 훽 + 푞 ! 2 ′ 훾̃푏,푝,푞(훽, 푥 , 휉) = ∏ . 훾̃푏,0,0(훽, 푥 , 휉). ( ) 훼푗 + 1 √훽! 훽 − 푝 + 푞 ! 푗=1 훤 + 훽 + 푞 ( 2 | (푗)| | (푗)|) 푘 ′ In the case 표푓 훽 − 푝 + 푞 ∉ 핫+we have 훾̃푏,푝,푞(훽, 푥 , 휉) = 0. The factor ′ 훾̃푏,푝,푞(훽, 푥 , 휉)can be expressed int he form

148

푚 2|훽(푗)|+|푞(푗)|−|푝(푗)|+2훼 −1 ( ′ ) 2 ( ) ( ) −푘 푗 훽, 푥 , 휉 = 훩훽퐴훽 휉 휒ℝ+ 휉 2 ∫ ∏ 푟푗 푚 푛−푘−1 ℝ+ ×ℝ ×ℝ+ 푗=1 2 ′ 2 −푥′+푦′ 푐휆푣휆 × 푒−2휉(|푟| +푣)−|푦 | 푏 (푟, , 휈 + |푟|2) 푟푑푟푑푦′ 푑 (16) 0 2√휉 4 푚 k k Let 훼 ∈ 핫+ be given and (p + q) ∈ 핫+ × 핫+. From Proposition (6.1.11) we conclude: ′ ′ 2 ∞ Corollary (6.1.12) )[38]: Let 훼 = 훼0(푟, 푦 , 퐼푚푧푛 − |휔 | ) ∈ 퐿 (퐷푛) be an 훼 − quasinilpotent quasiradial function. Under the assumption |p(j)| for all 푗 = 1, 2, … , 푚we have 푇훼푇휉푝휉̅ 푞 = 푇휉푝휉̅ 푞푇훼 = 푇훼휉푝휉̅ 푞 (17) on each weighted Bergman space. Proof: The first equality in (17) is a direct consequence of Theorem (6.1.10). If 푒(푧) ≡ ′ 1 then 푇푒 = 퐼푑, and thus γ̃b,0,0(β, x , ξ) ≡ 1. Hence, Proposition (6.1.11) implies that in the case of a symbol bξpξ̅q with |p(j)| = |q(j)|, for all 푗 = 1,2, . . . , 푚, one has 푚 훼푗 + 1 ( ) 훤 ( + |훽(푗)|) ′ 훽 + 푞 ! 2 훾̃푏,푝,푞(훽, 푥 , 휉) = ∏ , (18) ( ) 훼푗 + 1 √훽! 훽 − 푝 + 푞 ! 푗=1 훤 + 훽 + 푞 ( 2 | (푗)| | (푗)|) 푘 푘 Whenever훽 − 푝 + 푞 ∈ 핫+(cf. Example(6.1.8) for the choice of 훼 = (1, … ,1) ∈ 핫+ and the 푘 ′ case 푝푗 = 푞푗, 푗 = 1, … , 푘). Moreover, if 훽 − 푝 + 푞 ∈ 핫+, then it holds 훾̃푏,푝,푞(훽, 푥 , 휉). Theorem (6.1.10), Proposition (6.1.11) and the assumption that |푝(푗)| = |푞(푗)|, for all푗 = 1, 2,··· , 푚, imply now that ′ ′ ′ 훾̃푎푏,푝,푞(훽, 푥 , 휉) = 훾̃푏,푝,푞(훽, 푥 , 휉) . 훾̃푎,0,0(훽, 푥 , 휉) ′ ′ = 훾̃푏,푝,푞(훽, 푥 , 휉) . 훾̃푎,0,0(훽 − 푝 + 푞, 푥 , 휉) This together with (11) and Theorem (6.1.7) yields the second equality in (17). We define commutative Banach algebras of Toeplitz operators which are induced by the 푚 푚 quasi-nilpotent group action. Given a pair of multi-indices (푝, 푞) ∈ 핫+ × 핫+ , we put 푝̃(푗) ≔(0, … , 푝(푗), 0, … , ) and 푞̃(푗)≔(0, … , 푞(푗), 0, … , ) so that 푝 = 푝̃(1) + 푝̃(2) + ⋯ + 푝̃(푚) and 푞 = 푞̃(1) + 푞̃(2) + ⋯ + 푞̃(푚) Consider the Toeplitz operators: 푇푗: = 푇휉푝̅(푗)휉̅ 푞̅(푗) (cf. Definition (6.1.6)). Now, we can prove that certain products of Toeplitz operators are Toeplitz operators again with the product symbol. Proposition (6.1.13) )[38]: Let us assume that |푝(푗)| = |푞(푗)| for all 푗 = 1, 2, . . . , 푚. Then the Toeplitz operators 푇푗 commute mutually. Moreover, 푚

∏ 푇푗: = 푇휉푝휉̅ 푞 (19) 푗=1 on each weighted Bergman space.

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푝̃(푗) ̅푞̃(푗) Proof: Let 푏푗 ≔ 휉 휉 , for 푗 = 1, … , 푚, We only prove the following product rule: 푇푗푇푖: = 푇휉푝̃(푖)휉̅ 푞̃(푖)+푞̃(푗) (20) ∗ for 푖, 푗 ∈ {1, … , 푚}and 푖 ≠ 푗. According to Theorem (6.1.7) the operator 푅푇푗푇푖푅 acts on the 푘 푛−푘−1 sequence space ℓ2 (핫+, 퐿2(ℝ × ℝ+)) by the rule: ∗ ′ 푅푇푗푇푖푅 {푐훽(푥 , 휉)} 푘 훽∈핫+ ∗ ( ′ ) ′ = 푅푇푗푅 {훾̃푏푗,푝̅(푗),푞̅(푗) 훽, 푥 , 휉 . 훾̃푏푖,푝̅(푖),푞̅(푖)(훽 − 푝̃(푗), 푞̃(푗), 푥 , 휉) × 훾̃ (푥′, 휉)} 푐훽−푝̃(푖)−푝̃(푗)+푞̃(푖)+푞̃(푗) 푘 훽∈핫+ Hence it is clear that (20) is equivalent to:

( ′ ) ′ ( ′ ) 훾̃푏푗,푝̃(푗),푞̃(푗) 훽, 푥 , 휉 . 훾̃푏푖,푝̃(푖),푞̃(푖)(훽 − 푝̃(푗), 푞̃(푗), 푥 , 휉) = 훾̃푏푖,푏푗,푝̃(푖),푝̃(푗),푞̃(푖),푞̃(푗) 훽, 푥 , 휉 (21) By (18) we have 훼 + 1 훤 ( 푗 + |훽 |) (훽(푗) + 푞̃(푗))! 2 (푗) 훾̃ (훽, 푥′, 휉) = , 푏푗,푝̃(푗),푞̃(푗) 훼 + 1 훤 ( 푗 + |훽 | + |푞̃ |) √훽(푗)! (훽(푗) − 푝̃(푗) + 푞̃(푗))! 2 (푗) (푗) and similar for 푖 replaced by 푗. Moreover, the function on the right hand side of (21) has the explicit form: ( ′ ) 훾̃푏푖,푏푗,푝̃(푖),푝̃(푗),푞̃(푖),푞̃(푗) 훽, 푥 , 휉 훼ℓ + 1 (훽 + 푞̃(푖) + 푞̃(푗))! 훤 ( + |훽(ℓ)|) = ∏ 2 훼 + 1 훤 ( ℓ + |훽 | + |푞̅ |) √훽! (훽 − 푝̃(푖) − 푝̃(푗) + 푞̃(푖) + 푞̃(푗))! ℓ∈{푖,푗} 2 (ℓ) (ℓ) Now, (21) can be easily checked from these identities. 푚 Let 훼 ∈ 핫+ with |훼| = 푘 as before and consider two α-quasi-nilpotent ∞ quasi-homogeneous functions 휑푗 ∈ 퐿 (퐷푛) where j = 1, 2. We express 휑푗 , for 푗 = 1,2 in the form ′ ′ ′ ′ 2 푝 푞 휑1(푧 , 휔 , 푧푛) = 훼1(푟1, … , 푟푚, 푦 퐼푚푧푛 − |휔 | )휁 휁̅ , ′ ′ ′ ′ 2 푝 푞 휑2(푧 , 휔 , 푧푛) = 훼2(푟1, … , 푟푚, 푦 퐼푚푧푛 − |휔 | )휁 휁̅ , 푘 푘 where (푝, 푞), (푢, 푣)휖핫+ × 핫+ with 푝 ⊥ 푞 and 푢 ⊥ 푣 are the degrees of φ1 and 휑2, respectively. Moreover, assume that |푝(푗)| = |푞(푗)|and |푢(푗)| = |푣(푗)|, for 푗 = 1, 2, . . . , 푚. Theorem (6.1.14) )[38]: The Toeplitz operators T휑1 and T휑2 commute on each weighted 2 Bergman space 𝒜휆 (퐷푛) if and only if for each ℓ=1,2, . . . , 푘 one of the conditions (i) − (iv) is fulfilled: (i) 푝ℓ = 푞ℓ = 0 (ii) 푢ℓ = 푣ℓ = 0 (iii) 푝ℓ = 푢ℓ = 0 (iv) 푞ℓ = 푣ℓ = 0 150

Proof: Similar to the argument in the proof of Proposition (6.1.13) it follows that the operators 2 푘 Tφ1 and Tφ2 commute on 𝒜λ(Dn) if and only if for all 훽 ∈ 핫+ : ( ′ ) ( ′ ) ( ′ ) ( ′ ) 훾̃휑1,푝,푞 훽, 푥 , 휉 . 훾̃휑2푢,푣 훽 − 푝 + 푞, 푥 , 휉 = 훾̃휑2푢,푣 훽, 푥 , 휉 . 훾̃휑1,푝,푞 훽 − 푝 + 푞, 푥 , 휉 Since |푝(푗)| = |푞(푗)|and |푢(푗)| = |푣(푗)|, for 푗 = 1, 2, . . . , 푚we can use the ( ′ ) ( ′ ) factorization of훾̃휑1,푝,푞 훽, 푥 , 휉 and 훾̃휑2푢,푣 훽, 푥 , 휉 in Proposition (6.1.11): ( ′ ) ( ) ( ′ ) 훾̃휑1,푝,푞 훽, 푥 , 휉 = 훷푝,푞 훽 . 훾̃휑10,0 훽, 푥 , 휉 , ( ′ ) ( ) ( ′ ) 훾̃휑2푢,푣 훽, 푥 , 휉 = 훷푢,푣 훽 . 훾̃휑20,0 훽, 푥 , 휉 , where we use the notation: 훼 + 1 푚 Γ 푗 + 훽 (훽 + 푞)! ( 2 | (푗)|) 훷푝,푞(훽) = ∏ (22) ( ) 훼푗 + 1 √훽! 훽 − 푝 + 푞 ! 푗=1 훤 + 훽 + 푞 ( 2 | (푗)| | (푗)|) Moreover, it follows from Theorem (6.1.10) and again by the conditions on (푝, 푞) and (푢, 푣) that ( ′ ) ( ′ ) 훾̃휑10,0 훽, 푥 , 휉 = 훾̃휑10,0 훽 − 푢 + 푣, 푥 , 휉 ( ′ ) ( ′ ) 훾̃휑20,0 훽, 푥 , 휉 = 훾̃휑20,0 훽 − 푝 + 푞, 푥 , 휉 Therefore we only need to verify that 훷푝,푞(훽). 훷푢,푣(훽 − 푝 + 푞) = 훷푢,푣(훽). 훷푝,푞(훽 − 푢 + 푣). By a straightforward calculation this is equivalent to: (훽−푝+푞+푣)! (훽−푢+푣+푞)! (훽 + 푞)! = (훽 + 푣)! . (훽−푝+푞)! (훽−푢+푣)! Varyingβ it can be seen that this equality holds if and only if for each ℓ = 1, 2, . . . , 푘 one of the conditions (i) − (iv) is fulfilled. 푘 푘 푚 푚 Let (푝, 푞) ∈ 핫+ × 핫+ and 훼 ∈ 핫+ such that |α| = k. Let ℎ ∈ 핫+ be given with the properties: (i) ℎ푗 = 0, if 훼푗 = 1,

(ii) 1 ≤ ℎ푗 ≤ 훼푗 − 1, if훼푗 > 1. In the case of 훼푗1 = 훼푗2 with 푗1 < 푗2 we assume that ℎ푗1 ≤

ℎ푗2 . In the following we assume that 푝(푗) and 푞(푗) for 푗 = 1, … , 푚 are of the particular form

푝(푗) = (푝푗,1 , … , 푝푗,ℎ푗 , 0, … ,0) and 푞(푗) = (0, … , 0, 푞푗,ℎ푗+1, … , 푞푗,훼푗). (23) below we will use the data 훼 and ℎ to define commutative Banach algebras of Toeplitz operators.The second assumption in (ii) serves to avoid repetition of the unitary equivalent algebras. Define ℛ훼(ℎ) to be the linear space generated by all bounded measurableα-quasi-nilpotent quasi-homogeneous functions ′ ′ ′ ′ 2 푝 푞 푏(푧 , 휔 , 푧푛) = 푏0(푟1, … , 푟푚, 푦 , 퐼푚푧푛 − |휔 | ). 휁 휁̅ (24) Moreover, in (24) we assume that p(j) and q(j) are of the form (23) with:

푝푗,1 + ⋯ + 푝푗,ℎ푗 = 푞푗,ℎ푗+1 + ⋯ + 푞푗,훼푗 . As a corollary to Theorem (6.1.14) we obtain: Theorem (6.1.15)[38]: The Banach algebra generated by Toeplitz operators with symbols from ℛ훼(ℎ) is commutative. 151

Finally, we remark: (i) For 푘 > 2 andα ≠ (1,1, … ,1) the commutative algebras ℛα(h) are just Banach algebras, while the C∗-algebras generated by them are noncommutative. 2 (ii) These algebras are commutative for each weighted Bergman space 𝒜휆 (퐷푛) with 휆 > −1. (iii) For 푘 = 0 (nilpotent case) or 푘 = 1, 2 these algebras collapse to the single C∗- algebras which are generated by Toeplitz operators with quasinilpotent symbols ′ ′ 푏(푟, 푦 , 퐼푚푧푛 − |푧 |). Let 0 ≤ 휀 ≤ 푛 − 3, 휀 ≥ 0. We rather use the notation (푥 + 2휀) = ((푥 + 2휀)′, (푢 + ′ ′ 푛−휀−2 ′ 푛−휀 2휀) , (푥 + 2휀)푛) for (푥 + 2휀) ∈ 퐷푛+푠−1where (푥 + 2휀) ∈ ℂ and (푢 + 2휀) ∈ ℂ . 1+휀 푛−휀−2 The quasi-nilpotent group 핋 × ℝ × ℝ acts on 퐷푛+푠−1, cf. [218], as follows: given 2+ε 푛−휀−2 (푡, 푏푠−2, ℎ) ∈ 핋 × ℝ × ℝ, we have: ′ 풯(푡, 푏푠−2, ℎ): ((푥 + 2휀), (푢 + 2휀) (푥 + 2휀)푛) ′ ′ 2 → (푡(푥 + 2휀) , (푢 + 2휀) + 푏푠−2, (푥 + 2휀)푛 + ℎ + 2푖휔. 푏푠−2 + 푖|푏푠−2| ). Note that in the case 휀 = 푛 − 2 , 휀 ≥ 0we obtain the quasi-parabolic group, while for 휀 = −2 the group action is called nilpotent. 1+ε 푛−휀−2 On the domain 풟푠−2 = ℂ × ℂ × ℝ × ℝ+we use the variables ((푥 + 2휀)′, (푢 + 2휀)′, 푢, (푢 + 휀)) and we represent 퐿2(풟푠−2, 휂휀−1) in the form: 1+ε 푛−휀−2 퐿2(풟푠−2, 휂(휀−1)) = 퐿2(ℂ ) ⊗ 퐿2(ℂ ) ⊗ 퐿2(ℝ) ⊗ 퐿2(ℝ+, 휂(휀−1)). (25) Let F be the Fourier transform on 퐿2(ℝ), and with respect to the decomposition (25) consider the unitary operators 푈푠 ∶= 퐼 ⊗ 퐼 ⊗ 퐹 ⊗ 퐼 acting on 퐿2(풟푠−2, 휂(휀−1)). With this notation we put 𝒜푠(풟푠−2) ∶= 푈푠(𝒜푠−1(풟푠−2)). ε+1 We introduce polar coordinates on ℂ and put 푟 = (푟1, … , 푟ε+1) = (|(푥 + ′ ′ ′ ′ ′ 2휀)1|, … , |(푥 + 2휀)ε+1|). In the following we write 푥 := Re (푢 + 2휀) and (푥 + 휀) ≔ ′ ′ ′ 2 퐼푚(푢 + 2휀) . Then one can check that 푟 , (푥 + 휀) and 퐼푚(푥 + 2휀)푛 − |(푢 + 2휀) | are invariant under the action of the quasi-nilpotent group. Following the ideas in [218] and ( ) with 푟 푑 = 푟1푑푟1 · · · 푟(1+휀)푑푟(1+휀) we represent 퐿2 풟푠−2, 휂ε−1 in the for (1+휀) 1+휀 푛−휀−2 퐿2 (ℝ+ , 푟1푑푟1) ⊗ 퐿2(핋 ) ⊗ 퐿2(ℝ ) 푛−휀−2 ⊗ 퐿2(ℝ ) ⊗ 퐿2(ℝ) ⊗ 퐿2(ℝ+, 휂ε−1) (26)

We define the unitary operator 푈푠+1 on 퐿2(풟푠−2, 휂휀−1) by 푈푠+1 = 퐼 ℱ(1+휀)  퐹(푛−휀−2)  퐼  퐼  퐼. Here ℱ(1+휀) = ℱ ⊗ … ⊗ ℱ is the (1 + 휀)-dimensional discrete Fourier transform and 퐹(푛−휀−2) = 퐹 …  퐹 denotes the (푛 − 휀 − 2)-dimensional Fourier 푛−휀−2 transform on 퐿2(ℝ ). Note that 퐿2(풟푠−2, 휂(1+휀)) is isometrically mapped by 푈푠+1 onto

(ε+1) 1+휀 푛−휀−2 푛−휀−2 ℓ2 ((ℤ , 퐿2(ℝ+ ) ⊗ 퐿2(ℝ )) ⊗ 퐿2(ℝ ) ⊗ 퐿2(ℝ)

⊗ 퐿2(ℝ+, 휂(휀−1)). ) (27)

152

′ We put 𝒜푠+1(풟푠−2): = 푈푠+1(𝒜푠(풟푠−2)) and we write elements in (27) as {푓훽(푟 , 푥 , (푥 + 휀)′, 휉, 푢 + 휀)} ,where (푟, 푥′, (푥 + 휀)′, 휉, 푢 + 휀) ∈ ℝε+1 × ℝ푛−휀−2 × ℝ푛−휀−2 × ℝ × 훽∈ℤ푘 + ℝ+. Next we recall the definition of the unitary operator 푈푠+2 which acts on (27) by: 푈 : {푓 (푟, 푥′, (푥 + 휀)′, 휁, 푢 + 휀)} 푠+2 훽 훽∈ℤ

′ ′ 1 ′ ′ ⟼ {푓훽 (푟, √휁(푥 , (푥 + 휀) ), (−푥 , (푥 + 휀) ), 휁, 푢 + 휀)} 2√휉 훽∈ℤ(1+휀)

−1 One immediately checks that the inverse 푈푠+2 has the form 푈−1 : {푓 (푟, 푥′, (푥 + 휀)′, 휁, 푢 + 휀)} 푠+2 훽 훽∈ℤ ′ ′ 푥 ′ 푥 ′ ⟼ {푓훽 (푟 , − √휁(푥 + 휀) , + √휁(푥 + 휀) , 휁, 푢 + 휀)} 2√휁 2√휁 훽∈ℤ(1+휀)

In the following we write ℤ+ = ℕ ∪ {0} = {0,1,2, … } for the nonnegative integers. In order to state the main result of Section 8 in [218] we need to introduce the operator 푅푠−2 , which (1+휀) 푛−휀−2 defines an isometric embedding of ℓ2 (핫+ , 퐿2(ℝ × ℝ+)) into (27). It is explicitly given by ′ 푅푠−2 : {푐훽(푥 , 휁)} (1+휀) 훽∈ℤ+ 2 2 |(푢+휀)′| 훽 −휁(|푟 | +(푢+휀))− ′ ⟼ {휒 (ε+1) (훽, 휉)(푅푠−2)훽(휁)푟 푒 2 푐훽(푥 , 휁)} ℤ+ ×ℝ+ 훽∈ℤ(1+휀) = {푔 (푟, 푥′, (푥 + 휀)′, 휁, 푢 + 휀)} 훽 훽∈ℤ(1+휀) (ε+1) ′ Here 휒 (ε+1) (훽, 휁)denotes the characteristic function of ℤ+ × ℝ+ and 푐훽(푥 , 휁) is ℤ+ ×ℝ+ extended by zero for 휁 ∈ (−∞, 0) and all 푥′ ∈ ℝ푛−휀−2. Moreover, we have used the abbreviation

푛−휀−2 − 2ε+3 (2휁)|훽|+2ε+1 퐴훽(휁): = 휋 4 √ (28) 퐶ε−1 훽!Γ(ε) ∗ The ad joint operator 푅푠−1 is given by: 푅∗ : {푓 (푟, 푥′, (푥 + 휀)′, 휁, 푢 + 휀)} ⟼ {(퐴 ) (휁) 푠−1 훽 훽∈ℤε+1 푠−2 훽 ′ 2 2 |(푢+휀) | 훽 −휁(|푟 | +(푢+휀))− ′ ′ ′ × ∫ 푟 푒 2 푓훽(푟 , 푥 , (푥 + 휀) , 휁, 푢 + 휀)푟 푑푟 푑(푥 + 휀) ε+1 푛−휀−2 ℝ+ ×ℝ ×ℝ+ (ε−1) C휆(푢+휀) 푑(푢 + 휀)} (1+휀) (29) 4 훽∈ℤ+ 2 We set 푈 푠−2: = 푈푠+2푈푠+1푈푠푈푠−1, which gives a unitary operator from 푅ε−1(풟푛+푠−1) onto 퐴푠+2(풟푠−2) ≔ 푈푠+3퐴푠+1(풟푠−2). The following result has been proved in [218], and it

153

provides a decomposition of the Bergman projection 퐵풟푛+푠−1,ε−1 in form of a certain operator product. ∗ Theorem (6.1.16)[270]: [218] The operator 푅 푠−2 ∶= 푅s−1푈푠−2 maps 퐿2(풟푛+푠−1, μ̅(ε−1)) 휀+1 푛−휀−2 onto the space ℓ2(핫+ , 퐿2(ℝ × ℝ+)), and the restriction 2 ε+1 푛−ε−2 푅푠−2|(𝒜2 ) (풟 )(𝒜푠−2)ε−1(풟푠−2) → ℓ2(핫+ , 퐿2(ℝ × ℝ+)) 푠−2 ε−1 푠−2 is an isometric isomorphism. The ad joint operator ∗ ∗ ε+1 푛−휀−2 2 푅푠−2 = 푈푠−2 푅푠−1: 퐿2(핫+ , 퐿2(ℝ × ℝ+)) → (𝒜푠−2)ε−1(풟푠−2) ⊂ 퐿2(풟푛+푠−1, μ̅ε−1) ε+1 푛−휀−2 is an isometric isomorphism of 퐿2(핫+ , 퐿2(ℝ × ℝ+)) onto the subspace 2 (𝒜푠−2)ε−1(풟푠−2) of 퐿2(풟푛+푠−1, μ̅ε−1). Furthermore one has: ∗ ε+1 푛−휀−2 휀+1 푛−휀−2 푅푠−2 푅푠−2 = 퐼: 퐿2(핫+ , 퐿2(ℝ × ℝ+)) → 퐿2(핫+ , 퐿2(ℝ × ℝ+)), 푅∗ 푅 = 퐵 : 퐿 (풟 , μ̅ ) → (𝒜2 ) (풟 ) 푠−2 푠−2 풟푛+s−1,ε−1 2 푛+푠−1 ε−1 푠−2 ε−1 푠−2 Consider an (훽 + 휀)-quasi-nilpotent quasi-homogeneous symbol ′ ′ 푏푠−2((푥 + 2휀) , (푢 + 2휀) , (푥 + 2휀)푛) (1+휀) (1+휀) as in[38] and of degree (푝, 푝 + 휀) ∈ 핫+ × 핫+ with 푝 ⊥ (p + ε) . Our next aim is to ∗ (1+휀) 푛−휀−2 calculate the operator 푅푠−2 푇푏푠−2푅 푠−2. On the domain 풟푠−2 = ℂ × ℂ × ℝ × ℝ+ we use the variables ((푥 + 2휀)′, (푢 + 2휀)′, 푢, 푢 + 휀). Moreover, we express (푢 + 2휀)′in ′ polar coordinates (푢 + 2휀) = 푡1푟1, … , 푡(1+휀)푟(1+휀) where 푟푠 ≥ 0 and 푡푠 ∈ 핊 = 핊 for 푠 = 1, … , (1 + 휀). Then we have the relations (푢 + 2휀)푗,ℓ = r푗휁푗,ℓ = 푡푗,ℓ푟푗,ℓ −1 for ℓ = {푠, … , (훽푗 + 휀)}and 푗 = 1, … , 푚. It follows that 휉푗,ℓ = 푡푗,ℓ푟푗,ℓr푗 in the case of r푗 ≠ 0 and therefore: 푚 ̅̅̅̅̅̅̅̅̅p+ε 푝 ̅p+ε 푝+p+ε −|푝푗|−|(p+ε)푗| (휁 + 휀) = 푡 푡 푟 ∏ r푗 (30) 푗=1 Note that the assignment (푥 + 2휀)′ ⟼ (휁 + 휀)푝(̅̅휁̅̅+̅̅̅휀̅̅)p+ε depends on the initial choice of 푚 (훽 + 휀) ∈ ℤ+ .Using Theorem (6.1.16) we can write: 푅 푇 푅∗ = 푅 퐵 푏 퐵 푅∗ 푠−2 푏푠−2 푠−2 푠−2 풟푛+푠−1,휀−1 푠−2 풟푛,휀−1 푠−2 = 푅 (푅∗ 푅 ) 푏 (푅∗ 푅 ) 푅∗ 푠−2 푠−2 푠−2 푠−2 푠−2 푠−2 푠−2 ∗ ∗ ∗ ∗ = (푅푠−2 푅푠−2 ) 푅푠−2 푏푠−2푅푠−2 (푅푠−2 푅푠−2 ) = 푅푠−2푏푠−2푅푠−2

∗ −1 −1 −1 −1 = 푅푠−1푈푠+2푈푠+1푈푠푈푠−1푏푠−2푈푠−1푈푠 푈푠+1푈푠+2푅푠−1 ∗ ′ = 푅푠−2 푈푠+2푈푠+1푈푠푏푠−1(푟1, … , 푟푚, (푥 + 휀) , Im(푥 + 2휀)푛 ′ 2 푝̅̅̅̅̅̅̅̅̅p+ε −1 −1 −1 −1 − |(푢 + 2휀) | )(휁 + 휀) (휁 + 휀) 푈푠−1푈푠 푈푠+1푈푠+2푅푠−1 −1 ′ ′ First we calculate the operator 푈푠−1푏푈푠−1. Let {푓훽(푟 , 푥 , (푥 + 휀) , 휁, 푢 + 휀)} (1+휀)be an 훽∈푍+ element in the space (27) and write 푟 ∶= (푟1, . . . , 푟푚). Since the ymbol ′ 2 푝̅̅̅̅̅̅̅̅̅p+ε ′ 푏푠−1(푟 , (푥 + 휀) , (푢 + 휀) + |푟 | )(휁 + 휀) (휁 + 휀) is independent of 푥 we obtain from (4.2) that:

154

′ 푈푠+1푏푠−1(푟 , (푥 + 휀) , (푢 + 휀) 2 푝̅̅̅̅̅̅̅̅̅p+ε −1 ′ ′ + |푟 | )(휁 + 휀) (휁 + 휀) 푈푠+1{푓훽(푟 , 푥 , (푥 + 휀) , 휁, (푢 + 휀))} 1+ε 훽∈ℤ+

′ 2 2p+ε = {푏푠−1(푟 , (푥 + 휀) , (푢 + 휀) + |푟 | )푟

푚 −|푝푗|−|p+ε푗| ′ ′ × (∏ r푗 ) 푓훽+ε(푟 , 푥 , (푥 + 휀) , 휁, (푢 + 휀))} (31) 푗=1 훽∈ℤ1+ε Combining (31) and (29) gives: ∗ ′ ∗ −1 −1 ( ) 1+휀 ( ) 푅푠−2 푇푏푠−2푅푠−2 : {퐶훽 푥 , 휁 } 1+휀 = 푅푠−2푈푠+2푈푠+1푏푠−2푈푠+1푈푠+1 {휒ℤ 훽, 휁 훽∈ℤ+ +×ℝ+ 2 |(푢+휀)′| 훽 −휁(|푟|2+(푢+휀))− ′ (퐴푠−2)훽(휁)푟 푒 2 푐훽(푥 , 휁)} 훽∈ℤ1+ε ∗ ∗ −1 훽 = 푅 푈 푈 푏 푈 {휒 1+ε (훽, 휁)(퐴 ) (휁)푟 푠−2 푠+2 푠+1 푠−2 푠+1 ℤ+ ×ℝ+ 푠−2 훽 2 2 1 1 ′ ′ −휁(|푟 | +(푢+휀))− | 푥 +√휁 (푥+휀) | 1 2 2√휁 ′ ′ × 푒 푐훽 ( 푥 − √휁 (푥 + 휀) , 휁)} 2√휁 훽∈ℤ1+ε ∗ 훽+2(p+ε) = 푅 푈 {휒 1+휀 (훽 + ε, 휁)(퐴 ) (휁)푟 푏 푠−2 푠+2 핫+ ×ℝ+ 푠−2 훽+ε 푠−1 ′ 2 × (푟푠−2, (푥 + 휀) , (푢 + 휀) + |푟 | ) 2 푚+푠−1 2 1 1 ′ ′ −휁(|푟 | +(푢+휀))− −| 푥 +√휁 (푥+휀) | −|푝푗|−|(p+ε)푗| 2 2√휁 × ( ∏ r푗 ) 푒 푗=1

1 ′ ′ × 퐶훽+휀 ( 푥 − √휁 (푥 + 휀) , 휁)} 2√휁 훽∈ℤ1+ε ′ ′ ∗ −푥 + (푥 + 휀) = 푅푠−2 {휒핫1+휀×ℝ (훽 + ε, 휁)(퐴푠−2)훽+ε(휁)푏푠−1 (푟 , , (푢 + 휀) + + 2√휁 푚 1 2 −|푝푗|−|(p+ε)푗| −휁(|푟|2+(푢+휀))− | (푥+휀)′| 2 훽+2(푝+휀) 2 + |푟 | ) × (∏ r푗 ) 푟 푒 퐶훽+ε 푗=1 ′ (푥 , 휁)}훽∈ℤ1+휀 ′ = {(퐴 ) (휁)(퐴 ) (휁)휒 1+휀 (훽 + 휀, 휁) 퐶 (푥 , 휁)} 1+휀 푠−2 훽 푠−2 훽+휀 ℤ+ ×ℝ+ 훽+ε 훽∈ℤ 푚 2(훽+p+ε) 푟 −|푝푗|−|(p+ε)푗| −휁(|푟|2+(푢+휀))−|푥+휀|2 × ∫ (∏ r푗 ) 푒 1+휀 푛−휀−2 ℝ+ ×ℝ ×ℝ+ 푗=1 ′ ( )′ ( )휀−1 −푥 + 푥 + 휀 2 ′ C휀−1 푢 + 휀 × 푏푠−1 (푟 , , (푢 + 휀) + |푟 | ) 푟 푑푟 푑(푥 + 휀) 푑(푢 + 휀)} 2√휁 4 1+휀 훽∈ℤ+ 155

Now put: ′ 훾푏,푝,푝+휀(훽, 푥 , 휁): = (퐴 ) (휉)(퐴 ) (휉)휒 1+휀 (훽 + 휀, 휁) 푠−2 훽 푠−2 훽+휀 ℤ+ ×ℝ+ 푚 −|푝푗|−|(p+ε)푗| 2(훽+p+ε) −2휁(|푟|2+푢+휀)−|푥+휀|2 × ∫ ∏ r푗 푟 푒 1+ε 푛−휀−2 ℝ+ ×ℝ ×ℝ+ 푗=1 −푥′+(푥+휀)′ C (푢+휀)휀−1 × 푏 (푟, , (푢 + 휀) + |푟|2) 푟푑푟푑(푥 + 휀)′ 휀−1 푑(푢 + 휀) (32) 푠−1 2√휁 4 Hence, we have proved: Theorem (6.1.17)[270]: Let 푏 be defined as in [38]. The operator 푅 푇 푅∗ acts on 푠−2 푠−2 푏푠−2 푠−2 1+휀 푛−휀−2 the Hilbert space ℓ2(ℤ+ , 퐿2(ℝ × ℝ+)) by the rule: ∗ ( ′ ) ( ′ ) ( ′ ) 푅푠−2 푇푏푠−2푅푠−2 : {퐶훽 푥 , 휁 } 1+휀 = {훾푏푠−2,푝,푝+휀 훽, 푥 , 휁 . 퐶훽+휀 푥 , 휁 } 1+휀 훽∈ℤ+ 훽∈ℤ Note that, in the case ε = 0, Theorem 4.5 reduces to Theorem 4.2. Example (6.1.18)[270]:We calculate 푅 푇 푅∗ more explicitly in the special case 푠−2 푏푠−2 푠−2 where 1+휀 1+휀 푏푠−1` ≡ 1 and we choose 휀 = 푚 − 1 , i.e. 훼 = (1, … ,1) ∈ 핫+ . Let (푝, 푝 + 휀) ∈ ℤ+ such that 푝 ⊥ ( 푝 + 휀) and put ′ ′ 푝̅̅̅̅̅̅̅̅̅p+ε 푝 p+ε 푏푠−2((푥 + 2휀) , (푢 + 2휀) , (푥 + 2휀)푛) = (휁 + 휀) (휁 + 휀) = 푡 푡̅ According to Theorem (6.1.17) it is sufficient to calculate the functions: ( ′ ) 훾푏푠−2,푝,푝+휀 훽, 푥 , 휁 : ( ) ( )( ) ( ) ( ) = 퐴푠−2 훽 휁 퐴푠−2 훽+ε 휁 휒ℝ+ 휁 2 2 2훽+ε −2휁(|푟 | +(푢+휀))−|푥+휀| ′ × ∫ 푟 푠−2 푒 푟 푑푟 푑(푥 + 휀) 1+휀 푛−휀−2 ℝ+ ×ℝ ×ℝ+ (휀−1) C( )(푢 + 휀) × 휀−1 푑(푢 + 휀) 4 1+휀 1+휀 for all 훽 ∈ ℤ+ with (훽 + ε) ∈ 핫+ . We use the identity: 2 푛−휀−2 ∫ 푒−2휁(푢+휀)−|푥+휀| 푑(푥 + 휀)′(푢 + 휀)휀−1푑(푢 + 휀) = 휋 2 Γ(휀)(2휁)−(휆+1) 1+휀 푛−휀−2 ℝ+ ×ℝ ×ℝ+ (see [97]) where 휁 > 0, which together with (28) shows that |p+ε|−|푝| |훽|+휀+1+ 1 2 ( ′ ) (1+휀)( ) 2 2훽+휀 −2휁−|(푥+휀)| 훾푏푠−2,푝,p+ε 훽, 푥 , 휁 = 2 2휁 ∫ 푟 푒 푑푟 푘 √훽! (훽 + ε)! ℝ+ (p + ε) − p ∏1+휀 Γ 훽 + j j + 1 푗=1 ( j 2 ) = . √훽! (훽 + 휀)! ( ′ ) ′ In particular, in this case 훾푏푠−2,푝,푝+휀 훽, 푥 , 휁 is independent of 푥 and 휁 푚 Fix (훽 + 휀) ∈ 핫+ with |훽 + 휀| = 휀 + 1 as before and let 푎푠−2 = 푎푠−1(푟1, … , 푟푚, (푥 + ′ ′ 2 휀) Im(푥 + 2휀) 푛 − |(푢 + 2휀) | ) be a bounded measurable (훽 + 휀)-quasi-nilpotent quasi- radial function on 퐷푛+푠−1. Consider the symbol:

156

′ ′ ′ 푏푠−2((푥 + 2휀) , (푢 + 2휀) , (푥 + 2휀)푛) = 푏푠−1(푟1, … , 푟푚, (푥 + 휀) , Im(푥 + 2휀)푛 − |(푢 + 2휀)′|2). (휁 + 휀)푝(̅̅휁̅̅+̅̅̅휀̅̅)p+ε. (33) We calculate the operator products 푅 푇 푇 푅∗ and 푅 푇 푇 푅 ∗ . 푠−2 푏푠−2 푎푠−2 푠−2 푠−2 푎푠−2 푏푠−2 푠−2 According to Theorem (6.1.7) and Theorem (6.1.1) we have ∗ ∗ ∗ 푅푠−2 푇푏푠−2푇푎푠−2푅푠−2 {푐훽} 푘 = (푅푠−2 푇푏푠−2푅푠−2 ) (푅푠−2 푇푎푠−2푅푠−2 ) {푐훽} 휀+1 훽∈ℤ+ 훽∈ℤ+ ∗ ( ′ ) ( ′ ) =(푅푠−2 푇푏푠−2푅푠−2 ){훾푏̅ 푠−2,푝,푝+휀 훽, 푥 , 휉 . 푐훽 푥 , 휉 } 휀+1 훽∈ℤ+ ( ′ ) ( ′ ) ( ′ ) = {훾푏̅ 푠−2,푝,푝+휀 훽, 푥 , 휉 훾푎푠−2,0,0 훽 + ε, 푥 , 휉 푐훽+ε 푥 , 휉 } 휀+1 (34) 훽∈ℤ+ On the other hand it follo

∗ ∗ ∗ 푅푠−2 푇푎푠−2푇푏푠−2푅 {푐훽} 휀+1 = (푅푠−2 푇푎푠−2푅푠−2 ) (푅푠−2 푇푏푠−2푅푠−2 ) 푠−2 훽∈ℤ+ ∗ ( ′ ) ( ′ ) = (푅푠−2 푇푎푠−2푅푠−2 ){훾푏̅ 푠−2,푝,푃+휀 훽, 푥 , 휁 . 푐훽 푥 , 휁 } 휀+1 훽∈ℤ+ ( ′ ) ( ′ ) ( ′ ) = {훾푎̅ 푠−2,0,0 훽, 푥 , 휁 훾푏̅ 푠−2,푝,푝+휀 훽, 푥 , 휁 푐훽+ε 푥 , 휁 } 휀+1 (35) 훽∈ℤ+

Hence, we conclude from (34) and (35) that both operators 푇푎푠−2 and 푇푏푠−2commute if and only if ( ′ ) ( ′ ) 훾푎̅ 푠−2,0,0 훽, 푥 , 휁 = 훾푎̅ 푠−2,0,0 훽 + ε, 푥 , 휁 1+휀 for all 훽 ∈ ℤ+ . According to (9) this is equivalent to:

′ ( )′ 1 −푥 + 푥 + 휀 2 ∫ 푎푠−1 (푟 , , (푢 + 휀) + |푟 | ) 훽! 휀+1 푛−휀−2 ℝ+ ×ℝ ×ℝ+ 2√휉 2 ′ 2 2훽 −2휁((푢+휀)−|푟 | )−|(푥+휀) | 휀−1 × 푟 푒 (푢 + 휀) 푟 푑푟 푑 ( )−|푝|+|푝+휀| ′ ( )′ 2휁 −푥 + 푥 + 휀 2 = ∫ 푎푠−1 (푟 , , (푢 + 휀) + |푟 | ) (훽 + 휀)! 휀+1 푛−휀−2 ℝ+ ×ℝ ×ℝ+ 2√휁 2 ′ 2 2훽+ε −2휁((푢+휀)+|푟 | )−|(푥+휀) | 휀−1 ′ × 푟 푒 (푢 + 휀) 푟 푑푟 푑(푥 + 휀) 푑(푢 + 휀) ′ 2 Since 푎푠−1(푟 , (푥 + 휀) , Im(푥 + 2휀)푛 − |(푢 + 2휀)| ) only depends on 푟 = (푟1, … , 푟푚) we can assume that the above integral has the form

′ ′ −푥 + (푥 + 휀) 2 ′ 2 2 2훽 −2휁((푢+휀)+|푟 | )−|(푥+휀) | ∫ 푎푠−1 (푟 , , (푢 + 휀) + |푟 | ) 푟 푒 휀+1 푛−휀−2 ℝ+ ×ℝ ×ℝ+ 2√휁 휀−1 ′ × (푢 + 휀) 푟 푑푟 푑(푥 + 휀) 푑(푢 + 휀) =: (∗),

휀+1 휀+1 Where 훽 ∈ ℤ+ . With 푒 = (1,1, … ,1) ∈ ℤ+ we obtain

157

′ ( )′ 1 −푥 + 푥 + 휀 2 (∗) = 휀+1 ∫ 푎푠−1 (푟, , (푢 + 휀) + |푟 | ) 2 휀+1 푛−휀−2 ℝ ×ℝ ×ℝ+ 2√휁 2 ′ 2 2훽 −2휁((푢+휀)+|푟| )−|(푥+휀) | 휀−1 ′ × |푟 |푒 (푢 + 휀) 푟 푑푟 푑(푥 + 휀) 푑(푢 + 휀) 1 −푥′ + (푥 + 휀)′ = 휀+1 ∫ ∫ 푎푠−1 (푟 , , (푢 + 휀) 2 푛−휀−2 푚 훽 +휀−1 훽 +휀−1 ℝ ×ℝ+ ℝ+ ×핊 1 ×…×핊 푚 2√휁 푚 2 ′ 2 2 2훽+푒 2|훽(푗)|+2(훽+휀)푗−1 −2휁((푢+휀)+|푟| )−|(푥+휀) | + |푟 | ) |휌 |. (∏ r푗 ) 푒 푗=1 휀−1 ′ × (푢 + 휀) 푑휎(휌(푚))푟 푑푟 푑(푥 + 휀) 푑(푢 + 휀) (훽+휀)푗−1 In the last integral we wrote 푑휎(휌(푗)) for the standard area measure on the sphere 핊 . The integral over the m-fold product 핊(훽+휀)1−1 × … × 핊(훽+휀)푚−1can be calculated explicitly by using the following well-known formula: Lemma (6.1.19)[270]:Let 푑휎 denote the usual surface measure on the (푛 − 1)-dimensional 푛 − 1 1+휀 sphere 핊 and let 휃 ∈ ℤ+ . Then 휃 − 1 휃 − 1 2Γ ( 1 ) … Γ ( n ) ∫ |푦휃|푑휎(1 + 휀) = 2 2 n − |휃| 핊푛−1 Γ ( 2 ) Using the formula in Lemma (6.1.19)we define: 2훽+푒 ⊖훽≔ ∫ |휌 |푑휎(휌(1)) … 푑휎(휌(푚)) 핊훽1+휀−1×…×핊훽푚+휀−1 푚 −1 훽푗 + 휀 + 1 = 2푚훽! ∏ 푟 ( + |훽 |) (36) 2 (푗) 푗=1 This finally gives: ⊖ 훽 1 ′ ′ 2 (∗) = 1+휀 ∫ 푎푠−1 (푟 , (−푥 + (푥 + 휀) ), (푢 + 휀) + |푟 | ) 2 푚 푛−휀−2 ℝ+ ×ℝ ×ℝ+ 2√휁 ( ) ( ) 2 ′ 2 2|훽(푠)|+2 훽+휀 1−1 2|훽(푚)|+2 훽+휀 푚−1 −2휁((푢+휀)+|푟 | )−|(푥+휀) | 휀−1 ′ × r1 … r푚 푒 (푢 + 휀) 푑푟 푑(푥 + 휀) 푑(푢 + 휀). Note that the last integral does not depend on the full multi-index 훽 but rather on the values

|훽(1)| for 푗 = 1, … , 푚. We denote this integral by 퐺푎푠−2(|훽(1)|, … , |훽(푚)|). Then the commutativity condition (35) can be written in the form: ⊖ ⊖ 훽 퐺 (|훽 |, … , |훽 |) = (2휁)−|푝|+|푝+휀| 훽+휀 퐺 (|훽 | − |푝 | + |(푝 + 훽! 푎푠−2 (1) (푚) (훽+휀)! 푎푠−2 (1) (1) 휀)(1)|, … , |훽(푚)| − |푝(푚)| + |(푝 + 휀)(푚)|). According to the definition (36) this is equivalent to

158

푚 −1 (훽 + 휀)푗 + 1 퐺 (|훽 |, … , |훽 |) = ∏ Γ ( + |훽 |) 푎푠−2 (1) (푚) 2 (푗) 푗=1 ( )−|푝|+|푝+휀| ( ) = 2휁 퐺푎푠−2(|훽(1)| − |푝(1)| + | 푝 + 휀 (푗)|, … , |훽(푚)| − |푝(푚)| 푚 −1 (훽 + 휀)푗 + 1 + |(푝 + 휀) |) ∏ Γ ( + |훽 | − |푝 | + |(푝 + 휀) |) (푚) 2 (푗) (푗) (푗) 푗=1 This equality can be only true simultaneously for all 훽 + 휀 -quasi-nilpotent quasi-radial 1+휀 functions 푎푠−2 and all 훽 ∈ ℤ+ if |푝(푗)| = |(푝 + 휀)(푗) | for 푗 = 1, … , 푚. Hence, we obtain see[38]: 1+휀 Theorem (6.1.20)[270]: Let (훽 + 휀) ∈ ℤ+ be given. Then the statements (a), (b) and (c) below are equivalent: (a) For each (훽 + 휀)-quasi-nilpotent quasi-radial function 푎푠−2 = 푎푠−1(푟1, (푥 + ′ ′ 2 ∞ 휀) , Im(푥 + 2휀)푛 − |(푢 + 2휀) | ) ∈ 퐿 (퐷푛+푠−1) and each (훽 + 휀)-quasi-nilpotent quasi-homogeneous function ′ ′ 2 푝̅̅̅̅̅̅̅̅̅푝+휀 푏푠−2 = 푏푠−1(푟1, … , 푟푚, (푥 + 휀) , Im(푥 + 2휀)푛 − |(푥 + 2휀) | ). (휁 + 휀) (휁 + 휀) ∞ ∈ 퐿 (퐷푛+푠−1) (37) 1+휀 1+휀 of degree (푝, 푝 + 휀) ∈ ℤ+ × ℤ+ the Toeplitz operators 푇푎푠−2 and 푇푏푠−2 commute on each 2 weighted Bergman space 𝒜휀−1(퐷푛+푠−1). ( ′ ) ( ′ ) 푘 (b) The equality 훾푎̅ 푠−2,0,0 훽, 푥 , 휉 = 훾푎̅ 푠−2,0,0 훽 + 휀, 푥 , 휉 holds for all 훽 ∈ ℤ+ and for each (훽 + 휀)-quasi-nilpotent quasi-radial functions 푎푠−2. (c) The equality |푝(푗)| = | (푝 + 휀)(푗)| holds for each 푗 = 1, . . . , 푚. ∞ Now, let us assume that 푏푠−2 ∈ 퐿 (퐷푛+푠−1) is of the form (37). Under the assumption |푝(푗)| = ( ) ( ′ ) | 푝 + 휀 (푗)|, for each 푗 = 1, … , 푚, we calculate 훾푏̅ 푠−2,푝,푝+휀 훽, 푥 , 휁 in (32) more explicitly by 1+휀 reducing the order of integration. Assume that 훽 + 휀 ∈ 핫+ . Then: 훾̅ (훽, 푥′, 휁) 푏푠−2,푝,푝+휀 ( ) ( )( ) ( ) ( ) 2(훽+푝+휀) = 퐴푠−2 훽 휁 퐴푠−2 훽+휀 휁 휒ℝ+ 휁 ∫ 푟 휀+1 푛−휀−2 ℝ+ ×ℝ ×ℝ+ 푚 2 −|푝( )|−|(푝+휀)( )| 2 ̃ ′ 푗 푗 −2휁(|푟 | +(푢+휀))−|(푥+휀) | × ∏ r푗 푒 푏푠−1 푗=1 −푥′ + (푥̃+ 휀)′ 푐 (푢 + 휀)휀−1 2 ̃ ′ 휀−1 × (푟 , , (푢 + 휀) + |푟 | ) 푟 푑푟 푑(푥 + 휀) 푑(푢 + 휀) 2√휁 4 ( ) ( )( ) ( ) ( ) −(1+휀) =⊝훽+푝+휀 퐴푠−2 훽 휁 퐴푠−2 훽+휀 휉 휒ℝ+ 휁 2 푚 2|훽 |+|(푝+휀) |−|푝 |+2훼 −1 2 ̃ ′ 2 (푗) (푗) (푗) 푗 −2휁(|푟 | +(푢+휀))−|(푥+휀) | × ∫ ∏ r푗 푒 푏푠−1 푚 푛−휀−2 ℝ+ ×ℝ ×ℝ+ 푗=1 −푥′ + (푥̃+ 휀)′ 푐 (푢 + 휀)휀−1 2 ̃ ′ 휀−1 × (푟 , , (푢 + 휀) + |푟 | ) 푟 푑푟 푑(푥 + 휀) 푑(푢 + 휀) 2√휁 4 159

⊝ 퐴 (휁) 훽+푝+휀 훽+휀 ( ′ ) = . 퐷푏푠−2 훽, 푥 , 휁 ⊝훽 퐴훽(휁) (훽 + 휀) + 1 푚 Γ 푗 + 훽 (훽 + 푝 + 휀)! ( 2 | (푗)|) ( ′ ) = ∏ . 퐷푏푠−2 훽, 푥 , 휁 , 훽! (훽 + 휀)! 훽푗 + 휀 + 1 √ 푗=1 Γ + 훽 + (푝 + 휀) ( 2 | (푗)| | (푗)|) ( ′ ) where 퐷푏푠−2 훽, 푥 , 휁 = 훾푏̅ 푠−2,푝,푝+휀, which can be seen by choosing 휀 =0 in the above equalities. Hence we have proved: 푚 ∞ Proposition (6.1.21)[270]:. Let (훽 + 휀) ∈ 핫+ be given. Assume that 푏푠−2 ∈ 퐿 (퐷푛+푠−1) is of the form (37) and let |푝(푗)| = |(푝 + 휀)(푗)|, for each 푗 = 1, . . . , 푚. Then in the case of 푚 훽 + 휀 ∈ 핫+ we have ( ′ ) 훾푏̅ 푠−2,푝,푝+휀 훽, 푥 , 휁 (훽 + 휀) + 1 푚 Γ 푗 + 훽 (훽 + 푝 + 휀)! ( 2 | (푗)|) ( ′ ) = ∏ . 훾푏̅ 푠−2,0,0 훽, 푥 , 휉 . √훽! (훽 + 휀)! (훽 + 휀)푗 + 1 푗=1 Γ + 훽 + 푝 + 휀 ( 2 | (푗)| | (푗)|) ( ) 1+휀 ( ′ ) In the case of 훽 + 휀 ∉ 핫+ we have 훾푏̅ 푠−2,푝,푝+휀 훽, 푥 , 휁 = 0. the factor ( ′ ) 훾푏̅ 푠−2,푝,푝+휀 훽, 푥 , 휁 can be expressed in the form ( ′ ) 훾푏̅ 푠−2,푝,푝+휀 훽, 푥 , 휁 푚 2|훽(푗)|+|푞(푗)|−|푝(푗)|+2(훽+휀) −1 2 ( ) ( ) −(1+휀) 푗 =⊝훽 퐴훽 휁 휒ℝ+ 휁 2 ∫ ∏ r푗 푚 푛−휀−2 ℝ+ ×ℝ ×ℝ+ 푗=1 ′ ′ 2 ′ 2 −푥 + (푥 + 휀) −2휁(|푟 | +(푢+휀))−|(푥+휀) | 2 × 푒 푏푠−1 (푟 , , (푢 + 휀) + |푟 | ) 2√휁 푐 (푢 + 휀)휀−1 × 푟푑푟푑(푥 + 휀)′ 휀−1 푑(푢 + 휀) (38) 4 푚 1+휀 1+휀 Let 훽 + 휀 ∈ 핫+ be given and (푝, 푝 + 휀) ∈ 핫+ × 핫+ . From Proposition (6.1.21) we conclude: ′ ′ 2 Corollary (6.1.22)[270]: Let 푎푠−2 = 푎푠−1(푟 , (푥 + 휀) , 퐼푚(푥 + 2휀)푛 − |(푢 + 2휀) | ) ∈ ∞ 퐿 (퐷푛+푠−1) be an (훽 + 휀) − quasi-nilpotent quasi-radial function. Under the assumption |푝(푗)| for all 푗 = 1, 2, … , 푚 we have 푇 푇 푝̅̅̅̅̅̅̅̅푝+휀 = 푇 푝̅̅̅̅̅̅̅̅푝+휀푇 = 푇 푝̅̅̅̅̅̅̅̅푝+휀 (39) 푎푠−2 (휁+휀) (휁+휀) (휁+휀) (휁+휀) 푎푠−2 푎푠−2(휁+휀) (휁+휀) on each weighted Bergman space. Proof: The first equality in (39) is a direct consequence of Theorem (6.1.20). If 푒(푥 + 2휀) ≡ ′ 1 then Te = Id, and thus 훾푒̅ ,0,0(훽, 푥 , 휉) ≡ 1. Hence, Proposition (6.1.21)implies that in the case 푝̅̅̅̅̅̅̅̅̅푝+휀 of a symbol 푏푠−2 = (휁 + 휀) (휁 + 휀) with |푝(푗)| = |(푝 + 휀)(푗)|, for all j= 1,2, . . . , 푚, one has

160

( ′ ) 훾푏̅ 푠−2,푝,푝+휀 훽, 푥 , 휁 (훽 + 휀) + 1 푚 Γ 푗 + 훽 (훽 + 푝 + 휀)! ( 2 | (푗)|) = ∏ , (40) √훽! (훽 + 휀)! (훽 + 휀)푗 + 1 푗=1 Γ + 훽 + (푝 + 휀) ( 2 | (푗) (푗)|) 1+휀 1+휀 Whenever (훽 + 휀) ∈ 핫+ (cf. Example (6.1.18) for the choice of (훽 + 휀) = (1, … ,1) ∈ 핫+ 1+휀 and the case 푝푗 = (푝 + 휀)푗,for 1, … , 1 + 휀) Moreover, if 훽 + 휀 ∈ 핫+ , then it holds ( ′ ) 훾푏̅ 푠−2,푝,푝+휀 훽, 푥 , 휁 . Theorem (6.1.20), Proposition (6.1.21) and the assumption that|푝(푗)| = |(푝 + 휀)(푗)|, for all 푗 = 1,2, . . . , 푚, imply now that ( ′ ) ( ′ ) ( ′ ) 훾푎̅ 푠−2푏푠−2,푝,푝+휀 훽, 푥 , 휁 = 훾푏̅ 푠−2,푝,푝+휀 훽, 푥 , 휁 . 훾푎̅ 푠−2,0,0 훽, 푥 , 휁 ( ′ ) ( ′ ) = 훾푏̅ 푠−2,푝,푝+휀 훽, 푥 , 휁 . 훾푎̅ 푠−2,0,0 훽 + 휀, 푥 , 휁 This together with (33) and Theorem 4.5 yields the second equality in (39). Proposition (6.1.23)[270]:Let us assume that |푝(푗)| = |(푝 + 휀)(푗)| for all j= 1,2, . . . , 푚,. Then the Toeplitz operators 푇푗 commute mutually. Moreover, 푚

∏ 푇푗: = 푇(휁+휀)푝(̅̅휁̅̅+̅̅휀̅̅)푝+휀 (41) 푗=1 on each weighted Bergman space. ̅̅̅̅̅̅̅̅ 푝̅(푗)̅̅̅̅̅̅̅̅̅(푝+휀)(푗) Proof: Let 푏푗 ≔ (휁 + 휀) (휁 + 휀) , for 푗 = 1,2, . . . , 푚, We only prove the following product rule:

푇푗푇푖: = 푇 푝̅ +푝̅ (̅̅푝̅̅+̅̅̅휀̅̅) +(̅̅푝̅̅+̅̅̅휀̅̅) (42) (휁+휀) (푖) (푗)(휁̅̅̅+̅̅휀̅) (푖) (푗) for 푖, 푗 ∈ {푗 = 1,2, . . . , 푚, } and 푖 ≠ 푗. According to Theorem (6.1.17)the operator ∗ 휀+1 푛−휀−2 푅푠−2 푇푗푇푖푅푠−2 acts on the sequence space ℓ2(핫+ , 퐿2(ℝ × ℝ+)) by the rule: ∗ ′ 푅푠−2 푇푗푇푖푅푠−2 {푐훽(푥 , 휁)} 휀+1 훽∈핫+ ∗ ′ ′ = 푅 푇 푅 {훾̃ ̅̅̅̅̅̅̅̅ (훽, 푥 , 휁). 훾̃ ̅̅̅̅̅̅̅̅ (훽 − 푝̅ (̅̅푝̅̅+̅̅̅휀̅̅) , 푥 , 휁) 푠−2 푗 푠−2 푏푗,푝̅(푗),(푝+휀)(푗) 푏푖,푝̅(푖),(푝+휀)(푖) (푗) (푗) ′ × 푐 ̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅ (푥 , 휁)} 훽−푝̅(푖)−푝̅(푗)+(푝+휀)(푖)+(푝+휀)(푗) 휀+1 훽∈핫+ Hence it is clear that (42) is equivalent to: ′ ′ 훾̃ ̅̅̅̅̅̅̅̅ (훽, 푥 , 휁). 훾̃ ̅̅̅̅̅̅̅̅ (훽 − 푝̅ , (̅̅푝̅̅+̅̅̅휀̅̅) , 푥 , 휁) 푏푗,푝̅(푗),(푝+휀)(푗) 푏푖,푝̅(푖),(푝+휀)(푖) (푗) (푗) ′ = 훾̃ ̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅ (훽, 푥 , 휁). (43) 푏푖,푏푗,푝̅(푖),푝̅(푗),(푝+휀)(푖),(푝+휀)(푗) By (40) we have ( ′ ) 훾̃푏푗,푝̅(푗),(푝+휀)(푗) 훽, 푥 , 휁 (훽 + 휀)푗 + 1 ̅̅̅̅̅̅̅̅̅ Γ ( + |훽 |) (훽(푗) + (푝 + 휀)(푗))! 2 (푗) = , (훽 + 휀) + 1 ̅̅̅̅̅̅̅̅̅ Γ 푗 + 훽 + (̅̅푝̅̅+̅̅̅휀̅̅) √훽(푗)! (훽(푗) − 푝(̅ 푗) + (푝 + 휀)(푗))! ( 2 | (푗) (푗)|) and similar for 푖 replaced by 푗. Moreover, the function on the right hand side of (43) has the explicit form: 161

′ 훾̃ ̃ ̃ (훽, 푥 , 휁) 푏푖,푏푗,푝̃(푖),푝̃(푗),(푝+휀)(푖),(푝+휀)(푗) ̃ ̃ (훽 + (푝 + 휀)(푖) + (푝 + 휀)(푗))! = ̃ ̃ √훽! (훽 − 푝̃(푖) − 푝̃(푗) + (푝 + 휀)(푖) + (푝 + 휀)(푗))! (훽 + 휀) + 1 Γ ℓ + 훽 ( 2 | (ℓ)|) × ∏ (훽 + 휀)ℓ + 1 { } ( ̃ ) ℓ∈ 푖,푗 Γ ( 2 + |훽(ℓ) + 푝 + 휀 (ℓ)|) Now, (43) can be easily checked from these identities. Theorem (6.1.24)[270]: The Toeplitz operators T휑푠 and T휑푠+1 commute on each weighted 2 Bergman space (𝒜푠−2)휀−1(퐷푛+푠−1) if and only if for each ℓ = 1,2, . . . , 휀 + 1 one of the conditions (a) − (d) is fulfilled: (a) 푝ℓ = (푝 + 휀)ℓ = 0 (b) 푢ℓ = (푢 + 휀)ℓ = 0 (c) 푝ℓ = 푢ℓ = 0 (d) (푝 + 휀)ℓ = (푢 + 휀)ℓ = 0 2 Proof: Similar hat the operators T휑 푠 and T휑푠+1 commute on (𝒜푠−2)휀−1(퐷푛+푠−1) if and only 휀+1 if for all ∈ 핫+ : ( ′ ) ( ′ ) ( ′ ) ( ′ ) 훾̃휑푠,푝,푝+휀 훽, 푥 , 휁 . 훾̃휑푠+1푢,푢+휀 훽 + 휀, 푥 , 휁 = 훾̃휑푠+1푢,푢+휀 훽, 푥 , 휁 . 훾̃휑푠,푝,푝+휀 훽 + 휀, 푥 , 휁 Since |푝(푗)| = |(푝 + 휀)(푗)| and |푢(푗)| = | (푢 + 휀)(푗)| for 푗 = 1, 2, . . . , 푚we can use the ( ′ ) ( ′ ) factorization of 훾̃휑푠,푝,푝+휀 훽, 푥 , 휁 and 훾̃휑푠+1푢,푢+휀 훽, 푥 , 휁 in Proposition (6.1.21): ( ′ ) ( ) ( ′ ) 훾̃휑푠,푝,푝+휀 훽, 푥 , 휁 = Φ푝,푝+휀 훽 . 훾̃휑푠0,0 훽, 푥 , 휁 , ( ′ ) ( ) ( ′ ) 훾̃휑푠+1푢,푢+휀 훽, 푥 , 휁 = Φ푢,푢+휀 훽 . 훾̃휑푠+10,0 훽, 푥 , 휁 , where we use the notation: (훽 + 휀) + 1 푚 Γ 푗 + 훽 (훽 + 푝 + 휀)! ( 2 | (푗)|) (훽) = ∏ (44) √훽! (훽 + 휀)! (훽 + 휀)푗 + 1 푗=1 Γ + 훽 + (푝 + 휀) ( 2 | (푗) (푗)|) Moreover, it follows from Theorem (6.1.20) and again by the conditions on (푝, 푝 + 휀) and (푢, 푢 + 휀) that ( ′ ) ( ′ ) 훾̃휑푠0,0 훽, 푥 , 휁 = 훾̃휑푠0,0 훽 + 휀, 푥 , 휁 ( ′ ) ( ′ ) 훾̃휑푠+10,0 훽, 푥 , 휁 = 훾̃휑푠+10,0 훽 + 휀, 푥 , 휁 Therefore we only need to verify that Φ푝,푝+휀(훽). Φ푢,푢+휀(훽 + 휀) = Φ푢,푢+휀(훽). Φ푝,푝+휀(훽 + 휀). By a straightforward calculation this is equivalent to: (훽+푢+2휀)! (훽+푝+2휀)! (훽 + 푝 + 휀)! = (훽 + 푢 + 휀)! . (훽+휀)! (훽+휀)! Varying 훽 it can be seen that this equality holds if and only if for each ℓ = 1,2, . . . , 휀 + 1 one of the conditions (a)-(d) is fulfilled.

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Section (6.2) Toeplitz Operators with Quasi-Radial Quasi-Homogeneous Symbols We study of commutative algebras generated by Toeplitz operators acting on the Bergman spaces over the unit ball. The fact of just an existence of such algebras was quite unexpected and its exploration for the unit disk case was started in [294, 296, 295]. The final result on classification and description of the C∗-algebras generated by Toeplitz operators being 2 푛 commutative on all weighted Bergman spaces 𝒜휆 (𝔻 ) on the unit disk was obtained in [98]. In an equivalent reformulation it states that, under some technical assumption on the “richness” of a class of generating symbols, a C∗-algebra generated by Toeplitz operators is commutative on each weighted Bergman space if and only if the corresponding symbols of Toeplitz operators are constant on the orbits of a maximal commutative subgroup of the M̈ obius transformations of the unit disk. This result was extended then to the case of the unit ball. As proved in [218, 219], given a maximal commutative subgroup of biholomorphisms of the unit ball, the C∗-algebra generated by Toeplitz operators, whose symbols are constant on the orbits of this subgroup, is commutative on each weighted Bergman space. There are five different pairwise non-conjugate model classes of such subgroups: quasi-elliptic, quasi-parabolic, quasi-hyperbolic, nilpotent, and quasi-nilpotent (the last one depends on a parameter, giving in total 푛 + 2 model classes for the n-dimensional unit ball). As a consequence, for the unit ball of dimension 푛, there are 푛 + 2 essentially different “model” commutative C∗-algebras, all others are conjugated with one of them via biholomorphisms of the unit ball.The next surprise came first in [193] and was developed then in [37,38,195]. As it turned out, for n > 1 there exist many other, not geometrically defined, classes of symbols which generate commutative Toeplitz operator algebras on each weighted Bergman space. These classes of symbols were always subordinated to one of the above model classes of the maximal commutative subgroup (with the exception of the nilpotent subgroup). The corresponding commutative operator algebras were Banach, and being extended to C∗-algebras they became non-commutative. We note that in all above cases of the commutative C∗-algebras generated by Toeplitz operators these algebras always come with an unitary operator (specific for each algebra) that reduces each operator from the algebra to a multiplication operator, giving thus, among other results, a complete spectral picture of the operators under study. As for the commutative Banach algebras generated by Toeplitz operators, the results obtained so far give just the description of these algebras in terms of their generators. The next challenging task is to develop their Gelfand theory, obtaining thus more detailed information on the operators forming the algebra. We study the case of a commutative Banach algebra generated by Toeplitz operators with quasi-radial quasi-homogeneous symbols (i.e. an algebra subordinated to the quasi-elliptic group). To simplify the considerations we restrict our attention to the lowest dimensional case 푛 = 2.The corresponding (unique) commutative Toeplitz operator algebra 풯(휆) is Banach (not C∗), and can be described as follows: 2 푛 2 Let 퐻: = 𝒜휆 (𝔹 ) be the weighted Bergman space over 𝔹 with parameter 휆 > −1, and write ∗ 풯푟푎푑(휆) for the commutative C -subalgebra of ℒ(퐻) generated by all Toeplitz operators 푇푎 163 with radial bounded measurable symbols a on 𝔹2 (i.e.훼(푧) = 훼(|푧|)). Further, we denote by 푇휑 the unital Banach Structure of A Commutative Banach Algebra algebra with a single 2 generator 푇휑, where φ is the “simplest” quasi-homogeneous symbol on 𝔹 .It is easy to see that operators in 풯rad(λ) and 풯휑 commute and, as an important observation, we remark that 풯(휆)is generated by these two algebras (cf. Corollary (6.2.5)). Our results on the structure of풯(휆)already reveal some important features which we expect to be useful under a further study of the higher dimensional case 푛 > 2, and in a situation where the quasi-elliptic group of automorphisms of 𝔹푛 is replaced by another group among the above model classes. The main theorem explicitly expresses the maximal ideals of 풯(휆) and the Gelfand map. Theorem (Theorem (6.2.28)) The compact set 푀 (풯(휆)) of maximal ideals of the algebra 풯(휆) has the form 1 푀(풯(휆) ) = ℤ × {0} ∪ 푀 (휆) × 퐷̅ (0, ) + ∞ 2 where 푀∞(휆) can be identified with the subset of all multiplicative functional of 풯rad(λ) that map compact operators to zero. The Gelfand transform is generated by the following mapping of the elements of a dense (non-closed) subalgebra of 풯(휆): 푛 훾0(푘) (푘, 0) ∈ ℤ+ × {0} 푗 푛 1 ∑ 퐷훾푗푇훷 ⟼ { ∑ 휇(퐷훾푗)휉푗 (휇, 휉) ∈ 푀∞(휆) × 퐷̅ (0, ) 푗=0 푗=0 2

Here Dγj ∈ 풯rad(λ) is a diagonal operator with respect to the standard orthonormal basis [푒 : 훼 ∈ ℤ2 ] of 퐻 with the sequence 훾 = {훾 (|훼|)} of the corresponding eigenvalues. 푎 + 푗 푗 훼 As an important ingredient of the proof we carefully analyze the structure of the algebras ∗ 풯푟푎푑(휆), 풯휑, and of C -algebras that are generated by just a finite number of Toeplitz operators with radial symbols. In order to identify the multiplicative functionals of the previous algebras we essentially employ the concept of the “joint spectrum” and the “joint approximate spectrum“of finite tuples of operators together with the Berezin transform on functions with respect to suitable subspaces of 퐻. It is important to note that the arguments are not purely algebraic but heavily rely on the analytic structure of the generating Toeplitz operators and the underlying Bergman space. Some important properties of 풯(휆) can be deduced by the help of the previous theorem. Since 풯(휆)is not invariant under the ∗ −operation of ℒ(퐻) the inverse closedness of this algebra is not obvious, and usually such a feature is hard to show. Here we can prove the inverse closedness of 풯(휆) from the explicit description of the maximal ideal space and by extending multiplicative functionals (in a multiplicative way) from commutative Banach algebras to an enveloping (non-commutative) C∗ −algebra. We show that 풯(휆) is not semi-simple, and in Lemma (6.2.7) we describe some of the elements in its radical 푅푎푑풯(휆). However, to calculate the radical precisely we again need Theorem (6.2.28) together with additional arguments (see Lemma (6.2.38) and Theorem (6.2.43)).

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Finally we wish to point out that the derivation of the maximal ideal space has important consequences for the operator theory of the elements in 풯(휆)We give some remarks on the essential spectrum and the Fredholm property of (certain) operators 퐴 ∈ 풯(휆) (cf. Theorem (6.2.32) and Corollary (6.2.36)) and solve a“zero-product-problem”for Toeplitz operators (cf. Corollary (6.2.10)), which holds true despite of a certain ambiguity in the representation of operators as a finite sum of products of elements in풯푟푎푑(휆) and 풯휑(see Lemma (6.2.8)). We recall the construction of commutative Banach Toeplitz algebras that are subordinate to the quasielliptic group.We obtain a set of generators for this algebra 풯(휆)in the lowest dimensional case and we study some of its subalgebras. contains preliminary results on the maximal ideals for certain finitely generated subalgebras of 풯(휆). Our main result (Theorem (6.2.28)) on the Gelfand theory of 풯(휆)is proved and we give some applications. Among them we prove the inverse closedness of 풯(휆)and calculate its radical. We write 𝔹n, (푛 ∈ ℕ) for the open Euclidean unit ball in ℂ푛, i.e. 푛 푛 2 2 2 𝔹 ≔ {푧 = (푧1, … , 푧2) ∈ ℂ : |푧| = |푧1| + ⋯ + |푧푛| < 1} Let 푑푣 denote the standard volume form on 𝔹푛. With 휆 > −1 we consider the one-parameter family of the standard weighted measures 2 휆 풹휇휆(푧) = 푐휆(1 − |푧| ) 풹휈 푛 where 푐휆> 0 is a normalizing constant such that 푣휆(𝔹 ). More precisely, cλ is explicitly given by the formula: 훤(푛 + 휆 + 1) 푐 ≔ 휆 휋푛훤(휆 + 1) 2 푛 푛 The weighted Bergman space 𝒜휆 (𝔹 ) is the closed subspace in 퐿2(𝔹 , 푑푣휆) consisting of all 푛 푛 functions analytic in 𝔹 . We write 𝔹휆 for the orthogonal Bergman projection from 퐿2(𝔹 , 푑푣휆) 2 푛 onto 𝒜휆 (𝔹 ). It is well-known that 퐵휆 can be expressed as the following integral operator: 휑(휉) [퐵 휑](푧) = ∫ , 푑푣 (휉). 휆 (1 − 〈푧, 휉〉푛+휆+1) 휆 𝔹푛 푛 ̅ ̅ ∞ 푛 where 휑 ∈ 퐿2(𝔹 , 푑푣휆) and 〈푧, 휉〉 ≔ 푧1휉1 + ⋯ + 푧푛휉푛 .Given a function 푔 ∈ 퐿 (𝔹 ) the 2 푛 Toeplitz operator 푇푔 with symbol g acting on 𝒜휆 (𝔹 ) is defined by: 2 푛 푇푔휑 ≔ 퐵휆(푔휑) 휑 ∈ 𝒜휆 (𝔹 ) For 푛 > 1, new classes of commutative Banach algebras generated by Toeplitz operators with specific bounded symbols on 𝔹푛 have been constructed in [37,38,193,195]. As we have remarked already these algebras remain commutative on each weighted Bergman spaces 2 푛 𝒜휆 (𝔹 ) with 휆 > −1, and are induced by the maximal commutative subgroups of the biholomorphisms of the unit ball: quasi-elliptic, quasi-parabolic, quasi-hyperbolic and quasi- nilpotent. 2 n These Toeplitz Banach algebras are not invariant under the ∗-operation of ℒ(𝒜λ(𝔹 )), and being extended to C∗-algebras they become non-commutative. We shortly recall now the definition of the commutative algebras that are subordinated to the quasi-elliptic group of biholomorphisms (for further details see [217, 192]). Let 푘 = (푘1, 푘2,· · ·, 푘푚) be a tuple of positive integers with |푘| = 푘1 + 푘2 +· · · +푘푚 = 푛. We divide

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푛 the coordinates of 푧 ∈ 𝔹 into m groups of kj entries, respectively by using the notation 푧 = 푛 푧(1), … , 푧(푚) ∈ ℂ with 푘푗 푧(푗) = (푧푗,1, … , 푧푗,푘푗) ∈ ℂ 2푘푗−1 푘푗 where 푗 = 1, . . . , 푚. Let 핊 ⊂ ℂ denote the (real) (2푘푗 − 1) –dimensionalunit sphere in 푘푗 ℂ . We express 푧(푗) = 0 in polar-coordinates 푧(푗) = 푟푗푧(푗)휉(푗) with 푧 (푗) 2푘푗−1 휉(푗) = ∈ 핊 and 푟푗 = ‖푧(푗)‖ ∈ ℝ+ (45) ‖푧(푗)‖ A bounded function 휑(푧) on 𝔹푛a 3 is called k-quasi-homogeneous if it has the form: 푝(1) 푝(2) 푝(푚) −푞(1) −푞(2) −푞(푚) 휑(푧) = 푎(푟1, … , 푟푚)휉(1) 휉(2) … 휉(푚) 휉(1) 휉(2) … 휉(푚) (46) 푛 푛 and 푎is a function of the m non-negative real variables 푟1, … , 푟푚.The tuple (푝, 푞) ∈ ℤ+ × ℤ+ with 푝 ⊥ 푞is called the quasi-homogeneous degree of 휑(z). 푚 Fix a tuple ℎ = (ℎ1, … , ℎ푚) ∈ ℤ+ ,with ℎ푗 = 0 if 푘푗 = 1 and 1 ≤ ℎ푗 ≤ 푘푗 − 1 if 푘푗 > 1. We denote by ℛ푘(ℎ) the linear space generated by all k-quasi-homogeneous functions of the form (46) such that

(i) For 푗 with 푘푗 > 1: 푝(푗) = (푝푗,1, … , 푝푗,ℎ푗 , 0, … , 0) and 푞(푗) = (0, … , 0, 푞푗,ℎ푗+1 , … , 푞푗,푘푗 ), with 푝푗,1, … , 푝푗,ℎ푗, 푞푗,ℎ푗+1, . . . , 푞푗,푘푗 ∈ ℤ+, and푝푗,1 + ⋯ + 푝푗,ℎ푗 = 푞푗,ℎ푗+1, . . . , 푞푗,푘푗 . (ii) If 푘푗′ = 푘푗" with 푗′ < 푗′′ thenℎ푗′ ≤ ℎ푗′′ . Recall that the quasi-elliptic group of biholomorphisms of 𝔹n is isomorphic with the n-torus (핋푛 (here 핋 = 핊1) and acts on 𝔹n as follows 푛 핋 ∋ 푡 = (푡1, … , 푡푛) ∶ 푧 = (푧1, … , 푧푛) ⟼ 푡푧 = (푡1푧1, … , 푡푛푧푛). 푚 Note that the functions from ℛ푘(ℎ) are invariant under the subgroup 핋 of the quasi-elliptic group 핋푛, which acts on𝔹푛 as follows 푚 핋 ∋ 푡 = (푡1, … , 푡푚) ∶ 푧 = (푧1, … , 푧푚) ⟼ (푡1푧(1), … , 푡푚푧(푚)). The main result in [193] states the following: Theorem (6.2.1)[303]: The Banach algebra ℬ푘(ℎ) generated by Toeplitz operators with symbols from ℛ푘(ℎ) is commutative. ∗ In the case of n >1 the algebras ℬ푘(ℎ) do not extend to commutative C -algebras. This effect arise from the multidimensional setting and has no counterpart in the case of 푛 = 1. Our next global plan is to study the internal structure of ℬ푘(ℎ) more precisely and, in particular, we wish to determine their maximal ideal spaces. We consider the simplest model case where 푛 = 2, and 푘 = 2 that is, we fix the dimension 푛 = 2 and we choose 푚 = 1. As a consequence we need to put ℎ = (1) = 1, and our main object to study, the commutative Banach algebra 풯(휆): = ℬ2(1) , is generated by the operators of the form 푇푎(푟)휉(푝,0)휉(0,푝), where 푎 ∈ 퐿∞[0,1) and 푝 ∈ ℤ+. By [192], for any bounded measurable function a(r) we have 훼 훼 2 푇훼푧 = 훾훼,휆(|훼|)푧 , 훼 ∈ ℤ+ Where

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1 훤(|훼| + 휆 + 3) 훾 (|훼|) = ∫ 훼(√푟 )(1 − 푟)휆푟|훼|+1푑푟. (47) 훼,휆 훤(휆 + 1)훤(|훼| + 2) 0 According to [192], for |푝| = |푞| we have 훼 훼1+푝 훼2−푝 2 푇휉(푝,0)휉̅ (표,푝)푧 = 훾푝̅ ,휆(훼)푧1 푧2 , 훼 ∈ ℤ+, Where 훼 (훼 − 1) … 훼 (훼 − 푝 + 1) 훾̅ (훼) = 2 2 2 2 (48) 푝,휆 (푝 + 1 + |훼|)(푝 + |훼|) … (2 + |훼|) We mention that 훾푝̅ ,휆 does not depend on the weight parameter 휆. By [193], for any bounded measurable 훼 = 훼(푟) and 푝 ∈ ℤ+ we have

푇훼푇휉(푝,0)휉̅ (표,푝) = 푇휉(푝,0)휉̅ (표,푝)푇훼 = 푇훼휉(푝,0)휉̅ (표,푝) (49) As a consequence the algebra 풯(휆) is generated by the operators 푇훼, with 푎 ∈ 퐿∞[0,1), and + 푇휉(푝,0)휉̅ (표,푝), where 푝 ∈ ℤ (see Corollary (6.2.3) for an even“smaller”set of generators). We start our analysis by studying separately the different types of Toeplitz operators that, according to generate 풯(휆). First we consider operators with radial and then with quasi- homogeneous symbols. + Let 훾 = {훾(|훼|)} |훼| ∈ ℤ be a bounded sequence. Denote by 퐷훾 the (bounded linear) 2 2 diagonal operator which acts on the weighted Bergman space 𝒜휆 (𝔹 ), by the rule 훼 훼 2 풟훼푧 = 훾(|훼|) 푧 , 훼 ∈ ℤ+ Of course each Toeplitz operator with bounded measurable radial symbol α(r) is diagonal, and 푇훼 = 퐷훾훼,휆. However, as the next lemma states, not all bounded diagonal operators 퐷훾 can be represented in such a form since the eigenvalue sequence 훾훼,흀 of 푇훼 is always slowly oscillating. Lemma (6.2.2)[303]:Let 훼(푟) ∈ 퐿∞[0,1)and 푘 = |훼|.Then 푙푖푚(훾훼,흀(푘) − 훾훼,흀 (푘 + 1)) = 0. 푘→∞ Proof: Let 푀 = 푒푠푠 − 푠푢푝 |훼(푟)|. By (27), we have |훾훼,휆(푘) − 훾훼,휆(푘 + 1)| 1 훤(푘 + 휆 + 3) = | ∫ 훼(√푟)(1 − 푟)휆푟푘+1푑푟 훤(휆 + 1)훤(푘 + 2) 0 1 훤(푘 + 휆 + 4) − ∫ 훼(√푟)(1 − 푟)휆푟푘+2푑푟| 훤(휆 + 1)훤(푘 + 3) 0 1 훤(푘 + 휆 + 3) = | ∫ 훼(√푟)(1 − 푟)휆+1푟푘+1푑푟 훤(휆 + 1)훤(푘 + 2) 0 1 휆 + 1 훤(푘 + 휆 + 4) − ∫ 훼(√푟)(1 − 푟)휆푟푘+2푑푟| 푘 + 휆 + 3 훤(휆 + 1)훤(푘 + 3) 0 167

휆 + 1 = 훾 (푘) − 훾 (푘 + 1)| 푘 + 휆 + 3 훼,휆+1 훼,휆 휆 + 1 ≤ 2푀 푘 + 휆 + 3 The last expression tends to 0 when 푘 → ∞. It follows from the lemma that the set of (partial) limit points of the sequence 훾훼,휆 is connected and compact. According to the results in [194] and for the unweighted case (휆 = 0) the algebra 풯푟푎푑(휆)is ∗ isomorphic and isometric to the C -algebra SO(0) which consists of all slowly oscillating sequences satisfying the condition 푙푖푚 |훾(푚) − 훾(푛)| = 0. 푚 →1 푛 We denote by 푀(풯푟푎푑(휆))the compact set of maximal ideals of the algebra 풯푟푎푑(휆) (or, which is the same, of the algebra 푆푂(휆)). Let 푀∞(휆) be the fiber of푀(풯푟푎푑(휆)) consisting of all multiplicative functionals 휓 such that 휓(퐷훾) = 0whenever 퐷훾 is compact (or whenever 훾 ∈ 푐0, where c0 denotes set of all sequences converging to zero). Each point 푘 ∈ ℤ+ defines a multiplicative functional 휓(푘) on 풯푟푎푑(휆): 휓(푘): 퐷훾 ⟼ 훾(푘), and thus the set ℤ+ can be considered as a part of M(풯푟푎푑(휆)). Moreover, 푀(풯푟푎푑(휆)) = ℤ+ ∪ 푀∞(휆) (50) and by [95], the set ℤ+ is densely and homeomorphically embedded into 푀(풯푟푎푑(휆)) with respect to the Gelfand topology on 푀(풯푟푎푑(휆)) Furthermore, by [250], the set 푀∞(휆) is connected. We mention for completeness that none of the points of 푀∞(휆) can be reached by subsequences of ℤ+; its topological nature requires to use nets (subnets of ℤ+). That is, for each point 휇 ∈ 푀 (휆) there is a net {푛 } , valued in ℤ , which tends to 휇 in the Gelfand topology ∞ 훽 훽∈훽 + ( ) { ( )} ( ) of 푀 풯푟푎푑(휆) . Or, in other words, for each 훾 = 훾 푛 푛∈ℤ+ ∈ 푆푂 휆 , we have that 푙푖푚 훾(푛훽) = 훾(휇) (51) 훽∈훽 where we identify 훾(휇) with 휇(훾), the value of the functional 휇 ∈ 푀∞(휆) on the element훾 ∈ 푆푂(휆). Consider now the special case of a radial symbol: (푟) = 푟2 . By (47) we have 1 (| | ) 훤 훼 + 휆 + 3 휆 |훼|+1 훾 2, 휆(|훼|) = ∫ 훼(√푟)(1 − 푟) 푟 푑푟 푟 훤(휆 + 1)훤(|훼| + 2) 0 1 훤(|훼| + 휆 + 3) = ∫ 훼(1 − 푟)휆+1푟|훼|+2푑푟 훤(휆 + 1)훤(|훼| + 2) 0 훤(|훼| + 휆 + 3) |훼| + 2 = 퐵(휆 + 1, |훼| + 3) = 훤(휆 + 1)훤(|훼| + 2) |훼| + 휆 + 3

168

The sequence 훾푟2,휆(푛), 푛 ∈ ℤ+, is real valued, strictly monotone (thus separatingpoints of ℤ+), and convergent when 푛 → ∞. Hence, by the Stone-Weierstrass theorem, the unital C∗-algebra generated by a single Toeplitz operator 푇푟2 coincides with the algebra of all diagonal operators 퐷훾 with 훾 ∈ 푐, where 푐 denotes the set of all convergent sequences. Corollary (6.2.3)[303]: Let 훾 ∈ 푐 , then Dγ ∈ 풯(λ). In particular, for all 푛 ∈ ℤ+ the orthogonal 2 2 훼 projection 푃푛 of 𝒜휆 (𝔹 ) onto span{푧 : |훼| = 푛} belongs to the algebra 풯(λ). In order to simplify formulas, and with the coordinates in (45) we will use the notation (푝,0) ̅(표,푝) 3 2 휙푝 = 휙푝(휉) = 휉 휉 , where 휉 = (휉1, 휉2) ∈ 핊 ⊂ ℂ and 푝 ∈ ℕ; for 푝 = 1 we simply write 휙 = 휙1. We start with some calculations based on (48): 훼 푇 푧훼1푧훼2 = 2 푧훼1+1푧훼2−1, 훼 ≥ 1 휙 1 2 2 + |훼| 1 2 2 훼 (훼 − 1) 푇2푧훼1푧훼2 = 2 2 푧훼1+2푧훼2−2, 훼 ≥ 2 휙 1 2 (2 + |훼|)2 1 2 2 훼 (훼 − 1) 푇 푧훼1푧훼2 = 2 2 푧훼1+2푧훼2−2, 훼 ≥ 2 휙2 1 2 (3 + |훼|)(2 + |훼|) 1 2 2 Thus 1 1 훼 (훼 − 1) 1 (푇2 − 푇 )푧훼1푧훼2 = ( − ) 2 2 푧훼1+2푧훼2−2 = 푇2푧훼1푧훼2, 휙 휙2 1 2 2 + |훼| 3 + |훼| 2 + |훼| 1 2 3 + |훼| 휙 1 2 or 2 푇휙2 = 퐷푑2푇휙, where 2 + |훼| 푑 (|훼|) = , 훼 ∈ ℤ2 . 2 3 + |훼| +

Due to the remark before Corollary (6.2.3) we conclude that the Toeplitz operator 푇휙2 belongs to the unital algebra generated by 푇푟2 and 푇휙. Similarly, for any 푝 ∈ ℕ, and 훼2 > 푝 we have 훼 (훼 − 1) … (훼 − 푝 + 1) 푇 푧훼1푧훼2 = 2 2 2 푧훼1+푝푧훼2−푝 휙푝 1 2 (푝 + 1 + |훼|) … (2 + |훼|) 1 2 훼 (훼 − 1) … (훼 − 푝 + 1)(훼 − 푝) (푇 푇 ) 푧훼1푧훼2 = 2 2 2 2 푧훼1+푝+1푧훼2−푝−1 휙 휙푝 1 2 (푝 + 1 + |훼|) … (2 + |훼|)(2 + |훼|) 1 2 훼 (훼 − 1) … (훼 − 푝) 푇 푧훼1푧훼2 = 2 2 2 푧훼1+푝+1푧훼2−푝−1 휙푝+1 1 2 (푝 + 1 + |훼|) … (2 + |훼|) 1 2 By comparing these relations we obtain: 푝 푇 푇 − 푇 푧훼1푧훼2 = 푇 푇 푧훼1푧훼2. 휙 휙푝 휙푝+1 1 2 푝+2+|훼| 휙 휙푝 1 2 Note that the last equality is also valid in the case of 0 ≤ 훼2 ≤ 푝 and hence

푇휙푝+1 = 퐷푑푝+1푇휙 푇휙푝 where 2 + |훼| 2 + |훼| 푑 (|훼|) = = , 훼 ∈ ℤ2 푝+1 푝 + 1 + |훼| (푝 + 1) + 1 + |훼| +

169

By induction we finally have 푝 푝 푇휙푝 = (∏ 퐷푑푘) 푇휙 , (52) 푘=1 { (| |)} where the eigenvalue sequence 푑푘 = 푑푘 훼 |훼|∈ℤ+ is given by 2 + |훼| 푑 (|훼|) = 푘 푘 + 1 + |훼| Note that 푑1 (|훼|) ≡ 1, as it should be. An alternative form of (52) is 푝 푇휙푝 = 퐷푑̅푝푇휙 (53) ̃ ̃ where the eigenvalue sequence 푑푝 = {푑푝|훼|} is given by |훼|∈ℤ+ (2 + |훼|)푝 ̃ 푑푝|훼| = , (54) (푝 + 1 + |훼|)(푝) and(푥)(푝) = 푥(푥 − 1) · · · (푥 − 푝 + 1) is a kind of Pochhammer symbol. ̃ We note that, for each 푝 ∈ ℕ , both sequences 푑푝 and 푑푝 tend to 1 when |훼| → ∞. Thus we have according to the remark before Corollary (6.2.3):

Theorem (6.2.4)[303]:For each 푝 ∈ ℕ, the Toeplitz operator 푇휙푝 belongs to the unital algebra generated by 푇푟2 and푇휙. Corollary (6.2.5)[303]:The Banach algebra 풯(휆) is generated, in fact, just by Toeplitz operators 푇푎 with bounded measurable radial symbols 푎(푟) and the single Toeplitz operator 푇휙 (with the simplest quasi-homogeneous symbol 휙(휉)). 훼 2 We normalize the monomials푧 to the standard orthonormal basis [푒훼: 훼 ∈ ℤ+]of the 2 2 Bergman space 𝒜λ(𝔹 ), i.e. 훤(|훼| + 휆 + 3) 푒 = √ 푧훼 , 훼 ∈ ℤ2 . (55) 훼 훼! 훤(휆 + 3) + 2 Then we have in the case of 훼 ∈ ℤ+ with 훼2 ≥ 푝: √(훼 + 1)훼 푇 푒 = 1 2 푒 (56) 휙 훼 2 + |훼| (훼1+1,훼2−1), √(훼 + 푝) … (훼 + 1)훼 … (훼 − 푝 + 1) 푇푝푒 = 1 1 2 2 푒 (57) 휙 훼 (2 + |훼|)푝 (훼1+푝,훼2−푝), 1 which implies that ‖푇푝‖ = 2−푝 for all 푝 ∈ ℕ, and thus the spectral radius of 푇 is equal to . 휙 휙 2 Note that (48) and (56) imply that the action of Tϕ does not depend on the weight parameter 휆. Thus the structure of the unital Banach algebra 풯휙 generated by 푇휙 does not depend on the weight parameterλ as well, and thus the spectrum of 푇휙 is independent of 휆. Consider now the case 휆 = 0. Since φ extends continuously to the boundary ∂𝔹2 of 𝔹2 it follows from the results in [298] that the essential spectrum of the operator Tϕ is given by

170

푟1푟2 1 2 ̅ ̅ 푒푠푠 − 푠푝푇휙 = 퐼푚푇휙(휉)|휕𝔹 = 퐼푚 ( 2 2 푡1푡2)| = 퐷 (0, ) 푟1 + 푟2 2 2 1 2 푟1 +푟2 =1,푡1,푡2∈핊 1 1 where 퐷̅ (0, ) is the closed disk centered at origin and with the radius . 2 2 Finally, 1 1 퐷̅ (0, ) = 푒푠푠 − 푠푝 푇 ⊂ 푠푝 푇 ⊂ 퐷̅ (0, ) 2 휙 휙 2 1 implies that 푠푝 푇 = 퐷̅ (0, ). 휙 2 By [75], the maximal ideal space 푀( 푇휙) of the commutative Banach algebra 푇휙 coincides 1 with the spectrum of the operator 푇 ,i.e. 푀( 풯 ) = 퐷̅ (0, ). 휙 휙 2 Theorem (6.2.6)[303]: The Banach algebra 풯휙 is isomorphic via the Gelfand transform to the 1 1 algebra 퐶 퐷̅ (0, ), which consists of all functions analytic in 퐷 (0, ) and continuous on 푎 2 2 1 퐷̅ (0, ) . 2 ∗ ∗ Proof: Consider two unital algebras: the Banach algebra 풯휙 and the C -algebra 풯ϕ , both are ∗ generated by Tϕ. The operator Tϕ commutes with its adjoint 푇휙 = 푇휙̅ modulo a compact ̂∗ ∗ ∗ operator, thus the quotient algebra 풯ϕ = 풯ϕ /(풯ϕ ∩ 풦). where 풦 denotes the ideal of all compact operators, is a commutativeC∗-algebra which is isomorphic and isometric to 1 퐶(푒푠푠 − 푠푝 푇 ) = 퐶(퐷 (0, )). As the spectrum of any compact operator is at most countable 휙 2 and having at most one limit point 0, which is not the case for any non-zero operator from 풯휙 , we have 풯휙 ∩ 풦 = {0}. Thus 1 풯 = 풯 /( 풯 ∩ 풦) ≅ ( 풯 + 풯∗ ∩ 풦)/(풯∗ ∩ 풦) ⊂ 풯∗/(풯∗ ∩ 풦) ≅ 퐶(퐷̅ (0, ) 휙 휙 휙 휙 휙 휙 휙 휙 2 That is, the algebra 풯휙, being isomorphic to the uniform closure of all polynomials of ζ defined 1 1 on (퐷̅ (0, ), is isomorphic to 퐶푎 (퐷̅ (0, )) . 2 2 2 2 2 For each 푘 ∈ ℤ+, we denote by 퐻푘 the following subspace of 𝒜휆 (𝔹 ): 2 퐻푘 = 푠푝푎푛 {푒푎: 훼 ∈ ℤ+, |훼| = 푘}, (38) 2 2 and, of course, we have an orthogonal decomposition of 𝒜λ(𝔹 ) into finite dimensional Hilbert spaces ∞ 2 2 𝒜λ(𝔹 ) = ⨁ 퐻푘 푘=0 each space 퐻푘 is obviously invariant for all operators from 풯(휆). The diagonal operator 퐷휆, restricted to Hk, is just the scalar operator 훾(푘)퐼, while the operator 푇휙 acts on 퐻푘 as a weighted shift operator. Moreover, the operator 푇휙, restricted to 퐻푘, is nilpotent, 푘+1 ( 푇휙|퐻푘 ) = 0. In particular, this implies that, for all 푝 ∈ ℕ,

171

푝−1

⨁ 퐻푘 ⊂ 푘푒푟 푇휙 (59) 푘=0 2 2 Recall as well that each orthogonal projection 푃푘 of 𝒜휆 (𝔹 ) onto 퐻푘 is a diagonal operator, which belongs to the unital subalgebra of 풯(휆)generated by 푇푟2. An important information on the structure of 풯(휆) gives the next lemma. A much stronger result can be found at the end of the section (cf. Lemma (6.2.38) and Theorem (6.2.43)): Lemma (6.2.7)[303]: The algebra 풯(휆)is not semi-simple. Its radical 푅푎푑풯(휆)contains, in particular, all operators of the form 푇휙푝 = 퐷휸 where 훾 ∈ 푐0 and p ∈ ℕ. Proof: In virtue of (53) it is sufficient to prove that 퐷훾 푇휙 ∈ 푅푎푑풯(휆) , or (see, for example, [75]) that the operator 퐴 = 퐷훾 푇휙 is topologically nilpotent, i.e., 1 푙푖푚 ‖퐴푘‖푘 = 0 . 푘⟶∞ we have 푘 푘 푘 푘 푘 푘 푘 퐴 = 퐷훾 푇휙 = 퐷훾 푇휙 (퐼 − (푃0 + ⋯ + 푃푘−1)) = [퐷훾(퐼 − (푃0 + ⋯ + 푃푘−1))] 푇휙 Thus 1 푘 k 푘 푘 ‖퐴 ‖ ≤ ‖퐷훾(퐼 − (푃0 + ⋯ + 푃푘−1))‖. ‖푇휙 ‖ = ‖푇휙 ‖. 푠푢푝|훾(푙)|. 1>푘 As γ ∈ c0, the last expression tends to 0 when 푘 ⟶ ∞. We denote by 퐷(휆) the dense (non-closed) subalgebra of 풯(휆) formed by finite sums of finite products of its generators: 푇푎 with bounded measurable radial symbols 푎(푟) and 푇휙푝 with푝 ∈ ℕ. An operator 퐴 from 퐷(휆) has the form 푚 푝 퐴 = ∑ 퐷훾푝푇휙 푝=0

We mention that for arbitrary diagonal operators 퐷훾푝 the above representation is not unique. To describe this ambiguity we will use the notation 퐾훾(푝), with 푝 ∈ ℤ+, for a finite dimensional diagonal operator, whose eigenvalue sequence has the form 훾 = {훾(0), 훾(1), … , 훾(푝 − 1), 0,0, … }, of course, for 푝 = 0, it is just the zero operator. Lemma (6.2.8)[303]:We have ∑푚 푝 푝=0 퐷훾푝 푇휙 = 0(60) if and only if 퐷훾푝= 푘훾푝(푝) , for each 푝 = 0,1, … , 푚. Proof: The part “if” follows from (59).

To prove the “only if” part, consider any n ≥ m and note that each 퐷훾푝is diagonal with respect to the basis (55) with (| |) 퐷훾푝푒훼 = 훾푝 푎 푒훼. 푝 푝 ( ) Moreover, by (57), the operator 푇휙 acts on (55) as 푇휙 푒(훼1,푎2) = 휏푝 훼 . 172

( ) 푒(훼1+푝,훼2−푝), where 0 ≠ 휏푝 훼 ∈ ℝif훼2 ≥ 푝. Then, by (60), we have 푚 푚 푝 ( ) ( ) 0 = ∑ 퐷훾푝 푇휙 푒(0,푛) = ∑ 훾푝 푛 휏푝 0, 푛 푒(푝,푛−푝). 푝=0 푝=0 Since{e(p,n−p): p= 0, 1,…,m}forms a system of orthogonal vectors and 휏푝(0, 푛)=0 we conclude that 훾푝(푛) = 0 for 푝 = 0, 1, … , 푚 and 푛 ≥ 푚.

Therefore 퐷훾푝 is finite dimensional for 푝 = 0, 1, … , 푚, and, in particular, 퐷훾푚 = 퐾훾푚(푚). 푚 ( ) 푚 It follows that 퐷훾푚푇휙 = 퐾훾푚 푚 푇휙 = 0, and thus we have 푚−1 푝 ∑ 퐷훾푝푇휙 = 0 푝=0 Repeating the above arguments m times (each time lowering the sum upperlimit), we have consequently ( ) ( ) ( ) 퐷훾푚−1 = 퐾훾푚−1 푚 − 1 , 퐷훾푚−2 = 퐾훾푚−2 푚 − 2 , … , 퐷훾0 = 퐾훾0 0 = 0 At the same time the situation is quite different for special finite sums of finite products of generators from 퐷(휆). To proceed with the result we define the“grade”for some operators by:

푔푟푎푑푒 (퐷훾): = 0, and 푔푟푎푑푒 (푇휙푝 ) ≔ 푝. 푚 Moreover, if ∏푘=1 퐴푘 is the product of the above operators, then we put 푚 푚

푔푟푎푑푒 (∏ 퐴푘) ≔ ∑ 푔푟푎푑푒(퐴푘) 푘=1 푘=1 Theorem (6.2.9)[303]: Let us assume that all summands of the operator 푚 푚푘 푛푘

퐴 = ∑ (∏ 푇푎 ∏ 푇휙 ) = 0 푘,푞 푝푘,푠 푘=1 푞=1 푠=1 have different grades. Then it follows that for each k at least one radial symbol 푎푘,푞 is identically zero. Proof: By (53) and (54), we have 푚 푚푘 푝푘,1+⋯+푝푘,푛푘 퐴 = ∑ (∏ 푇푎푘,푞 퐷훾푘 푇휙 ) = 0, 푘=1 푞=1 where each 퐷훾푘 is invertible and its eigenvalue sequence tends to 1. Thus, by Lemma (6.2.8), ∏푚푘 we obtain that each diagonal operator 푞=1 푇푎푘,푞 is finite dimensional. Then the result follows by [319], Theorem (6.2.1), and Theorem (6.2.6). As a corollary to the previous theorem we give a result on the so-called zero-product problem (see, for example, [168,169,319]).

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Corollary (6.2.10)[303]:For the operator 푚 푛 퐴 = ∏ 푇 ∏ 푇 푎푞 휙푝푠 푞=1 푠=1 the following statements are equivalent: (i) 퐴 = 0, (ii) 퐴 is finite dimensional, (iii) At least one radial symbol 푎푞 is identically zero. We start from recalling some known facts and definitions: Let 𝒜 = 𝒜(푥1, … 푥푛) be a unital commutative Banach algebra generated by the elements 푥1, … 푥푛, and let 푀(𝒜) denote the compact set of its maximal ideals. Then (cf. [167]) the joint spectrum 휎(푥1, … 푥푛) of 푥1, … , 푥푛 is the subset of ℂ푛 defined by 휎(푥1, … 푥푛) = {푚(푥1), 푚(푥2), … . 푚(푥푛): 푚 ∈ 푀(𝒜)}. (61) In (61)we identify maximal ideals in 𝒜 and multiplicative functional on 𝒜in the usual way. As is well known, the mapping 푚 ∈ 푀(𝒜) → (푚(푥1), 푚(푥2), … . 푚(푥푛)) ∈ 휎(푥1, … 푥푛) defines a homeomorphism between 푀(𝒜) and 휎(푥1, … 푥푛) If푒 ∈ 𝒜 denotes the unit element then we can also write (cf.[318]) 푛 휎(푥1, … , 푥푛) = {(휇1, … 휇푛) ∈ ℂ : 퐽(푥1 − 휇1푒, … , 푥푛 − 휇푛푒) ≠ 𝒜} (62) where 퐽(푥1 − 휇1푒, … , 푥푛 − 휇푛푒) denotes the smallest ideal in the algebra 𝒜 which contains the elements 푥푗 − 휇푗푒, with 푗 = 1, … , 푛. Let 퐻 be a complex Hilbert space and (퐴1, 퐴2) be a tuple of (bounded) commuting operators on 퐻. 2 We say (cf. [67]) that (휇1 , 휇2) ∈ ℂ is in the joint approximate point spectrum 휎휋(퐴1, 퐴2) of (퐴1, 퐴2) if and only if, for all 퐵1, 퐵2 ∈ ℒ(퐻), 퐵1(퐴1 − 휇1퐼) + 퐵2(퐴2 − 휇2퐼) ≠ 퐼 Now, let 𝒜 = 𝒜(퐴1, 퐴2) be the Banach algebra in ℒ(퐻) generated by the commuting operators A1, A2 and the identity element. The next statement is well-known and quite standard. Lemma (6.2.11)[303]:The following inclusions hold: 휎휋(퐴1, 퐴2) ⊂ 휎(퐴1, 퐴2) ≅ 푀(𝒜) ⊂ 푀(𝒜1) × 푀(𝒜2) where 𝒜1 and 𝒜2 denote the unital Banach algebras generated by 𝒜1 and 𝒜2, respectively. Proof:Note that, by restriction, each element 휑 ∈ 푀(𝒜) defines a functional in 푀(𝒜푗), 푗 = 1,2 which proves the second inclusion. The first inclusion directly follows from the characterization (62). Below we need a more concrete characterization of the joint approximate point spectrum which has been given by A.T. Dash: 2 Proposition (6.2.12)[303]: [67] A tuple (μ1 , μ2) ∈ ℂ is in σπ(A1, A2) if and only if there is a sequence {fn}n ⊂ H of unit vectors such that ‖(퐴1 − 휇1퐼)푓푛‖ → 0 and‖(퐴2 − 휇2퐼)푓푛‖ → 0 as n tends to infinity. It is instructive to consider first the unital algebra with just two generators:

174 a diagonal operator 퐷훾 and 푇휙. The result will already give a good approximation to what one can expect for the algebra 풯(휆). We fix 휆 ∈ (−1, ∞) and a diagonal operator 퐴1 = 퐷훾 ∈ 풯푟푎푑(휆), whose eigenvalue sequence 훾 = {훾(푘)} ∈ ℤ+ satisfies the conditions: (i)훾(푘1) ≠ 훾(푘2), for all 푘1 ≠ 푘2, (ii) none of the eigenvalues 훾(푘) belongs to the set of limit points of 훾. We note that the aim of these conditions is to separate“as much as possible” the points of the compact set of maximal ideals of the unital algebra generated by 퐷훾. Furthermore they will be important in the proof of Lemma (6.2.14). Let 퐴2 = 푇휙. We consider the unital Banach algebra 𝒜 = 𝒜(퐴1, 퐴2) generated by two commuting elements 퐴1 and 퐴2. If we denote by퐿푖푚(훾) the set of all limit points of the sequenceγ, then the spectrum of the operator 퐷훾 is given by 휎(퐷훾) = 푐푙표푠 퐼푚(훾) = {훾(푘): 푘 ∈ ℤ+} ∪ 퐿푖푚 (훾) Recall that the operators A1 and A2 act on the basis elements (55) as follows 퐴1푒훼 = 퐷훾푒훼 = 훾|훼|푒훼 , √(훼 + 1)훼 퐴 푒 = 푇 푒 = 1 2 푒 . 2 훼 휙 훼 2 + |훼| (훼1+1,훼2−1) Lemma (6.2.13)[303]: We have (퐷훾) × {0} ⊂ 휎휋(퐴1, 퐴2) . Proof: Fix first(푘) ∈ 훾 , and observe that 푒(푘,0) ∈ 푘푒푟 푇휉(1,0)휉̅ (0,1). If we put 푓푛 ≡ 푒(푘,0) , for 푛 ∈ ℕ, then we have

‖(퐷훾 − 훾(푘)퐼)푓푛‖ = 0 and ‖(푇휉(1,0)휉̅ (0,1) − 0퐼)푓푛‖ = 0. That is, 퐿푖푚(훾) × {0} ⊂ 휎휋(퐴1, 퐴2).

Lemma (6.2.14)[303]:None of the points (훾(푘), 휁),where k ∈ ℤ+ and 푘 ∈ ℤ+ (휁 ∈ 1 퐷(0, )\{0}belong to the joint spectrum 휎(퐴 , 퐴 ) . 2 1 2 Proof: We fix a pair(훾(푘), 휁), with 휁 = 0, and show that the ideal in 𝒜(퐴1, 퐴2), which is generated by the operators 퐴1 − 훾(푘0)퐼 = 퐷훾 − 훾(푘0)퐼, and 퐴2 − 휁퐼 = 푇휙 − 휁퐼, coincides with the whole algebra 𝒜(퐴1, 퐴2).

Consider the finite dimensional space 퐻푘0 in (58). Then it can be easily seen that 퐻푘0 and its orthogonal complement ⊥ 2( 2) are invariant under the operators 퐻푘0 = 𝒜휆 𝔹 ⊝ 퐻푘0

퐷훾 and 푇휙. Moreover, the restriction of 𝒜2 = 푇휙 to 퐻푘0 is nilpotent, 푘0+1 ((𝒜2|퐻푘0) = 0. The operator 𝒜1 − 훾(푘0)퐼 is diagonal, and its eigenvalue sequence is of the form 훾̅ = { ( ) ( )} 훾 푘 − 훾 푘0 푘∈ ℤ+ .The above conditions (i) and (ii) guarantee that 푖푛푓 |훾(푘) − 훾(푘0)| > 0 푘≠푘0 for each 푘0 ∈ ℤ+. Thus the diagonal operator 퐷훾̅(−1) with the eigenvalue sequence

175

1 훾̅−1 = {(1 − 훿 ) } 푘,푘0 (푘) − 훾(푘 ) 0 푘∈ ℤ+ is well defined, it obviously belongs to 𝒜(퐴1) ⊂ 𝒜(퐴1, 퐴2), and ( ( ) ) 퐷훾̅(−1) 퐴1 − 훾 푘0 퐼 = 퐼 − 푃푘0, 2( 2) where 푃푘0 is the orthogonal projection of 𝒜휆 𝔹 onto 퐻푘0. From this relation we also conclude that 훾(푘0) defines an element in 𝒜(퐴1, 퐴2). The operator ( ( ) ) ( ) 퐷훾̅(−1) 퐴1 − 훾 푘0 퐼 + 퐴2 − 휁퐼 푃푘0 belongs to 𝒜(퐴1, 퐴2) and is invertible. Its inverse belongs to 𝒜(퐴1, 퐴2) as well and has the form 푘 ( ( ) ) −1 −1 −2 2 −푘0 0 퐷훾̅(−1) 퐴1 − 훾 푘0 퐼 − 휁 (휉 퐴2 + 휁 퐴2 + ⋯ + 휁 퐴2 )푃푘0 which implies that the ideal generated by the operators 퐴1 − 훾(푘0)퐼 = 퐷훾 − 훾(푘0)퐼 and 퐴2 − 휁퐼 = 푇휙 − 휁퐼 coincides with the whole algebra 𝒜(퐴1, 퐴2). To finish the description of the joint spectrum 휎(퐴1, 퐴2) we need first some preliminary 2 2 facts on the Berezin transform corresponding to certain subspaces of 𝒜휆 (𝔹 ). Let 푆 ⊂ ℤ+ be infinite. We introduce the Hilbert space 2 2 퐻푆 ≔ 푠푝푎푛̅̅̅̅̅̅̅{푒훼: |훼| ∈ 푆} ⊂ 𝒜휆 (𝔹 ). Its reproducing kernel function 퐾푆(푧, 휔) has the form of a power series converging uniformly on compacts of 𝔹2 × 𝔹2: ̅̅̅̅̅̅̅̅ 퐾푆(푧, 휔) = ∑ 푒훼(푧) 푒훼(휔). |훼|∈푆 2 Lemma (6.2.15)[303]:Let {휔푛}푛 ⊂ 𝔹 be such that the sequence (‖휔푛‖)푛of real numbers is 2 increasing and 푙푖푚 휔푛 = 푣 푘 ∈ 휕𝔹 . Then 푛→∞ 2 2 ‖퐾푆(. , 휔푛)‖ = ∑ |푒훼(휔푛)| → ∞, as 푛 → ∞. |훼|∈푆 Moreover, for all 푧, 휔 ∈ 𝔹2 the following estimate holds: 1 |퐾 (푧, 휔)| ≤ . (63) 푆 (1 − |〈푧, 푤〉|)3+휆 Proof: According to the multinomial theorem we have for 푗 ∈ ℤ+: 훤(푗 + 휆 + 3) 푗! 훤(푗 + 휆 + 3) ∑ |푒 (휔)|2 = ∑ |(휔 )|2훼1|(휔 )|2훼2 = ‖휔‖2푗. 훼 훤(휆 + 3)푗! 훼! 1 2 훤(휆 + 3)푗! |훼|∈푆 |훼|∈푆 Since휆 > −1 it follows with |훼| = 푗 → ∞ that 훤(푗 + 휆 + 3) 훤(푗 + 2) > = 푗 + 1 → ∞. 푗! 푗! As {휔푛}푛 is increasing and 푆 is infinite, the first assertion follows from 1 훤(푗 + 휆 + 3) ‖퐾 (. , 휔 )‖2 = ∑ ‖휔 ‖2푗 푆 푛 훤(휆 + 3) 푗! 푛 푗∈푆

176 and the monotone convergence theorem. The inequality(63) is a consequence of: 1 훤(푗 + 휆 + 3) 1 훤(푗 + 휆 + 3) 퐾 (푧, 휔) ≥ ∑ |〈푧, 휔〉|푗 ≥ ∑ |〈푧, 휔〉|푗 푆 훤(휆 + 3) 푗! 훤(휆 + 3) 푗! 푗∈푆 푗∈ℤ+ 1 = (1 − |〈푧, 휔〉|)3+휆 and the lemma is proved. Let 𝔹̅̅̅2̅ be the closed unit ball in ℂ2. Given a function 휓 ∈ 퐶𝔹̅̅̅2̅, we define its Berezin transform with respect to 퐻푆 as 1 2 퐵푆[휓](푧) ≔ 2 〈휓퐾푆(. , 푧), 퐾푆(. , 푧)〉, 푧 ∈ 𝔹 . ‖퐾푆(. , 휔푛)‖ Now, we can prove: Proposition (6.2.16)[303]:Let 휓 ∈ 퐶𝔹̅̅̅2̅ be invariant under the componentwise 핊1-action on 𝔹̅̅̅2̅, i.e. for all (휆, 푧) ∈ 핊1 × 𝔹̅̅̅2̅ we have: 휓(휆푧) = 휓(푧) (64) 2 푛−1 Let 푣 ∈ 휕𝔹 and put 휔푛 = 푣 such that lim 휔푛 = 푣. Then, we have: 푛 푛→∞ 푙푖푚 퐵푆[휓](휔푛) = 휓(푣). 푛→∞ 1 ̅̅̅2̅ Proof: Let 휀 > 0 and consider the orbit 푂푣 ≔ {휆푣: 휆 ∈ 핊 } ⊂ 휕𝔹 .Let훿 > 0 and define a 훿- neighborhood of 푂푣 by 훿 ̅̅̅2̅ 푂푣 ≔ {푧 ∈ 𝔹 ∶ 푑푖푠(푧, 푂푣) < 훿}. 훿 1 Let 푧 ∈ 푂푣 , then there is 휆0 ∈ 핊 such that |푧 − 휆0푣| < 훿.Since 휓 is continuous up to the boundary of 𝔹̅̅̅2̅ and due to the invariance (64) it follows that one can choose 훿 > 0 sufficiently small with 휀 > |휓(푧) − 휓(휆0푣)| = |휓(푧) − 휓(푣)|. (65) ̅̅̅2̅ 훿 Fix 훿 > 0 with (65). Then there is 훾 ∈ (0,1) such that for all 푧 ∈ 𝔹 \푂푣 and 푛 ∈ ℕ one has |〈푧, 휔푛〉| ≤ 훾 < 1. Hence it follows that: 1 훤(푗 + 휆 + 3) 1 훤(푗 + 휆 + 3) |퐾 (푧, 휔 )|2 ≤ ∑ |〈푧, 휔〉|푗 ≤ ∑ |〈푧, 휔〉|푗훾푗 푆 푛 훤(휆 + 3) 푗! 훤(휆 + 3) 푗! 푗∈푆 푗∈ℤ+ = 푐훿. Now we calculate |휓(푣) − 퐵푆[휓](휔푛)| 1 ≤ 2 ∫|휓(푣) ‖퐾푆(. , 휔푛)‖ 훿 푂푣 2 1 − 휓(푧)| |퐾푆(푧, 휔푛)| 푑푣휆(푧). + 2 ∫ |휓(푣) ‖퐾푆(. , 휔푛)‖ 2 훿 𝔹 \푂푣 | | ∞ 2 | 휓 |ℒ 𝔹̅̅̅2̅ − 휓(푧)| |퐾푆(푧, 휔푛)| 푑푣휆(푧) ≤ 휀 + 2푐훿 2. ‖퐾푆(. , 휔푛)‖ 177

The last term on the right tends to zero as 푛 → ∞ (by Lemma (6.2.15)), and hence the proposition is proved. 1 Lemma (6.2.17)[303]:We have 퐿푖푚(훾) × 퐷̅ (0, ) ⊂ 휎휋(퐴 , 퐴 ). 2 1 2 1 1 Proof: Let (휇, 휁) ∈ 퐿푖푚 (훾) × 퐷̅ (0, ). Recall that 퐷̅ (0, ) = 퐼푚휙| 2 . Then we choose 2 2 휕𝔹 2 푣 ∈ 휕𝔹 such that 휉 = 휙(푣)and fix a sequence{푚푒}푒 ⊂ ℤ+such that 푙푖푚 훾(푚푒) = 휇. 푒→∞ { } Consider the set 푆휇 ≔ 푚ℓ: ℓ ∈ 푁 ⊂ ℤ+ and define the Hilbert space 퐻푆휇. { | | } 퐻푆휇푠푝푎푛̅̅̅̅̅̅̅ 푒훼: 훼 = 푚ℓ, ℓ ∈ ℕ . 2 Let {휔푛}푛 ⊂ 𝔹 be a sequence as in Lemma (6.2.15)with 푙푖푚 휔푛 = 푣 . We define a 푛→∞ corresponding sequence {푓푛}푛 of unit vectors by 퐾푆(. , 휔푛) 푓푛 = (66) ‖퐾푆(. , 휔푛)‖ Then, we have 2 1 2 2 ‖(퐷훾 − 휇퐼)푓푛‖ = 2 ∑ ∑ |훾(푚ℓ − 휇)| |푒훼(휔푛)| . ‖퐾푆(. , 휔푛)‖ ℓ∈ℕ |훼|=푚ℓ 2 Given 휀 > 0 we choose ℓ0 ∈ ℕ such that |γ(mℓ − μ)| < 휀 for ℓ ≥ ℓ0. Then, ℓ0 2 1 2 2 ‖(퐷훾 − 휇퐼)푓푛‖ ≤ 2 ∑ ∑ |훾(푚ℓ − 휇)| |푒훼(휔푛)| + 휀. ‖퐾푆(. , 휔푛)‖ ℓ=1 |훼|=푚ℓ by Lemma (6.2.15) the first term on the right hand side tends to zero as 푛 → ∞, and therefore 퐿푖푚‖(퐷훾 − 휇퐼)푓푛‖ = 0 (67) 푒→∞ Using 0 ≤ 푇휙̅ 푇휙 ≤ 푇|휙|2 together with Proposition (6.2.16) we have 2 〈 〉 ̅ [ ]( ) [̅̅̅̅]( ) | |2 ‖(푇휙 − 휉퐼)푓푛‖ = 푇휙̅ 푇휙푓푛, 푓푛 − 휉퐵푆휇 휙 휔푛 − 휉퐵푆휇 휙 휔푛 + 휉 [| |2]( ) ≤ 퐵푆휇 휙 − 휉 휔푛 (68) → |휙 − 휉|2(푣) = 0, as 푛 → ∞ Finally, (67), (68) and Proposition (6.2.12)imply that (휇, 휉) ∈ 휎휋(퐷훾, 푇휙). Theorem (6.2.18)[303]: We have 1 푀(𝒜(퐴 , 퐴 )) = 휎(퐷 ) × {0}⋃퐿푖푚(훾) × 퐷̅ (0, ). 1 2 훾 2 1 = {훾(푘): 푘 ∈ ℤ } × {0}⋃퐿푖푚(훾) × 퐷̅ (0, ) + 2 To uniform the result and make it independent of a concrete choice of the diagonal operator 퐴1 = 퐷훾 we proceed as follows. Given the operator 퐷훾, the compact set of maximal ideals of the C∗-algebra 𝒜(A1) was identified with 휎(퐷훾) = {훾(푘): 푘 ∈ ℤ+} × ⋃ 퐿푖푚(훾).

178

The last set is homeomorphic to a certain compactification of ℤ+, and this homeomorphism is given by 훾(푘) ∈ 훾 ⟼ 푘, ∗ 훾 = 푙푖푚 훾(푘푛) ∈ 퐿푖푚(훾) ⟼ class of equivalence containing {푘푛}푛∈푘. 푒→∞ We say that two subsequences {푘푛} and {푘푚} are equivalent if and only if the following limits exist and coincide: 푙푖푚 훾(푘푛) = 푙푖푚 훾(푘푚) 푒→∞ 푒→∞ We denote by 푀∞(훾) the part of the maximal ideals of 𝒜(퐴1) which is homeomorphic to퐿푖푚(훾). Now Theorem (6.2.18) reads as follows. Theorem (6.2.19)[303]:We have 1 푀(𝒜(퐴 , 퐴 )) = ℤ × {0} ⋃ 푀 (훾) × 퐷̅ (0, ). 1 2 + ∞ 2 The difference among different choices of the generating operator 퐷훾 is reflected now in the different corresponding compactifications of ℤ+, i.e. in different sets 푀∞(훾). We describe first multiplicative functionals of the 퐶∗-algebra generated by a finite number of 2( 2) diagonal operators: Let 퐷훾1 , … , 퐷훾푛 be bounded diagonal operators on 퐻 = 𝒜휆 𝔹 acting on elements of the basis (55) as (| |) 2 퐷훾푗 푒훼 = 훾푗 훼 푒훼, for 훼 ∈ ℤ+ ∗ and whose eigenvalue sequence 훾푗 belongs to 푆푂(휆). Consider then the unital C −algebra ∗ ( ) 𝒜퐷 ≔ 𝒜(퐷훾1, … , 퐷훾푛) ⊂ ℒ 퐻 (69) which is generated by elements of 푫 = (퐷훾1, … , 퐷훾푛).

Recall that the joint spectrum 휎(퐷훾1, … , 퐷훾푛) of the operators 퐷훾1 , … , 퐷훾푛, which is identified ∗ ∗ with the maximal ideal space 푀(𝒜퐷) of 𝒜D, has the form: ( ) 푛 ∗ 휎(퐷훾1, … , 퐷훾푛 ) = { 휇1, … , 휇푛 ∈ ℂ : 퐽(퐷훾1 − 휇1퐼, … , 퐷훾푛 − 휇푛퐼) ≠ 𝒜퐷} ∗ where퐽(퐷훾1 − 휇1퐼, … , 퐷훾푛 − 휇푛퐼)denotes the smallest ideal in the algebra 𝒜D containing the ( ) elements 퐷훾푗 − 휇푗퐼 , for 푗 = 1, … , 푛. 휇1, … , 휇푛 ∈ 휎(퐷훾1, … , 퐷훾푛). Then ∗ ∗ 퐷 = (퐷훾1 − 휇̅̅1̅퐼)(퐷훾11 − 휇1퐼) + ⋯ + (퐷훾푛 − 휇̅̅푛̅퐼)(퐷훾푛 − 휇푛퐼) ∗ ∗ ∗ is an element of this ideal, and hence 퐷 is not invertible in 𝒜D. Since 𝒜D (as a C -algebra) is inverse closed, the operator 퐷 is not invertible in ℒ(H) either. Note that 퐷 is diagonal with the eigenvalues 2 푛 훾(|훼|) = |훾1(|훼|) − 휇1| + ⋯ + |훾푛(|훼|) − 휇푛| ≥ 0. Corollary (6.2.20)[303]: Either there is 푘 ∈ ℤ+ such that 훾푗(푘) = 휇푗 for all 푗 = 1, … , 푛, or there is a sequence {푚ℓ} ⊂ ℤ+ such that for all푗 = 1, … , 푛: 푙푖푚 훾푗(푚ℓ) = 휇푗. (70) 푒→∞ ( ) Let 휇1, … , 휇푛 ∈ σ(퐷훾1, … , 퐷훾푛). According to Corollary (6.2.20), we assume first that there is 푘 ∈ ℤ+ such that for all푗 = 1, … , 푛: 훾푗(푘) = 휇푗 ∗ Then we define a multiplicative functional 휑(푘)on𝒜퐷 by: 휑(푘)(퐷) ≔ 〈퐷푒(푘,0), 푒(푘,0)〉 (71) 179

∗ for all 퐷 ∈ 𝒜퐷. Note that 휓(푘)(퐷훾푗) = 휇푗 for all j = 1, … , n. If such k ∈ ℤ+ does not exists, then, by the second option of Corollary (6.2.21), there is a ∗ sequence {푚ℓ: ℓ ∈ 푁} ⊂ ℤ+ having the property (70). Define the functional on 𝒜퐷 by: ( ) 〈 〉 휓{푚ℓ} 퐷 : = 퐷푒(푘,0), 푒(푘,0) (72) ∗ Lemma (6.2.21)[303]: The limit (72) exists for all퐷 ∈ 𝒜퐷, and the functional 휓{푚ℓ}is multiplicative with 휓{푚ℓ}(퐷훾푗) = 휇푗, for all 푗 = 1, … , 푛 . Proof: Similar to the proof of Lemma (6.2.22) below. Now we modify the definition (72) so that the right hand side extends to a larger algebra (see Lemma (6.2.23) below). Consider the infinite set 푆 = {푚ℓ: ℓ ∈ ℕ} ⊂ ℤ+ and define the Hilbert space 퐻푆 = 푠푝푎푛̅̅̅̅̅̅̅{푒훼: 훼 ∈ ℤ+, |훼| ∈ 푆} 2 Let 퐾푆 be the reproducing kernel of 퐻푆, i.e. for all 푧, 휔 ∈ 𝔹 we have ̅̅̅̅̅̅̅̅ 퐾푆 = (푧, 휔) ≔ ∑ 푒훼(푧)푒훼(휔). (73) |훼|∈푆 1 Let 휉 ∈ 퐷̅ (0, ) and 푣 ∈ 휕𝔹2 such that 휙(푣) = 휉. Let {휔 } ⊂ 𝔹2 be a sequence with 2 푘 푘 2 휔푘 ⟶ 푣 ∈ 휕𝔹 as 푘 ⟶ ∞, and assume that {ωk}k is increasing. Define a sequence {푓푘}푘 of unit vectors in 퐻 by 퐾푆(. , 휔푘) 푓푘 = ∈ 퐻. (74) ‖퐾푆(. , 휔푘)‖

Lemma (6.2.22)[303]:. The multiplicative functional 휓{푚ℓ} in (72) can be also defined as ∗ 휓{푚 }(퐷훾) = 퐿푖푚〈퐷훾푓푘, 푓푘〉 , where 퐷훾 ∈ 𝒜퐷 (75) ℓ 푒→∞

Proof: Since the functional 휓{푚ℓ} is continuous and due to Lemma (6.2.21) it is sufficient to 푛 show that for all (푖1, … , 푖푛) ∈ ℤ+: 푖1 푖2 푖푛 푖1 푖2 푖푛 퐿푖푚〈퐷γ 퐷γ … 퐷γ 푓푘, 푓푘〉 = 휇γ 휇γ … 휇γ . 푒→∞ 1 2 푛 1 2 푛 A simple argument using the Cauchy-Schwarz and triangle inequality together with‖푓푘‖ = 1 shows that it is sufficient to prove for푗 = 1, . . , 푛that 푖푗 푖푗 퐿푖푚 ‖(퐷 − 휇 퐼) 푓푘‖ = 0. (76) 푘→∞ 훾푗 훾푗 But this has been already shown for the above choice of {푓푘}푘 in the proof of Lemma (6.2.17) by using the convergence (70). ∗ Further, for each operator 퐷훾 ∈ 𝒜퐷, the limit along the subsequence {푚ℓ} of its eigenvalue sequence γ exists and is equal to the value of the functional 휓{푚ℓ} on the diagonal operator 퐷훾: 푙푖푚 훾(푚 ) = 휓{푚 }(퐷훾). 푒→∞ ℓ ℓ 1 Let 휉 ∈ 퐷̅ (0, ) be as above, with corresponding sequences 푤 ⟶ 푣 ∈ 휕𝔹2 such that 휙(푣) = 2 푘 휉and {푓푘} is of the form (74).

Lemma (6.2.23)[303]:The functional 휓{푚ℓ} extends to the functional 휓 = (휓{푚ℓ}, 휉) on the ∗ algebra generated by elements of 𝒜퐷 and 푇휙 via 푗 푗 휓 (퐷푇 ) = 푙푖푚 〈퐷푇 푓푘, 푓푘〉 휙 푘→∞ 휙 180

∗ 푚 푗 with 푗 ∈ ℤ+ and 퐷 ∈ 𝒜퐷. Moreover, for elements of the form ∑푗=0 퐷푗푇휙we have 푚 푚 푗 푗 휓 (∑ 퐷푗푇휙) = ∑ 휓{푚ℓ}, (퐷푗)휉 . 푗=0 푗=0 Proof: Similar to the proof of Lemma (6.2.22) and using the convergence 푙푖푚‖(푇휙 − 푒→∞ 휉퐼)푓푘‖ = 0 (see (68)). Let 𝒜 be a unital commutative Banach algebra which is generated by two of its unital subalgebras 𝒜1 and 𝒜2 sharing the same identity, and let 푀(𝒜), 푀(𝒜1), and 푀(𝒜2) be their respective sets of maximal ideals (≡multiplicative functionals). Recall that, since the restrictions of a multiplicative functional 휓 ∈ 푀(𝒜) onto subalgebras 𝒜1 and 𝒜2 are multiplicative functional 휓1 ∈ 푀(𝒜1) and 휓2 ∈ 푀(𝒜2), correspondingly, we have a natural continuous mapping 휅: 휓 ∈ 푀(𝒜) ⟼ (휓1, 휓2) ∈ 푀(𝒜1) × 푀(𝒜2) As 𝒜 is generated by 𝒜1 and 𝒜2, the mapping 휅 is obviously injective, and thus its range can be identified with 푀(𝒜). The unital Banach algebra 풯(휆), we are interested in, is generated by two algebras sharing the ∗ same identity: the C -algebra 풯푟푎푑(휆), which is generated by all Toeplitz operators Ta with ∞ radial symbols a ∈ L [0,1), and the Banach algebra 풯ϕ, which is generated by a single Toeplitz operator Tϕ, where 휙(휉) = 휉(1,0)휉(0,1). Thus, by (50) and the last paragraph of the section, 1 the mapping κ identifies 푀(풯(휆)) with a subset of (ℤ × ⋃푀 (휆)) × 퐷̅ (0, ). + ∞ 2 1 Lemma (6.2.24)[303]:None of the points of the set ℤ × (퐷̅ (0, ) \{0}) belongs to 푀(풯(휆)). + 2 1 Proof: Let us assume that a point (푘, 휉) ∈ ℤ × (퐷̅ (0, ) \{0}) belongs to 푀(풯(휆)). Then, + 2 for the operator 퐴 = 푃푘푇휙 ∈ 풯(휆), where 푃푘is the orthogonal projection onto Hk (see (58)), we have휓(퐴) = 1, 휉 = 0. At the same time, by Lemma (6.2.7), the operator 퐴 belongs to the radical of the algebra 풯(휆), and thus 휓(퐴) = 0. Contradiction. Lemma (6.2.25)[303]:The set ℤ+ × {0} belongs to 푀(풯(휆)). Proof: Let 휓 = (푘, 0) ∈ ℤ+ × {0}. Denote by 휓(푘) the multiplicative functional on 풯푟푎푑(휆) (see (71)) given by: 휓(푘)(퐷훾) ≔ 〈퐷훾푒푘,표, 푒푘,표〉 = 훾(푘), where 퐷훾 ∈ 풯푟푎푑(휆). Then the functional 휓 = (푘, 0) = (휓(푘), 0) is defined on a dense subalgebra 퐷훾 of 풯(휆) as 푚 푝 follows: for any 퐴 = ∑푗=0 퐷푝푇휙 ∈ 퐷(훾) where 퐷푝 ∈ 풯푟푎푑(휆) we put: ( ) 〈 〉 ( ) ( ) ψ A ≔ Ae(k,0), e(k,0) = ψ(k) D0 = γD0 k . (77) m p Note that the functional ψ is well-defined, since ∑j=0 DpTϕ = 0 implies D0 = 0, according to Lemma (6.2.8). Moreover, we have 푚 푚 푚 푝 푝 푝 |휓 (∑ 퐷푝푇휙 )| = |〈∑ 퐷푝푇휙 푒(푘,0), 푒(푘,0)〉| ≤ ‖∑ 퐷푝푇휙 ‖ 푝=0 푝=0 푝=0 Hence 휓 is continuous and extents to a multiplicative functional on 풯(휆). Recall that 푀∞(휆) denotes the multiplicative functionals on 풯푟푎푑(휆) that map compact operators to zero. 181

1 Lemma (6.2.26)[303]:The set 푀 (휆) × 퐷̅ (0, ) belongs to M(풯(휆)). ∞ 2 1 Proof: We define the functional 휓 = (휇, 휉) ∈ 푀 (휆) × 퐷̅ (0, ) on a dense subalgebra 풟 of ∞ 2 훾 푚 푝 풯(휆) as follows: for any 퐴 = ∑푝=0 퐷푝푇휙 ∈ 풟(휆) , where 퐷푝(휆) ∈ 풯푟푎푑(휆), we put: 푚 푚 푚 푝 푝 푝 휓 (∑ 퐷푝푇휙 ) : = ∑ 휇(퐷푝)휉 = ∑ 훾퐷푝(휇)휉 푝=0 푝=0 푝=0 First we need to show that 휓 is well-defined. Indeed, according to Lemma (6.2.8), the equality 푚 푝 퐴 = ∑푝=0 퐷푝푇휙 = 0 implies that Dp is compact for all 푝 = 0, … , 푚. But 휇 ∈ 푀∞(휆), and thus 휇(퐷푝) = 0 for all 푝 = 0, … , 푚,which implies that 휓(퐴) = 0. The functional 휓is obviously multiplicative on 퐷(훾), and thus it remains to show that it is continuous and therefore extents to a multiplicative functional on 풯(휆). 푚 푝 ∗ ∗ Fix now any 퐴 = ∑푝=0 퐷푝푇휙 ∈ 퐷(휆) and consider the unital C -algebra 𝒜Dgenerated by ̂ ∗ ∗ 퐷0, … , 퐷푚. Clearly, the restriction 휓 of 휓 to 𝒜퐷 defines a multiplicative functional on 𝒜D. Note that ( 휇0, … , 휇푚) = (휓(퐷0), … , 휓(퐷푚)) ∈ 휎(퐷0, … , 퐷푚). ̂ ∗ Since ψ maps compact operators in 𝒜Dto zero and because of Lemma (6.2.23), we have 휓̂has the form (75): ̂ ∗ 휓(퐷훾) = 푙푖푚〈퐷훾푓푘, 푓푘〉 , 퐷훾 ∈ 𝒜퐷 푘→∞ where {푚ℓ: ℓ ∈ 푁} ⊂ ℤ+ is a suitable sequence which is induced by ( 휇0, … , 휇푚) as was explained, and 푓푘, with 푘 ∈ ℕ, are given by (74) with 휉 = 휙(푣). Now from Lemma (6.2.23) it follows that 푚 푚 푚 푝 푝 푝 |휓 (∑ 퐷푝푇휙 )| = |휓 〈∑ 퐷푝푇휙 푓푘, 푓푘〉| ≤ ‖∑ 퐷푝푇휙 ‖, 푝=0 푝=0 푝=0 and thus 휙 is continuous on 퐷훾 and extends to a multiplicative functional on 퐷(휆). Theorem (6.2.27)[303]:The compact set 푀(풯(휆)) of maximal ideals of the algebra 풯(휆) has the form 1 푀(풯(휆)) = ℤ × {0} ∪ 푀 (휆) × 퐷̅ (0, ), + ∞ 2 (i) The Gelfand image of the algebra 풯(λ) is isomorphic to 풯(휆)/푅푎푑 풯(휆) and coincides with the algebra 1 푆푂(휆) ∪ [퐶(푀 (휆)) ⊗̂ 퐶 (퐷̅ (0, ))], ∞ 휀 푎 2 which is identified with the set of all pairs 1 (훾, 푓) ∈ 푆푂(휆) × [퐶(푀 (휆)) ⊗̂ 퐶 (퐷̅ (0, ))] ∞ 휀 푎 2 satisfying the following compatibility condition 훾(휇) = 푓(휇, 0), for all 휇 ∈ 푀∞(휆). ̂ Here ⊗휀 denotes the injective tensor product, and we identify γ(μ) with the value of the functional 휇 ∈ 푀∞(휆) on the element 훾 ∈ 푆푂(휆). (ii) The Gelfand transform is generated by the following mapping of elements of 푆푂(휆): 182

푛 훾0(푘) 푖푓 (푘, 0) ∈ ℤ+ × {0} 푝 푛 1 ∑ 퐷훾푗 푇휙 ↦ { 푗 . ∑ 훾푗(휇)휉 , 푖푓 (휇, 휉) ∈ 푀∞(휆) × 퐷̅ (0, ) 푗=0 푗=0 2 Proof: Follows directly from Lemmas (6.2.24), (6.2.25), and(6.2.26), Theorem (6.2.6) and the injective tensor product description (see [256]). Our next aim is to show that the algebra 풯(λ) is inverse closed, that is: each operator 퐴 ∈ 풯(휆) 2 2 −1 which is invertible in ℒ(𝒜λ(𝔹 )) is invertible in 풯(휆), i.e., 퐴 ∈ 풯(휆). The proof of this fact essentially relies on Theorem (6.2.27).

Lemma (6.2.28)[303]:Let 휑 = (푘, 0) ∈ ℤ+ × {0} ⊂ 푀(풯(휆)), and assume that 퐴 ∈ 풯(휆) is 2 2 invertible in ℒ(𝒜λ(𝔹 )). Then 휓(퐴) = 0. Proof: Recall that the functional휓 = (푘, 0) on 풯(휆) is defined on the dense subalgebra D(λ) by (77). Clearly, it extents by the same formula (77) to a continuous (not necessarily 2 2 multiplicative) functional on 𝒜λ(𝔹 ). 푚 푗 2 2 Let ∑푝=0 퐷훾푗푇휙 ∈ 퐷(휆), and let 퐵 ∈ ℒ(𝒜휆(𝔹 ))be arbitrary.Then: 휓(퐵퐴) = 〈퐵퐴푒(푘,0), 푒(푘,0)〉 = 〈퐵퐷훾0푒(푘,0), 푒(푘,0)〉 = 훾0(푘)〈퐵푒(푘,0), 푒(푘,0)〉 = 휓(퐴)휓(퐵) = 휓(퐵)휓(퐴) By continuity it follows that 휓(퐵퐴) = 휓(퐵)휓(퐴) for all 퐴 ∈ 풯(휆). In particular, if 퐴 is 2 2 invertible in 𝒜λ(𝔹 ) , then 1 = 휓(퐼) = 휓(퐴−1퐴) = 휓(퐴−1)휓(퐴), and we conclude that 휓(퐴) ≠ 0. ( ) ∗ ∗ Given 퐷 = (퐷훾1, … , 퐷훾푛 ) ⊂ 풯푟푎푑 휆 , we consider the unital C -algebra 𝒜D generated by ̃ ̃∗ ∗ 퐷훾1 , … , 퐷훾푛 (see (69)). Let 𝒜퐷 and 𝒜퐷 be the Banach algebra and the C -algebra which are ∗ ∗ generated by elements of 𝒜퐷⋃푇휙 and of 𝒜퐷⋃푇휙, respectively. Clearly we have 𝒜̃퐷 ⊂ 𝒜̃퐷. 1 Consider now the functional 휓 = (휇, 휉) ∈ 푀 (휆) × 퐷̅ (0, ) ⊂ 푀(풯(휆)). Its restriction to the ∞ 2 ∗ algebra 𝒜퐷 maps compact operators to zero. Thus, according to Lemmas(6.2.23) and(6.2.24), we can construct the sequence of unit vectors {푓푘}푘 by (54) with 휉 = 휙(푣) such that 푗 푗 ∗ 휓 (퐷훾푇 ) = 푙푖푚 〈퐷훾푇 푓푘, 푓푘〉 , 퐷훾 ∈ 𝒜퐷 (78) 휙 푘→∞ 휙 Moreover, by continuous extension the right hand side of (78) defines a multiplicative 1 continuous functional on 𝒜̃ (which coincides with the restriction of 휓 ∈ 푀 (휆) × 퐷̅ (0, )to 퐷 ∞ 2 the algebra 𝒜̃퐷. We show now that the restriction of 휓to 𝒜̃퐷 extends further to a multiplicative and continuous ∗ ∗ functional on the (non-commutative) C -algebra 𝒜̃퐷. The nature of such an extension is very 2 2 simple. Let 풦 be the ideal of all compact operators on 𝒜휆 (𝔹 ). Two quotient algebras ∗ ∗ ∗ 𝒜̂퐷 = 𝒜̃퐷(𝒜̃퐷 ∩ 풦) 푎푛푑 𝒜̂퐷 = 𝒜̃퐷/(𝒜̃퐷 ∩ 풦) 2 푛 will be involved. For an algebra 𝒜 ⊂ ℒ(𝒜휆(𝔹 )), we denote by 푝푟 the natural projection 푝푟: 𝒜 → 𝒜̂ = 𝒜̃/ (𝒜̃ ∩ 풦) ∗ As [푇휙, 푇휙̅ ] ∈ 풦 and both operators 푇휙and 푇휙̅ commute with diagonal operators 퐷훾, the C - ∗ ∗ algebra𝒜̃퐷 is commutative, and furthermore 𝒜̂퐷 ⊂ 𝒜̃퐷. 183

The functional 휓, restricted to 𝒜̂퐷, maps compact operators from 𝒜̂D to zero, and thus it admits the natural decomposition. 푝푟 휓̂ 휓: 𝒜̃퐷 → 𝒜̂퐷 → ℂ for a suitable multiplicative functional ψ̂ on 𝒜̂D. We extend now the functional 휓̂ from 𝒜̂퐷 to the functional (one-dimensional representation) ̂∗ ̂∗ ∗ ̂∗ ̂∗ 휓 on 𝒜퐷 on the C -algebra 𝒜퐷 defining it on the extra generator [푇휙̅ ] = 푇휙̅ + 𝒜퐷 ∩ 풦 as it should be: ̂∗ ̅̅̅̅̅̅̅̅̅̅̅ 휓 ([푇휙̅ ]) = 휓([푇휙̅ ]) ∗ ∗ The extension 휓 of the functional 휓 from 𝒜̂D onto 𝒜̂D is thus given by 푝푟 휓̂∗ ∗ 휓 : 𝒜̃퐷 → 𝒜̂퐷 → ℂ As the next lemma shows the functional 휓∗has the same form as in (78): ∗ 푗1 푗2 푗1 푗2 휓 (퐷훾푇 푇 ) = 푙푖푚 〈퐷훾푇 푇 푓푘, 푓푘〉 (79) 휙 휙 푘→∞ 휙 휙 ∗ Lemma (6.2.29)[303]:The limit on the right hand side of (79) exists for all Dγ ∈ 𝒜D and j1, j2 ∈ ℤ+. Moreover, it has the value: 푗2 푗1 푗2 푗1̅̅̅̅̅̅̅̅̅̅̅ 푙푖푚 〈퐷훾푇 푇 푓푘, 푓푘〉 = 휓(퐷훾)휓(푇휙) 휓([푇휙̅ ]) 푘→∞ 휙 휙 ∗ ∗ In particular, by linearity and continuity it induces a multiplicative functional 휓 on 𝒜̃D which extends 휓. Proof: Similar to the proof of Lemma (6.2.22) or Lemma (6.2.23) and by using the convergence ̅ 푙푖푚‖(푇휙̅ − 휉퐼)푓푘‖ = 0, where 휉 = 휓(푇휙). 푘→∞ 1 Corollary (6.2.30)[303]:let휓 = (휇, 휉) ∈ 푀 (휆) × 퐷̅ (0, ) ⊂ 푀(풯(휆)), and let 𝒜 ∈ 𝒜̃ be ∞ 2 퐷 2 푛 invertible as an element in ℒ (𝒜휆 (𝔹 )). Then 휓(𝒜) ≠ 0. ∗ 2 n ̃ Proof. As each C -subalgebra of ℒ(𝒜λ(𝔹 )) , the algebra 𝒜D is inverse closed and thus we −1 ∗ have 𝒜 + 𝒜̃퐷. According to the previous lemma the functional ψextends to a multiplicative ∗ ∗ functional 휓 on 𝒜̃D , and therefore 1 = 휓∗(퐴퐴−1) = 휓(퐴)휓(퐴−1), which shows that 휓(퐴) ≠ 0. 2 푛 Now, we deal with the general case: Let 퐴 ∈ 풯(휆) be invertible as an element in ℒ(𝒜휆 (𝔹 )), ∗ then we wish to show that 퐴 ∈ 풯(휆). Choose a sequence {퐴푘}푘 ⊂ 퐷(휆) such that 푙푖푚 퐴푘 = 퐴 푘→∞ Since the group of invertible elements is open, we can assume that 퐴푘 is invertible for all 푘 ∈ ℕ. Moreover, by the continuity of inversion we have −1 −1 퐴 = 푙푖푚 퐴푘 푘→∞ −1 ( ) and hence it is sufficient to show that 퐴푘 ∈ 풯 휆 for each 푘. Fix 퐷 = (퐷훾1, … , 퐷훾푛) ⊂ 풯푟푎푑(휆), such that (with our notation above) 퐴푘 ∈ 𝒜̃퐷

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From Theorem (6.2.27), Lemma (6.2.28) and Corollary (6.2.30), together with the fact that 퐴푘 is invertible, we have 휓(퐴푘) ≠ 0 −1 For all multiplicative functionals on 𝒜̃퐷 , and thus 퐴푘 ∈ 𝒜̃퐷 ⊂ 풯(휆). Hence we have shown: Theorem (6.2.31)[303]:The commutative Banach algebra 풯(λ) is inverse closed, and, in particular, for each 퐴 ∈ 풯(휆) , 푠푝 퐴 = 푠푝 2 푛 퐴 풯(휆) ℒ(𝒜휆(𝔹 )), The next assertions give, in particular, some information on the spectra of elements of the algebra 풯(휆). 푝 Lemma (6.2.32)[303]: The difference 푇휙푝 − 푇휙 , where p ∈ ℕ, belongs to the radical of the algebra 풯(λ). Proof: By (53) we have 푇 − 푇푝 = 퐷 푇푝, and the assertion follows from Lemma (6.2.7) 휙푝 휙 푑̃푝−1 휙 and the convergence 푙푖푚 푑푝 (푘) = 1 (see (54)). 푘→∞ Corollary (6.2.33)[303]:The operators 푚 푚푘 푛푘 푚 푚푘 푝 +⋯+푝 푘,1 푘,푛푘 ∑ (∏ 푇푎 ∏ 푇휙 ) and ∑ (∏ 푇푎 푇 ) 푘,푝 푝푘,푠 푘,푝 휙 푘=1 푞=1 푠=1 푘=1 푞=1 differ by an element in the radical and thus have the same Gelfand images and the same spectra. Theorem (6.2.35)[303]: With our previous notation we have: (i) The Calkin algebra 풯̂(휆) = 풯(휆) / (풯(휆) ∩ 풦) is semi-simple and isomorphic to the 1 injective tensor product (푀 (휆)) ×⊗̂ 퐶 퐷̅ (0, ). ∞ 휀 푎 2 (ii) The Calkin algebra 풯̂∗(휆) = 풯∗(휆)/(풯∗(휆) ∩ 풦) of the C∗-extension 풯∗(휆) of the Banach 1 algebra 풯(λ) is isomorphic and isometric to 퐶(푀 (휆)) × 퐷̅ (0, ). ∞ 2 (iii) Both natural homomorphisms 1 풯(휆) → 풯̂(휆) ≅ 퐶(푀 (휆)) ×⊗̂ 퐶 (퐷̅ (0, )), ∞ 휀 푎 2 1 풯∗(휆) → 풯̂(휆) ≅ 퐶(푀 (휆)) × (퐷̅ (0, )). ∞ 2 are generated by the following mapping: 푛 푛 1 푗 ( ) 푗 ( ) ( ) ̅ ∑ 퐷훾푗 푇휙 ↦ ∑ 훾푗 휇 휉 , 휇, 휉 ∈ 푀∞ 휆 × 퐷 (0, ) 푗=0 2 푗=0 (iv) The essential spectrum of all elements of both algebras 풯(λ) and 풯∗(λ) is connected. Proof: Follows from Theorem (6.2.27) and by arguments similar to the ones used in the proof of Theorem (6.2.31) The last assertion is a consequence of the connectedness of both sets 1 푀 (휆) and 퐷̅ (0, ) . ∞ 2

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∑푛 푗 Corollary (6.2.35)[303]:For all elements of the form 퐼 + 푗=1 퐷훾푗 푇휙 , where 휂 ∈ ℂ, and, in particular, for finite sums of finite product of generators of the algebra 풯(휆). 푛 푚푘 푛푘

휂퐼 + ∑ (∏ 푇푎 ∏ 푇휙 ), 푘,푝 푝푘,푠 푘=1 푞=1 푠=1 where η ∈ ℂ and none of nk is equal to zero, their spectrum and essential spectrum coincide. In particular, for such operators the spectrum is connected, and being Fredholm this operator is invertible. ∑푛 푗 ( ) Corollary (6.2.36)[303]:An element 푗=1 퐷훾푗 푇휙 from the dense subalgebra D λ is compact if and only if each diagonal operator 퐷훾푗 is compact, or if and only if each eigenvalue sequence 훾푗 belongs to 푐0. Our last aim is to describe explicitely the radical of the algebra 풯(λ). Consider the multiplicative functional 휓(푘, 0) ∈ ℤ+ × {0} ⊂ 푀(풯(휆)). First we would like to analyze the structure of operators A ∈ 풯(λ) satisfying the following property 휓(푘,0)(퐴) = 0 for all 푘 ∈ ℤ+. (80) ∑푝 푗 ( ) ( ) For an element 퐴 = 푗=0 퐷훾푗푇휙 from the dense subalgebra 퐷 휆 of 풯 휆 , we have 푝 ( ) 푗 휓(푘,0) 퐴 = 휓(푘,0) (∑ 퐷훾푗 푇휙) = 훾0(푘) 푗=0 ∑p j That is, the operator A = j=0 D훾푗 Tϕ satisfies property (80) if and only if 훾0 ≡ 0, i.e. 퐴 has to be of the form 푝 푗 퐴 = ∑ 퐷훾푗 푇휙 = 푇휙퐶, (81) 푗=0 푝 푗−1 where 퐶 = ∑푗=0 퐷훾푗푇휙 ∈ 풯(휆). The description of 푅푎푑 풯(휆) ∩ 퐷(휆) is straightforward: Lemma (6.2.37)[303]:We have 푅푎푑 풯(휆) ∩ 퐷(휆) = {푇휙퐶: 퐶 ∈ 퐷(휆) ∩ 퐾} Proof: Observe first that 퐴 ∈ 푅푎푑 푇(휆) ∩ 퐷(휆) if and only if 퐴 satisfies property (80) and 퐴 is compact. That is 퐴 is of the form 퐴 = 푇휙퐶 in (81) with 퐶 ∈ 퐷(휆); and by Corollary (6.2.36) compactness of 퐴 is equivalent to compactness of 퐶. The description of 푅푎푑 풯(휆) ∩ (풯(휆)\퐷(휆)) is more elaborated. We start with an easy lemma. Lemma (6.2.38)[303]:Each operator 퐴 ∈ 풯(휆)\퐷(휆) satisfying the property (80) can be approximated in norm by the operators of the form (81). Proof: Given 퐴 ∈ 풯(휆)\퐷(휆), there is a sequence of operators 푝푛 ̃ ( ) 푗 ( ) 퐴푛 = ∑ 퐷훾푗 푛 푇휙 ∈ 퐷 휆 (82) 푗=0 186

1 such that ‖퐴 − 퐴̃ ‖ < . Then, for each k ∈ ℤ , we have 푛 푛 + 1 |훾(푛)(푘)| = |휓 (퐴̃ )| = |휓 (퐴̃ − 퐴)| ≤ ‖퐴 − 퐴̃ ‖ < 0 (푘,0) 푛 (푘,0) 푛 푛 푛 1 That is, ‖퐷 (푛)‖ ≤ for all푛 ∈ ℕ. Then for the operators 훾0 푛 푝푛 ̃ ( ) 푗 퐴푛 = ∑ 퐷훾푗 푛 푇휙 = 푇휙퐶푛 푗=1 1 we have ‖퐴̃ − 퐴 ‖ = ‖퐷 (푛)‖ ≤ . 푛 푛 훾0 푛 2 Finally, the inequality ‖퐴 − 퐴 ‖ ≤ ‖퐴 − 퐴̃ ‖ + ‖퐴̃ − 퐴 ‖ ≤ implies that the sequence 푛 푛 푛 푛 푛 {An} converges to the operator A. 2 2 Let 퐻 ≔ 𝒜휆 (𝔹 ). We consider two of its subspaces 푉 and 푊 defined by: 푉 = 푠푝푎푛̅̅̅̅̅̅̅{푒(푘,0): 푘 ∈ ℤ+} and 푊 = 푠푝푎푛̅̅̅̅̅̅̅{푒(0,푘): 푘 ∈ ℤ+} ⊥ It is easy to see that 푉 = 푘푒푟 푇휙 and 푊 = 푐표푘푒푟 푇휙 = (퐼푚푇휙) . We introduce as well the orthogonal projections 푃 ∶ 퐻 → 푉 and 푄 ∶ 퐻 → 푊. Corollary (6.2.39)[303]: For each operator 퐴 ∈ 풯(휆) satisfying the property (80), 퐼푚퐴 ⊥ 푊. Proof: For operators 퐴 ∈ 풯(휆) it follows from (81). Then, by Lemma (6.2.38), each operator 퐴 ∈ 풯(휆)\퐷(휆) with (80) can be uniformly approximated by operators whose range is orthogonal to W. Thus the limit operator A obeys the same property. The following result is known (see, for example, [48,253]). Lemma (6.2.40)[303]:Let 퐻 be a Hilbert space and 𝒜 α C∗-algebra in ℒ(H). Let A ∈ 𝒜 have a closed range. Then: (1)The orthogonal projection 푃 onto 푘푒푟퐴 = 푘푒푟퐴∗퐴 belongs to the algebra 𝒜. (2) There exists 퐵 ∈ 𝒜 , namely 퐵 = (푃 + 퐴∗퐴)−1, 퐴∗ ∈ 𝒜, such that (i)푃 = 퐼 − 퐵퐴 is the orthogonal projection onto ker 𝒜, (ii) 푄 = 퐼 − 퐴퐵 is the orthogonal projection onto (퐼푚 𝒜)⊥, (iii) 퐴퐵퐴 = 퐴 and 퐵퐴퐵 = 퐵, i.e. B is a relative inverse of A. 2 With 푛 ∈ ℤ+ we recall the definition 퐻푛 = 푠푝푎푛{푒훼: 훼 ∈ ℤ+, |훼| = 푛}, and let: 푛

푃̃푛: 퐻 → ⨁ 퐻푗 = 퐻̃푛 푗=0 denote the orthogonal projection. Recall that the finite dimensional spaces H̃n and hence H̃n are invariant for all elements of 풯∗(λ). Consider the sequence of C∗-algebras ∗ ∗ ∗ 풯푛 (휆) = {퐴푃̃푛 ∶ 퐴 ∈ 풯 (휆)} ⊂ 풯 (휆), ∗ where we have used that 푃̃푛 ∈ 풯푟푎푑(휆) ⊂ 풯 (휆). Then the restriction of 푇휙 to 퐻̃푛 defines an ∗ element in 풯푛 (휆) for all 푛 ∈ ℤ+ (we keep denoting the restriction by 푇휙 and do not indicate the 푛-dependence). Since 푑푖푚퐻̃푛 < ∞ , the range of 푇휙 in 퐻̃푛 is closed. Let ∏푛 퐻̃푛 → 푘푒푟 푇휙 = 푉 ∩ 퐻̃푛 ⊂ 퐻̃푛 denote the orthogonal projection.

187

Letting ∏푛 ≡ 0 on 퐻 ⊖ 퐻̃푛, we can consider ∏푛 푎푠 a (finite dimensional) orthogonal projection on 퐻. ∗ ∗ By Lemma (6.2.40)applied to 푇휙 ∈ 풯푛 (휆) ⊂ ℒ(퐻̃푛), we have ∏푛 ∈ 풯푛 (휆). As 퐻̃푛 and 퐻 ⊖ 퐻̃푛 are invariant under 푃 we remark that the projections 푃 and 푃̃푛 commute, and, in particular, ∏푛 = 푃̃푛푃푃̃푛. Now we give a generalization of (a slightly weaker version of) Lemma (6.2.37). ∗ Lemma (6.2.41)[303]:Let 퐴 ∈ 푅푎푑 풯(휆), then there is 퐶 ∈ 풯 (휆) ∩ 풦 such that 퐴 = 푇휙퐶. Proof: Recall that if 퐴 ∈ 푅푎푑 풯(휆) then A is compact and satisfies property (80). According to Corollary (6.2.39), we have (퐼푚 퐴 ∩ 퐻̃푛) ⊥ (푊 ∩ 퐻̃푛), for all 푛 ∈ ℕ, According to Lemma −1 ̃ ∗ (6.2.40) (1) the operator Bn = (푃 + 푇휙̅ 푇휙) 푇휙̅ 푃푛 belongs to 풯푛 (휆), and Lemma (6.2.40), (ii) gives 퐴푃̃푛 = 퐼푛푃̃푛 = (푄푛 + 푇휙퐵푛)퐴푃̃푛 = 푇휙퐵푛퐴푃̃푛 (83) ∗ where In is the identity element in ℒ(퐻̃푛) and 푄푛 = 퐼푛 − 푇휙퐵푛 ∈ 풯푛 (휆)is the orthogonal projection onto 푊 ∩ 퐻̃푛. Since 푃̃푛 → 퐼 as 푛 → ∞ in the strong topology, it immediately follows that −1 퐵푛 → 퐵 = (푃 + 푇휙̅ 푇휙) 푇휙̅ in the strong sense as 푛 → ∞. Since 퐴 is compact it follows that 퐵푛퐴 → 퐵퐴 and 퐴푃̃푛 → 퐴 as n → ∞ with respect to the norm topology. The inequality 0 ≤ ‖퐵푛퐴푃̃푛 − 퐵퐴‖ ≤ ‖퐵푛퐴푃̃푛 − 퐵퐴푃̃푛‖ + ‖퐵퐴푃̃푛 − 퐵퐴‖ ≤ ‖퐵푛퐴 − 퐵퐴‖ + ‖퐵‖‖퐴푃̃푛 − 퐴‖ implies that ‖퐵푛퐴푃̃푛 − 퐵퐴‖ → 0 as 푛 → ∞. If n on both sides of (83) tends to infinity then we ∗ ∗ conclude from 퐵푛퐴푃̃푛 ⊂ 풯푛 (휆) ⊂ 풯 (휆) for all n ∈ ℕ that 퐵퐴 ∈ 푇∗(휆), and, in addition, 퐴 = ∗ 푇휙퐵퐴 = 푇휙퐶 with 퐶 = 퐵퐴 ∈ 풯 (휆). Finally the simple arguments based on results of Theorem (6.2.34) show that the operator 퐴 = ∗ ∗ 푇휙퐶 ∈ 풯 (휆) is compact if and only if the operator 퐶 ∈ 풯 (휆) is compact. Summarizing our previous results we obtain the following description of 푅푎푑 풯(휆): Theorem (6.2.42)[303]:We have Rad 풯(λ) = {TϕC: C ∈ T∗(λ) ∩ K} ∩ 풯(λ) (84) If 퐴 ∈ 푅푎푑 풯(휆) ∩ 풟(휆), then 퐴 can be even expressed in the form 퐴 = 푇휙퐶, where 퐶 is chosen in 풟(휆) ∩ 풦. Proof: The first assertion follows directly from Lemma (6.2.41) and the simple observation that the elements on the right hand side of (84) do belong to the radical. The last statement has been shown in Lemma (6.2.37). Remark(6.2.43)[303]:Lemma (6.2.41) says that 퐴 ∈ 푅푎푑 풯(휆) has the form 퐴 = 푇휙퐶, where 퐶 ∈ 풯∗(휆) ∩ 풦, while Lemma (6.2.37) gives a more precise information in case of 퐴 ∈ 풟(휆) ∶ 퐶 can be taken from 풟(휆) ∩ 풦. Let us comment this ambiguity for 퐴 ∈ 푅푎푑 풯(휆) ∩ 풟(휆). We start from the representation 퐴 = 푇휙퐶1, with 퐶1 ∈ 풟(휆) ∩ 풦given by Lemma (6.2.37). In ∗ turn, Lemma (6.2.41) gives a different representation 퐴 = 푇휙퐶2 of 퐴, where 퐶2 ∈ 풯 (휆) ∩ 풦. In particular, from 0 = 푇휙(퐶1 − 퐶2) it follows that 퐼푚(퐶1 − 퐶2) ⊂ 퐾푒푟푇휙. More precisely, the operators 퐶1 and 퐶2 are related in the following way: 188

−1 −1 −1 퐶2 = 퐵퐴 = (푃 + 푇휙̅ 푇휙) 푇휙̅ 퐴 = (푃 + 푇휙̅ 푇휙) 푇휙̅ 푇휙퐶1 = (1 − (푃 + 푇휙̅ 푇휙) 푃) 퐶1. −1 The operator 푃 obviously commutes with (푃 + 푇휙̅ 푇휙) , and (we recall that) 푃 is the −1 orthogonal projection onto 퐾푒푟푇휙. Thus 퐶2 = 퐶1 − 푃(푃 + 푇휙̅ 푇휙) 퐶1, and we arrive again to the previous two different representations of the same operator 퐴: 퐴 = 푇휙퐶2 = 푇휙퐶1 We mention as well that the projection 푃 does not belong to the algebra 풯∗(휆), while the −1 operator푃(푃 + 푇휙̅ 푇휙) 퐶1 does. Section (6.3) Toeplitz Operators on the Unit Ball Let 𝔹푛 denote the complex 푛-dimensional open unit ball in ℂ푛. We introduce the standard 2 2 푛 weighted 퐿 −spaces 퐿 (𝔹 , 푑푣휆), where the family of probability measures 2 푑푣휆(푧) = 푐휆(1 − |푧| 휆푑푣(푧) is parameterized by 휆 ∈ (−1, ∞). The normalizing constant 휆 is given in (85) below and by dv n 2 푛 we denote the usual volume form on 𝔹 . We write 𝒜휆 (𝔹 ) for the classical weighted Bergman 2 푛 space, being the closed subspace of 퐿 (𝔹 , 푑푣휆) that consists of all complex-valued analytic 푛 2 푛 functions. The Toeplitz operator 푇푎 with symbol 푎 ∈ 퐿∞(𝔹 ) acting on 𝒜휆 (𝔹 ) is defined as 2 n the compression of a multiplication operator on L (𝔹 , dvλ) on to the Bergman space, i.e., 2 푛 푇푎푓 = 퐵휆(푎푓), where 퐵휆 is the Bergman (orthogonal) projection of 퐿 (𝔹 , 푑푣휆) on to 2 푛 𝒜휆 (𝔹 ). 푛 For a generic subclass 푆 ⊂ 퐿∞(𝔹 ) of symbols the algebra 풯(푆) generated by Toeplitz operators 푇푎 with 푎 ∈ 푆 is non-commutative and practically nothing can be said on its 푛 properties .However, if 푆 ⊂ 퐿∞(𝔹 ) has a more specific structure (e.g. induced by the geometry of 𝔹푛, or with a certain smoothness properties) the study of operator algebras 풯(푆) is quite important and has attracted lots of interest during the last decades. The reason of such an interest is caused, in particular, by the fact that such algebras provide rather simple but tractable examples of non-commutative algebras and play an important role in the applications. At the turn of this century it was observed (see [192] that, unexpectedly and contrary to the case of Toeplitz operators acting on the Hardy space over the circle, there exist many non- trivial algebras 풯(푆) that are commutative on each standard weighted Bergman space. The detailed structural analysis of such commutative algebras became then an important task sit provides an explicit information on many essential properties of Toeplitz operators such as compactness, boundedness, invariant subspaces, spectral properties, etc. We continue a project on the classification and structural analysis of commutative Banach algebras that are generated by Toeplitz operators with a specific class of the so-called quasi- 2 푛 homogeneous symbols acting on the weighted Bergman space 𝒜휆 (𝔹 ). The classification of these algebras has been given earlier in [193]. Subsequently in [303], and as a model case, we have analyzed the examples example 풯(푆) of such type. More precisely, 풯(푆) is the unique commutative Banach algebra in the above classification geverated by Toeplitz operators over the two- dimensional unit ball 𝔹2. As it turned out 풯(휆) is generated, in fact, by all Toeplitz operator with bounded measurable complex-valued radial symbols a on𝔹2 (i.e. 푎(푧) = 푎(|(푧)|)togather with a single Toeplitz operator 푇휙 having a certain quasi-homogeneous 189 symbol 휙. Among other results we explicitly described the space of maximal ideals of 풯(λ), results we were able to prove the inverse closeness of 풯(휆) and state some spectral properties of the elements in 풯(휆). Our next plan is to extend the results in [303] to the case of all commutative Toeplitz Banach algebras in [193] that are induced by the quasi-elliptic group of biholomorphisms of the unit ball 𝔹n. In [193] these algebras have been described in terms of their generators. Thus developing the Gelf and theory for these algebras will permit us to obtain more detailed (and, in particular, spectral) information on their elements. As it turned out, the algebra 풯(λ). studied in [303] indeed was the simplest in many respects. Let us single out some of the principal difficulties that bring the general multi-dimensional case studied here compared to the analysis of 풯(휆). We mention first that in all cases the algebra under study is generated by two mutually commuting commutative subalgebras: the C∗-algebra generated by Toeplitz operators with bounded (quasi-) radial symbols and the Banach algebra generated by Toeplitz operators with quasi-homogeneous symbols. In the case of [303] this last Banach algebra was generated by a single operator Tϕ with the simplest quasi-homogeneous symbol 휙. In the general case we can as well essentially reduce the number of generators having quasi- homogeneous symbols. However, still a finite number 푁 ≥ 1 of them remains. Due to this reduction some (bounded) Toeplitz operators with unbounded quasi-radial symbols come into play, and therefore immediately: realize whether these Toeplitz operators with unbounded symbols belong or do not belong to the C∗-algebra generated by Toeplitz operators with bounded quasi-radial symbols. Fortunately we manage to prove that the answer is positive, and so we do not need any further extension of the generator set. In order to describe the compact set of maximal ideals of the Banach algebra generated by Toeplitz operators with quasi-homogeneous symbols we need to calculate the joint spectrum of its generators (not just the spectrum of the unique generator, as in [303]). In addition this task be-comes more difficult as, contrary to the case of [303], the quasi-homogeneous functions in general do not extend continuously to the boundary 휕𝔹푛 of the unit ball. As a consequence our previous approach in [303] does not apply anymore in its full power. Finally we mention that C∗-algebra generated by Toeplitz operators with bounded quasi-radial symbols has a more complicated structure and involves a new class of slowly oscillating sequences which is defined. We recall the construction in [193] of a class of commutative Banach algebras ℬ푘(ℎ) 2 푛 that are generated by Toeplitz operators on the weighted Bergman space 𝒜휆 (𝔹 ) and have k −quasi-radialquasi-homogeneous symbols. These algebras are indexed by pair (푘, ℎ) of multi-indexes that fulfill certain relations. We point out that this class of Banach algebras is subordinated to the group of quasi-elliptic automorphisms of the unit ball since the k-radial part of the symbols of the generating operator is invariant under the action of this quasi-elliptic group, defined by the n-torus action on the ball 𝔹푛 (see [245,192]). ∞ We devoted to the analysis of the subalgebra 풯휆(퐿푘−푞푟) of ℬ푘(ℎ), which is generated ∞ byToeplitz operators withbounded k-quasi-radial symbols .All elements 푇 ∈ 풯휆(퐿푘−푞푟) are 2 푛 operators that are diagonal with respect to the standard monomial basis in 𝒜휆 (𝔹 )). Therefore 190

∞ ∗ we can identify 푇 with its eigenvalue sequence and interpret 풯휆(퐿푘−푞푟) as a C -subalgebra of 푚 all bounded complex-valued sequences on ℤ+ (here 푚 ∈ ℕ is the length of the multi-indexk). As for bounded Toeplitz operators with k-quasi-radial symbols we give an explicit integral form for their eigenvalues, and show that the eigenvalue sequences for all operators from ∞ 풯휆(퐿푘−푞푟) slowly oscillate in a specific sense.Quite a delicate task in this section is the proof that some Toeplitz operators with unbounded quasi-radial symbols belong to the algebra ∞ ∞ 풯휆(퐿푘−푞푟). We prove as well that 풯휆(퐿푘−푞푟) contains the orthogonal projections onto certain 2 푛 types of closed subspaces of 𝒜휆 (𝔹 ). These two assertions are crucial for reducing the set of generators of ℬ푘(ℎ)in Theorem (6.3.20). With an analysis of the maximal ideal space ∞ ∞ 푀(풯휆(퐿푘−푞푟)) of the algebra 풯휆(퐿푘−푞푟). In particular, we give a useful stratification of ∞ 푀(풯휆(퐿푘−푞푟))in Lemma (6.3.15).This stratification will be important in the description of the maximal ideals of the full algebra ℬ푘(ℎ). We define a second subalgebra 풯휆(휀푘(ℎ)) of ℬ푘(ℎ), which is generated by a finite set of Toeplitz operators with so-called elementary k-quasi-homogeneous symbols. Contrary to the case, the generators of 풯휆(휀푘(ℎ)) are not anymore diagonal with respect to the standard monomial basis and their action on the elements of this basis is independent of the weight parameter λ > −1. We show that the union of generators of both above sub algebras gives in fact a reduced set of generators for ℬ푘(ℎ). As was already mentioned, a special care has to be taken since the elementary k-quasi- homogeneous functions do not admit continuous extensions to the boundary 휕𝔹푛, unless 푚 = 1. The main result is description of the maximal ideal space of 풯λ(εk(h)) and the corresponding Gelf and transform. First we treat the case where 푚 = 1 and in the final part we generalize the result to the case m > 1 by using an appropriate tensor product description. We list several open problems closely related to the results.Finally we wish to remark that in the case of dimension n ≥ 3 adscription of the maximal ideal space of the full commutative algebra ℬ푘(ℎ), the Gelf and map, a characterization of the radical, and a spectral analysis of its elements can be achieved. 푛 푛 2 2 2 n Consider the open complex unit ball 𝔹 : = {푧 ∈ ℂ : |푧| = |푧1| +··· +|푧푛| < 1} in ℂ equipped with the standard weighted measure. 2 휆 푑푣휆(푧) = 푐휆(1 − |푧| ) 푑푣(푧), Where 휆 > −1 is fixed. Here we write dv for the usual volume form on Bn. Recall that due to n the assumption 휆 > −1 the measure vλ(𝔹 ) of the unit ball is finite and we chose 푐휆 > 0 such 푛 that 푣휆(𝔹 ) = 1. In fact this is realized by defining 훤 (푛 + 휆 + 1) 푐 : = . (85) 휆 휋푛훤 (휆 + 1) 2 푛 We write 퐿 (𝔹 , 푑푣휆) for the Hilbert space of all squareintegrable functions with respect to 푣휆. The corresponding norm and inner product are denoted by‖·‖휆 and 〈·,·〉λ, respectively. 2 푛 n The weighted Bergman space 𝒜휆 (𝔹 ) over 𝔹 consists of all complex-valued analytic functions that are squareintegrable with respect to the measured 푣휆. It is a standard fact that 2 푛 2 푛 𝒜휆 (𝔹 ) is a closed sub space of 퐿 (𝔹 , 푑푣휆) and that the orthogonal projection (Bergman 2 푛 2 푛 projection) Bλ from 퐿 (𝔹 , 푑푣휆) on to 𝒜휆 (𝔹 ) has the following explicit form 191

휑(휔) [퐵 휑](푧) = ∫ 푑푣휆(휔), where 휑 ∈ 퐿2(𝔹푛, 푑푣 ). 휆 (1 − 〈푧, 푤〉)푛+휆+1 휆 𝔹푛 In the following we write 〈푧, 휔〉: = 푧1휔̅1 +··· +푧푛휔̅푛 for the usual Euclidean inner product. Let ℤ+ be the set of all non-negative integers. Recall as well that the standard orthonormal basis 푛 2 n [푒훼: 훼 ∈ ℤ+] of 𝒜λ(𝔹 )is given by the monomials 훤 (푛 + |훼| + 휆 + 1) 푒 (푧): = √ 푧훼. (86) 훼 훼! 훤 (푛 + 휆 + 1)

Although the functions 푒훼 depend on the particular choice of the weight 휆 we do not indicate this dependence in our notation and we assume that 휆 ∈ (−1, ∞) is arbitrary but fixed. Given 푔 ∈ 퐿 (𝔹푛), the Toeplitz operator푇 with symbol g on 𝒜2(𝔹푛) is defined by: ∞ 푔 휆 푔(푤)푓(휔) [푇 푓 ](푧): = [퐵 (푔푓)](푧) = ∫ 푑푣 (휔). 푔 휆 (1 − 〈 푧, 휔〉)푛+휆+1 휆 𝔹푛 Recall that the algebra generated by the set of all Toeplitz operators with bounded measurable symbols is non-commutative. However, after a restriction of the symbol class to a certain type of functions it turns out that the induced Toeplitz algebra becomes commutative. In sense, commutative Banach algebras generated by Toeplitz operators of such type are subordinate to (some of) the maximal abeliansub groups of the automorphism group 퐴푢푡 (𝔹푛) of 𝔹푛. The classification of commutative algebras interms of the generating symbols has been given in [37,38,193,195]. Here we are interested in a class of commutative Toeplitz Banach algebras that is induced by the quasi-elliptic group of biholomorphisms of 𝔹푛, cf.[193].For completeness were call some notation and results from [193]: 푚 Let 푚 ∈ {1, . . . , 푛}, and fix a tuple 푘 = (푘1, . . . , 푘푚) ∈ ℤ+ with |푘| = 푘1 +··· +푘푚 =n. Then we can interpret ℂ푛 as a product space ℂ푛 = ℂ푘1 × ℂ푘2 ×···× ℂ푘푚, and we use the notation 푛 푘푗 푧 = (푧(1), . . . , 푧(푚)) ∈ ℂ , where 푧(푗 ) = (푧푗,1, . . . , 푧푗, 푘푗 ) ∈ ℂ . 2푘푗−1 푘푗 We will frequently employ polar coordinates. Let us write 푆 1 ⊂ ℂ for the (real) (2푘푗 − kj 푘푗 1)-dimensional units herein ℂ . We express non-zero vectors 푧(푗) ∈ ℂ in the form푧(푗) = 푟푗휉(푗),where 푧(푗) 2푘푗−1 휉(푗) = ∈ 푆 and 푟푗 = |푧(푗)| ∈ ℝ+. |푧(푗)| Let us recall now the notion of 푘-quasi-homogeneous functions on the unit ball 𝔹푛 in [303,193]: 푛 푛 Definition (6.3.1)[302]: Fix (푝, 푞) ∈ ℤ+ × ℤ+ with 푝 ⊥ 푞(i. e. 〈푝, 푞〉 = 0). Abounded function 휑(푧)on 𝔹n is called “k-quasi-radialquasi-homogeneous” with the quasi-homogeneous degree (p, q) if it has the form 푝(1) 푝(2) 푝(푚) 푞(1) 푞(2) 푞(푚) 휑(푧) = 푎(푟1, … , 푟푚)휉(1) 휉(2) … 휉(푚) 휉(1) 휉(2) … 휉 (푚) (87) and 푎 = 푎(푟1, … , 푟푚) is a function of them non-negative real variables 푟1, … , 푟푚. A function that can be expressed in the form 푎 = 푎(푟1, … , 푟푚) is called “k-quasi-radial”.

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∞ In what follows we denote by 퐿푘−푞푟 the Banach space of all bounded measurable 푘-quasi-radial functions on 𝔹n. In order to define a class of c commutative Toepliz Banach algebras we choose a second tuple 푛 ℎ ∈ ℤ+ which is subordinate of 푘 in the sense that it fulfills the following conditions: ℎ푗 = 0 if푘푗 = 1 and 1 ≤ ℎ푗 ≤ 푘푗 − 1 if 푘푗 > 1. We denote by ℛ푘(ℎ) the linear space generated by all k-quasi-radialquasi-homogeneous functions (87) which satisfy (i) and (ii): 푘푗 (i) For j with 푘푗 > 1 the tuples 푝(푗), 푞(푗) ∈ ℤ+ have the form:

푝(푗) = (푝푗, 1, … , 푝푗,ℎ푗 , 0, … , 0) and 푞(푗 ) = (0, … , 0, 푞푗,ℎ푗+1, … , 푞푗,푘푗 ) (88) and, in addition, |푝(푗)| = |푞(푗)|. (ii)If 푘푗′ = 푘푗′′ with 푗′ < 푗′′, thenℎ푗′ ≤ ℎ푗′′. n For a given set ℱ of bounded measurable complex-valued functions on 𝔹 we denote by 풯휆(퐹) 2 푛 the unital Banach algebra in ℒ(퐴휆(𝔹 )) which is generated by all Toeplitz operators with ∗ symbols in ℱ. Note that 풯휆(퐹) has the structure of a C -algebra in the case when ℱ = ℱ̅, i.e. ℱ is invariant under complex conjugation.[22] states: 2 푛 Theorem (6.3.2)[302]:The Banach subalgebra ℬ푘(ℎ): = 풯휆(ℛ푘(ℎ)) of ℒ(퐴휆(𝔹 )) which is generated by Toeplitz operators with symbols from ℛ푘(ℎ) is commutative for all 휆 > −1. We study separately two commutative subalgebras of ℬ푘(ℎ)which together generate ℬ푘(ℎ)The first one has the structure of a commutativeC∗ −algebra and is generated by Toeplitz operators with k −quasi-radial symbols. The second one is a commutative Banach algebra generated by a finite number of Toeplitz operators with certain quasi-homogeneous symbols. The analysis of these subalgebras serves as an important tool for studying the complete Banach algebra structure of ℬ푘(ℎ) in an upcoming work. As is shown [192], a Toeplitz operator with a bounded measurable k-quasi-radial symbol 푎 = 훼 푛 푎(푟1, … , 푟푚) has the monomials푧 (or the normalizedmonomials eα(z) (66)), where 훼 ∈ ℤ+, as eigenfunctions. In what follows, for each multi-indexα = (훼1, … , 훼푛) = (훼(1), … , 훼(푚)) ∈ 푛 ℤ+, 푚 we denote by 휅 = 휅(훼) = (휅1, … , 휅푚) ∈ ℤ+ the multi-index with the entries 휅푗 = |훼(푗)|, for α all 푗 = 1, … , 푚. In particular, |훼| = |휅| and the eigenvalue γa,k,λ(κ) of Ta with respect to z depends only on κ = κ(α), i.e., 훼 훼 푇푎푧 = 훾푎,푘,휆(휅)푧 . (89) Moreover, we have the following explicit expression of 훾푎,푘,휆(휅) 푚 푚 ( | | ) 2 훤 푛 + 휅 + 휆 + 1 2 2푘푗−1 훾푎,푘,휆(휅) = 푚 ∫ 푎(푟) (1 − |푟| )휆 ∏ 푟푗 푑푟 (90) 훤(휆 + 1) ∏푗=1(푘푗– 1 + 푘푗) ! 휏(𝔹푚) 푗=1 푚 푚 The integration is taken over the base 휏(𝔹 ) = {푟 = (푟1, … , 푟푚) ∈ ℝ+ : 0 ≤ |푟| < 1}of the unit ball 𝔹m considered as a Reinhardt domain.

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푛 푚 Corollary (6.3.3)[302]: Let 훼 ∈ ℤ+, and κ = (|훼(1)|, … , |훼(푚)|) ∈ ℤ+ as above. Then we have

푚 푚 2푘 +2푘 −1 훤(휆 + 1) ∏푗=1(푘푗– 1 + 푘푗) ! ∫ 푎(푟) (1 − |푟|2)휆 ∏ 푟 푗 푗 푑푟 = (91) 푗 푗 2푚훤(푛 + |휅| + 휆 + 1) 휏(𝔹푚) 푗=1 m Proof: Put 푎(푟) ≡ 1, then Ta is the identity operator. Thus훾푎,푘,휆(휅) = 1 for all κ ∈ ℤ+ , and the assertion follows from the expression (90) of 훾푎,푘,휆(휅). 푚 Let 훾 = {훾(휅)}휅∈ℤ+ bean arbitrary bounded sequence of complex numbers. Denoteby 퐷훾 the 2 n diagonal operator which acts on the weighted Bergman space 𝒜λ(𝔹 ) by the rule 푛 퐷훾푒훼(푧) = 훾 (휅)푒훼(푧), 훼 ∈ ℤ+. (92) According to the above remarks each Toeplitz operator with bounded measurable k-quasi- radial symbol 푎(푟1, . . . , 푟푚) is diagonal, and 푇푎 = 퐷훾푎,푘,휆. However, as we will show later on, not all bounded diagonal operators 퐷훾 can be represented in such a form since the eigenvalue sequence 훾푎,푘,휆 of Ta possesses certain specific features (cf.Lemma (6.3.6)). Now consider a particular case of k-quasi-radial symbols:

m m ( | | ) 2 Γ n + κ + λ + 1 2 2(kj+δj,1)+2kj−1 γ 2 (κ) = ∫ a(r) (1 − |r| )λ ∏ r dr r1,k,λ m j Γ(λ + 1) ∏j=1(kj– 1 + kj) ! τ(𝔹m) j=1 푚 푚 2 훤(푛 + |휅| + 휆 + 1) 훤(휆 + 1) ∏푗=1(푘푗– 1 + 푘푗 + 훿푗,1)! = 푚 . 푚 훤(휆 + 1) ∏푗=1(푘푗– 1 + 푘푗) ! 2 훤(푛 + |휅| + 휆 + 1) 푘 + 휅 = 푙 푙 , 푛 + |휅| + 휆 + 1

Where 훿푗,1 is the standard Kronecker delta. We mention that neither of the sequences {훾 2 (푘)} , where 푙 = 1, . . . , 푚, has a limit 푟1 ,푘,휆 휅 when휅 → ∞ (or, which is the same, |푘| → ∞), though all of them possess many partial limit m values. In particular, given any m-tuples = (s1, . . . , sm) ∈ ℤ+ , we introduce the subsequence m {푘푠(푛)}푛∈ℤ+ofℤ+ , where 푘푠(푛) = (푠1푛, 푠2푛, . . . , 푠푚푛). Then, for each 푙 = 1, . . . , 푚, 푠푙 푙푖푚 훾푟2,푘,휆 (휅푠(푛)) = . 푛→∞ 1 |푠| 푚 In general, let {푘(푛)}푛∈ℤ+, where 푘(푛) = (푘1(푛), 푘2(푛), . . . , 푘푚(푛))be subsequence of ℤ+ . Then the limit of the sequence 훾 2 along the subsequence {푘(푛)} exists if and only if 푟1 ,푘,휆 푛∈ℤ+ 푘 (푛) the sequence { 푙 } has a limit, and in the last case both limits coincide. |푘(푛)| 푛∈ℤ+ At the same time, regardless of the way how κ tend so infinity, we always have 푙푖푚 훾푟2,푘,휆(푘) +··· +훾푟2 ,푘,휆(푘) = 1. |휅|→∞ 1 푚 We denote by ∆푚−1 the standard (푚 − 1)- dimensional simplex with the vertices 푚 (1,0, . . . ,0), (0,1,0, . . . ,0), . . . , (0, . . . ,0,1) ∈ ℝ+ . Summarizing the above results to:

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푚 Lemma (6.3.4)[302]:For any subsequence {푘(푛)}푛∈ℤ+ of ℤ+ , all limits 푙푖푚 훾 2 (푘(푛)) = 훽푙, 푙 = 1, . . . , 푚, |휅(푛)|→∞ 푟퐼 ,푘,휆 푚−1 Exist if and only if (훽1, . . . , 훽푚) ∈ ∆ and 푘푙(푛) 푙푖푚 = 훽푙, 푙 = 1, . . . , 푚. |휅(푛)|→∞ |푘(푛)| 푚−1 Moreover, for each point(훽1, . . . , 훽푚) ∈ ∆ , there is a subsequence {푘(푛)}푛∈ℤ+ with the above properties. In order to make our considerations geometrically more transparent we proceed as follows. 푚 푚 푚 푚 Denote by ℝ1 = (ℝ , ‖·‖1) the Banach space ℝ equipped with the norm‖푥‖1 = ∑푗=1 |푥푗|, 푚 푚−1 푚 where 푥 = (푥1, . . . , 푥푚) ∈ ℝ . Let 푆1 (0, 푅) be the sphere in ℝ1 of radius 푅 centered at the 푚 푚 푚 origin. We interpret ℤ+ = ℤ+ ∩ ℤ1 as a metric space with the metric 휌1(휅′, 휅′′) = ‖푘′ − 푘′′‖1 m 푚−1 푚−1 푚 in herited from ℝ1 . Then |푘| = 휌1(0, 휅), and ∆ = 푆1 (0,1) ∩ ℝ+ is nothing but the 푚−1 corresponding part of the unit sphere 푆1 (0,1). 푚 푚 푚−1 We denote by ℝ̅1 the compactification of ℝ1 by the “infinitely far” sphere푆1 (0, ∞) consisting of rays through the origin. Each such ray and its intersection with the unit sphere 푚−1 푆1 (0,1) are identified. This yields a parametrization of the points of the “infinitely far” 푚−1 푚−1 sphere 푆1 (0, ∞) by elements of the unit sphere 푆1 (0,1), identifying these two objects. 푚 푚 푚 푚 Let now ℤ̅+ be the closure of ℤ+ inℝ̅1 , being the compactification of ℤ+ by the corresponding 푚−1 푚−1 푚 푚 푚−1 part ∆ (∞) of the “infinitely far” sphere 푆1 (0, ∞). That is ℤ̅+ = ℤ+ ∪ ∆ (∞), where we also parameterize the points of ∆푚−1(∞) by the points of ∆푚−1, identifying them. Lemma (6.3.4) states that each sequence 훾 2 , 푙 = 1, . . . , 푚, admits continuous extension to 푟푙 ,푘,휆 ∆푚−1(∞). ̅푚 ∗ 푚 We denote by 푐(ℤ+ ) the unital C -algebra which consists of all bounded sequences {훾(푘)}휅∈ℤ+ that admit a continuous extension to ∆푚−1(∞). The elements of them- tuple (훾 2 , … , 훾 2 ) are real-valued sequences and separate the 푟1 ,푘,휆 푟푚,푘,휆 푚 points of ℤ+ . Then, by the above discussion and the Stone–Weierstrass theorem, the unital ∗ 2 2 2 C -algebra풯휆({푟1 , 푟2 , … , 푟푚})coincides with the algebra of all diagonal operators 퐷훾 with 훾 ∈ 푚 푚 푐(ℤ̅+ ). Furthermore, its maximal ideal space coincides with ℤ+ . 푚 For each 푘 = (푘1, … , 푘푚) ∈ ℤ+ introduce the finite dimensional space퐻푘 defined by 푛 퐻푘 ≔ 푠푝푎푛{푒훼: 훼 ∈ ℤ+, |훼(푗 )| = 푘푗, 푗 = 1, … , 푚}(93) And let 2 푛 푃푘: 퐴휆(𝔹 ) → 퐻푘 (94) Be the orthogonal projection onto 퐻푘. Then we have: 푚 ∞ m Corollary (6.3.5)[302]: Let 훾 ∈ 푐(ℤ̅+ ), then 퐷훾 ∈ 풯휆(퐿푘−푞푟). In particular, for all κ ∈ ℤ+ : 2 2 2 ∞ 푃푘 ∈ 풯휆({푟1 , 푟2 , … , 푟푚}) ⊂ 풯휆(퐿푘−푞푟 ⊂ 퐵푘(ℎ). 푚 For each푗 = 1,2, . . . , 푚,introducethestandardunit vector풆푗: = (0, … ,0, 푗 ↓ 1,0, . . . ,0) ∈ ℤ+ , 푚−1 being also the 푗-th vertex of ∆ . We denote by 푑1(푚) these to fall bounded sequences 훾 = 푚 {훾(휅)}휅∈ℤ+ that satisfy the following condition

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푚 휅푠 푠푢푝 |푘| |훾 (푘) − ∑ 훾 (푘 + 풆푠 )| < +∞. (95) 휅≠0 |휅| 푠=1 휅 휅 Note that the point ( 1 , … , 푚) ∈ ∆푚−1corresponds to the point of ∆푚−1(∞)defined by the |푘| |푘| m ray passing through κ ∈ ℤ+ . 푚 Lemma (6.3.6)[302]: For each 푎 = 푎(푟1, … , 푟푚) ∈ 퐿∞(휏(𝔹 )),the eigenvalues equence 훾푎,푘,휆of the Toeplitz operator 푇푎 belongs to 푑1(푚). Proof: For all푘 ≠ 0 we have by (90) 푚 푚 ( | | ) 2 훤 푛 + 휅 + 휆 + 1 2 휆+1 2푘푗−2푘푗−1 훾푎,푘,휆 = 푚 ∫ 푎(푟) (1 − |푟| ) ∏ 푟푗 푑푟 훤(휆 + 1) ∏푗=1(푘푗– 1 + 푘푗) ! 휏(𝔹푚) 푗=1 푚 푚 ( | | ) 2 훤 ` 휅 + 휆 + 2 2 휆 2푘푗−2푘푗−1 = 푚 [ ∫ 푎(푟) (1 − |푟| ) ∏ 푟푗 푑푟 훤(휆 + 1) ∏푗=1(푘푗– 1 + 푘푗) ! 휏(𝔹푚) 푗=1 푚 푚 2 휆 2푘푗−2푘푗−1+2훿푗,푠 − ∑ ∫ 푎(푟) (1 − |푟| ) ∏ 푟푗 푑푟] 푠=1 휏(𝔹푚) 푗=1 푚 푛 + |휅| + 휆 + 1 푘 + 휅 = 훾 (푘) − ∑ 푠 푠 훾 (푘 + 풆 ) 휆 + 1 푎,푘,휆 휆 + 1 푎,푘,휆 푠 푠=1 Or, 푚

푁(푎, 휆, 푘): = (휆 + 1)[훾푎,푘,휆+1(푘) − 훾푎,푘,휆(푘)] + ∑ 푘푠훾푎,푘,휆(푘 + 풆푠) − 푛훾푎,푘,휆(푘) 푠=1 푚 푘푠 = |푘| [훾푎, 푘, 휆(휅) − ∑ 훾 (푘 + 풆 )]. |푘| 푎,푘,휆 푠 푠=1 But |푁(푎, 휆, 푘)| ≤ 2(푛 + 휆 + 1)‖푎‖∞, And the result follows. Some important comments to the definition of d1(m) and Lemma (6.3.6) have to be added. First of all we mention that the condition (95) pacifies the form in which the sequences 훾 ∈ 푑1(푚) may oscillate at infinity. In particular, (95) implies that 푚 푘푠 푙푖푚 [훾푎,푘,휆(푘) − ∑ 훾푎,푘,휆(푘 + 풆푠)] = 0. 푘→∞ |푘| 푠=1 Then, for 푚 = 1, the class 푑1(1) coincides with the class which in the work of Suárez [290] was denoted by 푑1 and is commonly used in Tauberian theory. For each 푚 ≠ 1, 푑1(푚) Is just a (non-closed) linear space, and only for 푚 = 1 it is an algebra Further, for 푚 = 1 ,i.e., in the case of radial symbols, and the un weighted Bergman spaces, it is known [271] (see as well [76,290] for the one-dimensional case 푛 = 1) that the set of Toeplitz operators with bounded radial symbolsis dense in the C∗-algebra generated by these 196 operators, and that the 푙∞-closure of the set of corresponding eigenvalue sequences 훾푎 coincides ∗ with the 푙∞- closure of 푑1. Moreover, by [271] this closure coincides with the C -algebra 푆푂1(ℤ+) of all slowly oscillating sequences introduced by Schmidt [246], i.e., of all bounded ∞ sequences 훾 = 훾(푝)푝=0such that 푙푖푚 |훾 (푝) − 훾 (푞)| = 0. 푝+1 →1 푞+1 This gives the exact characterization of the algebra According to [194] the set SO1(ℤ+) is apropersubset of the standard SO(ℤ+) Recall in this m connection (see, for example, [293]) that the algebra SO(ℤ+ ) consists of all sequence 훾 = 푚 {훾(푘)}휅∈ℤ+ such that 푚 푙푖푚(훾 (푘) − 훾 (푘 + 휌)) = 0 푓표푟 푎푙푙 휌 ∈ ℤ+ . 휅→∞ For 푚 = 1and anarbitrary weight parameter 휆 ∈ (−1, ∞) Lemma (6.3.6) just ensures that the ∞ algebra 풯λ(Lrad). is isomorphic to a subalgebra of 푆푂1(ℤ+) . ∗ ∞ It is clear that for 푚 > 1 the C -algebra 풯휆(퐿푘−푞푟) is isomorphic to a certain subalgebra of the ∗ C -algebra generated by sequences in 푑1(푚).But the exact description of this subalgebra is ∞ 푚 unknown. As partial information we mention that the algebra 풯휆(퐿푘−푞푟) intersects 푆푂(ℤ+ ), for example, by diagonal operators whose eigenvalue sequences have a limit as 푘 = ∞ (푘1, . . . , 푘푚) → ∞. At the sametime, contrary to the case 푚 = 1, 풯휆(퐿푘−푞푟)is not a subalgebra 푚 of 푆푂(ℤ+ ),, as will be shown in Lemma (6.3.13). It is unclear whether the set of all Toeplitz operators with bounded k-quasi-radial symbols is ∞ dense in 풯휆(퐿푘−푞푟), as it is the case for 푚 = 1 and 휆 = 0. The fact that 푑1(푚)(contrarytod1) is not an algebra suggests that the answer might be negative. We list the above open questions among others.Then extcorollaries to Lemma (6.3.6) give a further characterization of the eigenvalue sequences 훾푎,푘,휆 of Toeplitz operators 푇푎 ∈ ∞ 풯휆(퐿푘−푞푟). Corollary (6.3.7)[302]:We have that 푚 휅푠 훾 (휅) − ∑ 훾 (휅 + 풆 ) = 풪(1/|휅|) 푎푠 휅 → ∞. 푎,푘,휆 |휅| 푎,푘,휆 푠 푠=1 For each 푗 = 1, … , 푚 let us fix the values of 휅푆 for all 푠 ≠ 푗arbitrarily, and define the “푗 − 푡ℎ coordinate” sequence κ̂j = {κ̂j(n)}n∈ℤ+, where 휅푗̂ (푛) = (휅1(푛), … , 휅푚(푛)) has the entries 푚−1 휅푗̂ (푛): = 푛 and 휅푠(푛): = 휅푠. By considering 풆푗 as the j-th vertex of ∆ (∞) it is clear that 푚−1 푙푖푚 휅̂푗(푛) = 푒푗 ∈ ∆ (∞). 푛→∞

Corollary (6.3.8)[302]:For each푗 = 1, … , 푚 the sequence {훾푎,푘,휆(휅푗̂ (푛))}푛∈ℤ+ be longs to 푑1, and thus to 푆푂1(ℤ+). Proof: Follows from Lemma (6.3.6) together with the observation that 휅푗̂ (푛) 푙푖푚 = 1, 푛→∞ |휅푗̂ (푛)| And that the sum ∑푠≠푗 |휅푠(푛)| does not depend on 푛 and is bounded.

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The nextimportant particular case of 푘-quasi-radial symbols is as follows: for each 푗 = 1, . . . , 푚, we introduce the unbounded k-quasi-radial function 2 2 (푗 ) 1 − |푧| 1 − |푟| 푎 (푧): = 2 = 2 ∈ 푅+ ∪ {+∞}. |푧(푗 )| 푟푗 Although this symbol is unbounded, it generates a bounded Toeplitz operator. Moreover, Corollary (6.3.3) yields the explicit formula for its eigenvalue sequence (see the notation in (89)): (푗 ) 휆 + 1 푛 훾푎 , 푘, 휆(휅(훼)) = , 훼 ∈ ℤ+. 휅푗 + 푘푗 − 1 (푗) 푛 In particular, since푘푗 > 1 we have that 푎 ∈ 퐿1(𝔹 , 푑휈휆) for 푗 = 1, . . . , 푚 with norm (푗 ) (푗 ) 휆 + 1 ‖푎 ‖ = 〈푎 , 1〉 = 〈푇 (푗 )1, 1〉 = 훾 (푗 ) (0) = . 퐿 (𝔹푛푑푣 ) 휆 푎 휆 푎 ,푘,휆 1 휆 푘푗 − 1 The last formula implies an interesting characterization of the first (i.e., for 휅 = 0) eigenvalue of a Toeplitz operator with quasi-radial symbol. 푛 Lemma (6.3.9)[302]:Let 푎(푟) ∈ 퐿1(𝔹 , 푑푣휆) be aquasi-radial symbol. Then the first eigenvalue (ground state) of a bounded, or densely defined unbounded Toeplitz operator Ta is given by

훾푎,푘,휆(0) = ∫ 푎(푟) 푑푣휆. 𝔹푛 ‖ ‖ 푛 In the case of a non-negative symbol a it coincides with 푎(푟) 퐿1(𝔹 ,푑푣휆). Recall that a finite positive measure ν on 𝔹n is called a Carleson measure with respect to 2 푛 𝒜휆 (𝔹 ) (shortly: Carleson measure) if the Toeplitz operator Tν with measure symbol ν defined by 푓 (휔) 푇 푓: = ∫ 푑휈(휔), 푤ℎ푒푟푒 푓 ∈ 퐻∞(𝔹푛), 휈 (1 − 〈푧, 휔〉)푛+1+휆 𝔹푛 has a bounded extension from the space 퐻∞(𝔹푛) of all bounded analytic functions on 𝔹푛 to 2 푛 the Bergman space 𝒜휆 (𝔹 ) (e.g. see [179], [65]). From the above discussion it follows that (푗) 휈푗: = 푎 푑푣휆 for 푗 = 1, … , 푚 is a Carleson measure. Let us recall the notion of the (ℓ, 휆)-Berezin-transform whereℓ ≥ λ (for the unweighted case 휆 = 0 see [148, 65], and for an arbitrary weight 휆 > −1 see [179]). If μ is a complex- 푛 푛 valued, Bore regular measure on 𝔹 and 푧 ∈ 𝔹 then we set 푛+1+ℓ 푐ℓ (1 − |휙푧(휔)|2) 퐵ℓ(휇)(푧): = ∫ 2 푛+1+ℓ 푑휇(휔) . (96) 푐휆 (1 − |휔| ) 𝔹푛 Here the constants 푐휆 and 푐ℓ were defined in (85) and 휑푧 denotes the (unique up to unitary n multiples) automorphism of 𝔹 with 휑푧 ∘ 휑푧 = 푖푑 and 휑푧(0) = 푧. When 휇 = 푎푑푣휆 with 푎 ∈ 푛 퐿1(𝔹 , 푑푣휆) then a change of variables shows that (76) takes the form

198

퐵ℓ(푎 푑푣휆)(푧) = ∫ 푎 ∘ 휑푧(휔)푑푣ℓ(휔). 𝔹푛 Note that the right hand side of this equation is independent of λ > −1. On the other hand, by inserting the well-known relation (1 − |푧|2)(1 − |휔|2) 1 − |휙푧(휔)| = |1 − 〈 푧, 휔〉|2 Into (96) we obtain the following expression for the (ℓ, 휆) − Berezin-transform, which will be most suitable for the considerations below, 푎(휔) 퐵 (푎 푑휈 )(푧) = (1 − |푧|2)푛+1+ℓ ∫ 푑휈 (휔). (97) ℓ 휆 (1 – 〈푧. 휔〉)2(푛+1+ℓ) 휆 𝔹푛 The following result has been proved in the unweighted situation 휆 =0 in [66], and in the general weighted case 휆 > −1 in [179]: Proposition (6.3.10)[302]: (See [65,179].) Let the positive measure ν be Carleson with respect n to the weighted measured 푣휆 on 𝔹 where 휆 > −1. Then: 푛 (i)The functions 퐵ℓ(휈) are bounded and continuous on 𝔹 for allℓ ≥ 휆. 2( 푛) (ii)The convergence 푇퐵ℓ(휈) → 푇휈 holds, as ℓ → ∞, in the uniform topology of ℒ(𝒜휆 𝔹 ). (j) Corollary (6.3.11)[302]: The Toeplitz operators 푇푎(푗) with 퐿1-symbol a , where 푗 ∈ ∞ {1, . . . , 푚}, belong to the algebra 풯휆(퐿푘−푞푟). (푗) Proof: Since the measures 휈푗: = 푎 푑푣휆, for j= 1, . . . , 푚, are Carleson we conclude from Proposition (6.3.10) that the functions 휈푗, ℓ: = 퐵ℓ(휈푗) are bounded and as ℓ → ∞ the norm (푗) convergence 푇휈푗,ℓ → 푇푎 holds. Hence it remains to show that 휈푗,ℓ(푧) for all j andℓ are k-quasi- radial. For any given 푟 ∈ ℕ let us denote by 푈(푟) the group of unitary 푟 × 푟-matrices. Then we have the natural embedding 퐺 ≔ 푈(푘1) ×···× 푈(푘푚) ⊂ 푈(푛) of groups. It can be easily seen from the expression (97) of the (ℓ, 휆) −Berezin transform and the transformation rule of the integral that in the case of a k-quasi-radial symbols 푎(푤) the integral transform 퐵ℓ(푎)(푧) is invariant under the action of 퐺. Hence 퐵ℓ(푎)(푧) defines a k-quasi-radial function, as well for all ℓ ≥ 휆. Since 푎(푗) is 푘-quasi-radial this observation finishes the proof.

The eigenvalues of the Toeplitz operator 푇푎(푗) are real-valued and monotonically decreasing (푗) when 휅푗 = |푎 | tends to infinity. By c we denoe the set of all converging sequences. The Stone– Weierstrass the ore directly leads to the following lemma. ∗ ∞ Lemma (6.3.12)[302]: The unital C -subalgebra in 풯휆(퐿푘−푞푟) which is generated by the operators 푇푎(푗), where 푗 = 1, … , 푚, coincides with the set of all diagonal operators 퐷γ(j) whose (j) (푗) 푚 eigenvalue sequences γ = {훾 (휅)}휅∈ℤ+ depend on 휅푗 only; more over being considered (푗) (푗) with respect to their dependence on 휅푗, i. e. , 훾 = {훾 (휅푗)}휅푗∈ℤ+, they belong to c.

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∞ We mention that the operators 푇푎(푗) ∈ 풯휆(퐿푘−푞푟) where 푗 = 1, . . . , 푚, commute among them- selves. Moreover, they commute with each Toeplitz operator 푇휑 having a k-quasi-radialquasi- homogeneous symbol 휑 ∈ 푅푘(ℎ). ∞ m The next lemma shows that 풯휆(퐿푘−푞푟) is not a subalgebra of SO(ℤ+ ). Lemma (6.3.13)[302]:For 푗 = 1, … . , 푚, the eigenvalue sequence 훾푎(푗),푘,휆of the Toeplitz operator 푇푎(푗) is not slowly oscillating when 휅 = (휅1, … , 휅푚) → ∞. 푚 Proof: The assumption that the sequence 훾푎(푗),푘,휆 belongs to 푆푂(ℤ+ ). Implies that the set of its partial limits is connected (see, [293]). At the same time this set has the form 휆 + 1 { } ∪ {0} 휅푗 + 푘푗 − 1 휅푗∈ℤ+ and hence is discrete, which leads to a contradiction. Let 푗 ∈ {1, . . . , 푚} be fixed and for each 푑 ∈ ℤ+ consider the closed subspace: (푗 ) (푗 ) 2 푛 퐻푑 : = 푠푝푎푛{푒훼: |훼 | = 휅푗 = 푑} ⊂ 𝒜휆 (𝔹 ). (98) (푗) 2 푛 (푗 ) 2 n (푗 ) By 푄푑 : 𝒜휆 (𝔹 ). → 퐻푑 we denote the orthogonal projection of 𝒜λ(𝔹 ) on to 퐻푑 . 2 Moreover, given (푎, 푏) ∈ ℝ+, we define family of diagonal operators (푗 ) 푎 + 휅푗 퐷푎,푏 푒훼: = 푒훼, (99) 푎 + 푏 + 휅푗 푛 (푗) where 훼 ∈ ℤ+, 휅푗 = |훼 |. These operators will appear in (107) below. Then Lemma (6.3.12) yields (푗 ) (푗 ) ∞ Corollary (6.3.14)[302]:The operators 푄푑 and 퐷푎,푏 belong to the algebra 풯휆(퐿푘−푞푟). ∞ Next we describe a fibration of the compact set 푀(풯휆(퐿푘−푞푟)) of maximal ideals of the ∗ ∞ commutative C -algebra 풯휆(퐿푘−푞푟). We identify a maximal ideal with a corresponding multiplicative functional in the standard way. First we introduce some notation: 푚 푐 Let 휃 = (휃1, . . . , 휃푚) ∈ {0,1} =: 훩, and with ퟏ = (1,1, . . . ,1) we write 휃 = 1 − 휃 for the “complementary” m-tuple. In particular, we have ퟏ푐 = ퟎ = (0,0, . . . ,0). Using the notation퐽휃 = {푗: 휃푗 = 1}, we introduce 휃 ℤ+ = ⊕ ℤ+(푗 ) 푎푛푑 휅휃 = {(휅푗1, . . . , 휅푗|휃| ): 푗푝 ∈ 퐽휃} . 푗∈퐽휃 Given 휃 ∈ 훩, let ∞ (푗 ) 0 for all 푑 ∈ ℤ+, 푖푓 휃푗 = 0 푀휃 = {휇 ∈ 푀(풯휆(퐿푘−푞푟)): 휇 (푄푑 ) = { } . 1 for some 푑 ∈ ℤ+, 푖푓 휃푗 = 1 Lemma (6.3.15)[302]:The following decomposition onto mutually disjoint sets holds ∞ 푀(풯휆(퐿푘−푞푟)) = ⋃ 푀휃 . 휃∈훩 Proof: From the definition of the sets푀휃 it follows that the union is disjoint and varying 휃 ∈ 훩 we cover all possible cases. We note that the set 푀1 admits a simple alternative description. For each 휇 ∈ 푀1 there is a κ ∈ m ℤ+ such that

200

푚 푚 휇(푃 ) = 휇 (∏ 푄(푗 )) = ∏ 휇 (푄(푗 )) = 1, 휅 휅푗 휅푗 푗=1 푗=1

Where 푃휅 is the orthogonal projection defined in (94). Thus, given any operator 퐷훾 = ∞ ∑ 푚 (휌)푃 ∈ 풯 (퐿 ), we have 휌∈ℤ+ 훾 휌 휆 푘−푞푟 휇(퐷훾) = 휇(푃휅)휇(퐷훾) = 휇(푃휅퐷훾) = 훾 (휅)휇(푃휅) = 훾 (휅). m m Identifying the functional μ with κ ∈ ℤ+ , we have that M1 = ℤ+ . Moreover, each functional 휇 ∈ 푀1 can be defined by the formula 〈 〉 휇(퐷): = 퐷푒훼휅 , 푒훼휅 휆, 푘1 푘푚 푛 Where 훼휅: = ((휅1, 0, … ,0), … . , (휅푚, 0, … ,0)) ∈ ℤ+ ×···× ℤ+ = ℤ+. ∞ We note as well that all functional from 푀(풯휆(퐿푘−푞푟))\푀1 map compact operators of the ∞ algebra 풯휆(퐿푘−푞푟) to zero. In order to analyze the sets 푀휃, with 휃 ≠ 1, we mention first that for each 휇 ∈ 푀휃 there is auniquetuple 휅 ∈ ℤ휃 such that 휇(푄(푗)) = 1 for all 푗 ∈ 퐽 . Therefore, we have the following 휃 + 휅푗 휃 decomposition of Mθ in to disjoint sets

푀휃 = ⋃ 푀휃 (휅휃 ), 휃 휅휃∈ℤ+ Where 푀 (휅 ) = {휇 ∈ 푀 : 휇(푄(푗)) = 1forall 푗 ∈ 퐽 }. 휃 휃 휃 휅푗 휃 Note that,we mention for the completeness that none of the points of 푀휃(휅휃) can be reached |휃푐| by subsequences; its topological nature requires to use nets (subnets of ℤ+ ). That is, for each |휃푐| point 휇 ∈ 푀휃(휅휃) there is a net {휅휃푐(훽)}훽∈퐵, valued in ℤ+ , such that 휅 = 휎(휅휃, 휅휃푐(훽)) ∞ tends to 휇 in the Gelfand topology of 푀(풯휆(퐿푘−푞푟)). Here σ is the permutation of 푚- tuples 푐 푚 such that (휅휃, 휅휃 (훽)) = 휎 − 1(휅1, … , 휅푚). In other words, for each 훾 = {훾(휅)}휅∈ℤ+ ∈ ∞ 풯휆(퐿푘−푞푟), we have that 푙푖푚 훾 (휅) = 훾 (휇), (100) 훽∈퐵 휃 푚 Where 휅 = 휎(휅휃, 휅휃푐 (훽)) with (휅휃, 휅휃푐 (훽)) ∈ ℤ+ × 푍휃푐+= ℤ+ , and where we identify 훾(휇) ∞ with 휇(훾), the value of the functional 휇 on the element 훾 ∈ 풯휆(퐿푘−푞푟). We consider a second intrinsic commutative subalgebra 풯λ(εk(h)) of Bk(h), which is generated by a finite set of elementary k-quasi-homogeneous symbols (see the definition in (105)). The structure of this algebra is independent of the weight parameter 휆 and different from ∞ 풯휆(퐿푘−푞푟). its elements are not diagonal operators with respect to the standard orthonormal 2 푛 basis of𝒜휆 (𝔹 ). We start by recalling some notation and results from [193]: 푛 푛 Let (푝, 푞) ∈ ℤ+ × ℤ+ be a pair of orthogonal multi-indices with |푝(푗)| = |푞(푗)| for all 푗 = 1,2, … , 푚. We use the notation in [293] and write 푝̃(푗 ): = (0, … , 0, 푝(푗 ), 0, … , 0) and 푞̃(푗 ) = (0, … , 0, 푞(푗 ), 0, … , 0), 201

Where the entries 푝(푗) and 푞(푗) are at the푗-th position, respectively. For each 푗 = 1, … , 푚 with 푝̅(푗) 푝̅(푗) 푘푗 > 1consider the Toeplitz operator 푇휉 휉 .Aswas shown in [293],we have 푚

푇 ̅ = ∏ 푇 푝̃ 푞̃ (101) 휉푝휉푞 휉 (푗)휉 (푗) 푗=1 훼 푛 Let us recall the action of T 푝̃ 푞̃ on the monomials 푧 , 훼 ∈ ℤ . According to [293] we have 휉 (푗)휉 (푗) + 0, if ∃ℓ ∈ {1, … , 푘푗 } such that훼푗,ℓ < 푞푗,ℓ − 푝푗,ℓ, 훼 푇 푝̃ 푞̃ 푧 = { , (102) 휉 (푗)휉 (푗) ( ) 훼+푝̃(푗 )−푞̃(푗 ) 훾̃푘,푝(j ),푞(j ),휆 훼 푧 , otherwise ( ) 푛 Where the numbers 훾̃푘,푝(푗 ),푞(푗 ),휆 훼 ∈ ℤ+ explicitly are given by ( ) 훾̃푘,푝(푗 ),푞(푗 ),휆 훼 푚 2 훤(푛 + |훼| + 휆 + 1)(훼 + 푢 − 푣 + 푝̃(푗))! = 훤(휆 + 1)(푘푗– 1 + |훼(푗) + 푝(푗)|)! ∏ℓ≠푗(푘ℓ − 1 + |훼(푗)|)! (훼(푗) + 푝̃(푗 ) − 푞̃(푗 )) ! 푚 2 2|훼(푗)|+2푘푗−1 × ∫ (1 − |푟| )휆 ∏ 푟ℓ 푑푟. 휏(𝔹푚) ℓ=1 The integral on the right hand side can be calculated by using Corollary (6.3.3) (훼 + 푝̃ )! (푘 − 1 + |훼( )|)! ( ) (푗) 푗 푗 훾̃푘,푝(푗 ),푞(푗 ),휆 훼 = (푘푗– 1 + |훼(푗) + 푝(푗)|)! (훼(푗) + 푝̃(푗 ) − 푞̃(푗 ))! (훼(푗) + 푝(푗))! (푘푗– 1 + |훼(푗) + 푝(푗)|)! = . (푘푗– 1 + |훼(푗) + 푝(푗)|)! (훼(푗) + 푝(푗) − 푞(푗)) Note that this expression only depends on the portion 훼(푗)of α. For simplicity 푝(1) 푝(푚) 푞(1) 푞(푚) 훷푝,푞(푧): = 휉(1) … 휉(푚) 휉(1) … 휉 (푚) (103) For a 푘-quasi-homogeneous function of degree (푝, 푞). Replacing 푧훼 by the normalized monomial 푒훼(푧) in (86) gives: Lemma (6.3.16)[302]: For 푗 = 1, … , 푚 with 푘푗 > 1the Toeplitz operator 푇훷 acts on the 푝̃(푗),푞̃(푗) n or honor-mal basis [eα: α ∈ ℤ+] by the rule 0, if ∃ℓ ∈ {1, … , 푘푗} such that 훼푗,ℓ < 푞푗,ℓ − 푝푗,ℓ, 푇훷 푒훼 = { ( ) ( ) 푝̃(푗),푞̃(푗) ( ) 훼+ 푝̃ 푗 –푞̃ 푗 휌푘,푝(푗 ),푞(푗 ),휆 훼 푧 , otherwise ( ) The factor 휌푘,푝(푗 ),푞(푗 ),휆 훼 is independent of the weight parameter 휆 and given by (훼(푗) + 푝(푗))! (푘푗 – 1 + |훼(푗)|)! 휌푘,푝(푗),푞(푗)(훼) = . (104) √(훼(푗)! (훼(푗) + 푝(푗) − 푞(푗))! (푘푗 − 1 + |훼(푗) + 푝(푗)|)! In particular, (104) only depends on 훼(푗). Now assume in addition that the tuples 푝(푗) and 푞(푗) are of the form (88), for 푗 = 1, . . . , 푚, and such that (by definition) 훷푝̃(푗),푞̃(푗) ∈ 푅푘(ℎ). Then we have 푝 푞 −푞 푝푗,1 푗,ℎ푗 푗,ℎ푗+1 푗,푘푗 훷푝̃( ),푞̃( )(푧): = 휉 … 휉 휉 … 휉푞 . 푗 푗 1,푗 푗,ℎ푗 푗,ℎ푗+1 푗,푘푗

202

Let 푘푗 > 1 and (ℓ푗, 푟푗) ∈ {1, … , ℎ푗} × {ℎ푗 + 1, … , 푘푗} =: 퐽ℎ,푗 be arbitrary. We define set of so-called elementary k-quasi-homogeneous functions by ( ) ̅ 훹ℓ푗,푟푗 푧 : = 휉푗,ℓ푗휉푗,푟푗 (105) Since |푝(푗)| = |푞(푗)| we can express 푝(푗) + 푞(푗) (non-uniquely) in the form

P(j) + q(j) = ∑ βℓj,rj 퐞ℓj,rj,

(ℓj ,rj)∈Jh,j kj where 퐞ℓ ,r = (0, … , 1, 0, … , 1, 0, … , 0) ∈ ℤ ,and 훽ℓ ,푟 ∈ ℤ+. We also write et for the t-th j j ↑ ↑ + 푗 푗 ℓj ℓj 푘푗 ( ) standard unit vector in ℝ . For each 푗 = 1, … , 푚we now can decompose 훷푝̃(푗),푞̃(푗) z (non- uniquely) in the form of a product

훽 ( ) ( ) ℓ푗,푟푗 훷푝̃(푗),푞̃(푗) 푧 = ∏ 휓ℓ푗,푟푗 푧 (106) (ℓ푗 ,푟푗)∈퐽ℎ,푗 After a straight forward computation using Lemma (6.3.17) we have 푇훹 푇푝̃ ,푞̃ 푒훼 ℓ푗,푟푗 (푗) (푗) 0, 푖푓 ∃푡 ∈ 1, … , 푘 푤푖푡ℎ 훼 < (−푒 – 푝 + 푒 + 푞 ) , 푗 푗,푡 ℓ푗 (푗) 푟푗 (푗 ) 푡

2 (훼 + 푝̃ + 풆̃ℓ )! [(푘푗 − 1 + |훼(푗)|!)] = (푗 ) 푗 , 푒훼+̅풆 +푝̃ –풆̃ –푞̃ ℓ푗 (푗 ) 푟푗 (푗 ) √훼! (훼 + 풆̃ + 푝̃ – 풆̃ − , 푞̃ )(푘 − 1 + |훼 + 푝 |)! (푘 − 1 + |훼 |)! ℓ푗 (푗 ) 푟푗 (푗 ) 푗 (푗) (푗) 푗 (푗) { otherwise.

On the other hand it is easy to verify that 푇훹 푇푝̃ ,푞̃ 푒훼 = 푇훷 푒훼 ℓ푗,푟푗 (푗) (푗) 푝̃(푗),+풆̃ℓ 푞̃(푗)+풆̃ 푗 푟푗 0. , if∃푡 ∈ 1, … , 푘 with훼 < (−푒 – 푝 + 푒 + 푞 ) 푗 푗,푡 ℓ푗 (푗) 푟푗 (푗 ) , 푡

(훼 + 푝̃ + 풆̃ ) ! (푘 − 1 + |훼 |)! = (푗 ) ℓ푗 푗 (푗) 푒훼+풆̃ +푝̅ –풆̃ −푞̅ ℓ푗 (푗 ) 푟푗 (푗 ) √훼! (훼 + 푝̃ – 풆̃ − , 푞̃ − 푒 ) (푘 + |훼 + 푝 |)! (푗 ) ℓ푗 (푗 ) 푟푗 푗 (푗) (푗) { otherwise. 푛 by comparing the action of the above operators on [푒훼: 훼 ∈ ℤ+] we conclude that |푝(푗)| 푇휓 푇훷 − 푇휓 푇훷 = 푇휓 푇훷 . ℓ푗,푟푗 푝̃(푗),푞̃(푗) ℓ푗,푟푗 푝̃(푗),푞̃(푗) ℓ푗,푟푗 푝̃(푗),푞̃(푗) 푘푗 + |훼(푗) + 푝(푗)| Or equivalently

203

푘푗+|훼(푗)| 푇휓ℓ ,푟 훷푝̃ ,푞̃ = 푇휓ℓ ,푟 푇훷푝̃ ,푞̃ 푒훼 (107) 푗 푗 (푗) (푗) 푘푗+|훼(푗)+푝(푗)| 푗 푗 (푗) (푗) using the notation in (99) we can rewrite (107) in the form (푗) 푇휓 푇훷 = 퐷 푇휓 푇훷 ℓ푗,푟푗 푝̃(푗),푞̃(푗) 푘푗 ,|푝(푗)| ℓ푗,푟푗 푝̃(푗),푞̃(푗) Note that according to Corollary (6.3.14) one has 퐷(푗) ∈ 풯 (퐿∞ ). 푘푗 ,|푝(푗)| 휆 푘−푞푟

Now, we return to symbols Φ푝̃(푗),푞̃(푗)of the form (106). With j ∈ {1, … , m} consider the sequences {훾̃푗(푟)}푟∈ℤ+defined by |푝(푗 )|−1 푘푗 + 푟 (푘푗 + 푟)|푝(푗)|(푘푗 − 1 + 푟)! 훾̃푗(푟) ≔ ∏ = , (108) 푘푗 + 푟 + ℓ (푘 − 1 + 푟 + |푝 |)! ℓ=1 푗 (푗) ( ) n and note that 푙푖푚푟→∞훾̃푗 푟 = 1. Let 퐷훾푗be the diagonal operator acting one α, for all α ∈ ℤ+, by

퐷훾푗 푒훼 = 훾̃푗(훼(푗))푒훼. ∞ then it follows from Lemma (6.3.12) that Dγj ∈ 풯λ(Lk−qr). Moreover, by induction it is clear from (87) that |푝(푗 )|−1 훽 ℓ푗,푟푗 푇 푝̃(푗) 푞̃(푗) = 푇훷 = 퐷훾 ∏ 푇 . (109) 휉 휉 푝̃(푗),푞̃(푗) 푗 푇훹 ℓ푗,푟푗 ℓ=1 We also need the exact action of a product of Toeplitz operators with elementary k-quasi- homogeneous symbols.

Lemma (6.3.17)[302]:Let 훷푝̃(푗),푞̃(푗) be given by (106) and define the operator A appearing in (89) by 훽ℓ ,푟 퐴 ≔ ∏ 푇 푗 푗 . (110) 푇훹 ℓ푗,푟푗 (ℓ푗 ,푟푗)∈퐽ℎ,푗 푛 Then A acts on the basis [푒훼: 훼 ∈ ℤ+]by the rule 0, if∃ℓ ∈ {ℎ + 1, … , 푘 }: 훼 < 푞 , 푗 푗 푗,ℓ 푗,ℓ

−1 퐴 푒훼 = 퐷훾 훷푝̃ ,푞̃ 푒훼 = 푗 (푗) (푗) 1 (훼(푗)+푝(푗))! √ . 푒 . otherwise. |푝 | 훼+푝̃(푗)−푞̃(푗) (푗) (훼(푗)+푞(푗))! {(푘푗 + |훼(푗)|) n In particular, the operator A acts on [eα: α ∈ ℤ+].in the form

퐴푒훼 = 푚(훼(푗))푒훼+푝̃(푗)−푞̃(푗) , (111) 푛 Where the scalar factors푚(훼(푗))only depend on the j-th portion α(j)of [푒훼: 훼 ∈ ℤ+]. Proof: The first assertion follows from Lemma (6.3.16), (109), and the relation (훼 + 푝̃(푗 ))! (훼(푗) + 푝(푗)) =

√훼! (훼 + 푝̃(푗) − 푞̃(푗))! √훼(푗)! (훼(푗) + 푝(푗) − 푞(푗))! The second statement is an immediate consequence of the first. 204

Remark(6.3.18)[302]:Recall that the decomposition (106) of 푇훷 is not unique. However, 푝̃(푗),푞̃(푗) Lemma (6.3.18) shows that the map 훽ℓ ,푟 훷 → ∏ 푇 푗 푗 . = 퐴 푝̃(푗),푞̃(푗) 푇훹 ℓ푗,푟푗 (ℓ푗 ,푟푗)∈퐽ℎ,푗 Is well-defined, i.e. the operator A is independent of there presentation (106) of 푇훷 . 푝̃(푗),푞̃(푗) We associate to the space 푅푘(ℎ) the following set ℇ푘(ℎ) of elementary 푘-quasi-homogeneous functions of quasi-homogeneous degree (1,1) 푚

ℇ푘(ℎ) ≔ ⋃ ℇ푘,푗(ℎ). (112) 푗=1 Where for each 푗 ∈ {1, … , 푚} with 푘푗 = 1 we putℇ푘,푗(ℎ) = ∅, and in the case where 푘푗 > 1wedefine ̅ ℇ푘,푗(ℎ) ∶= {휓ℓ ,푟 (푧) = 휉푗 , 휉 푗 , ∶ (ℓ푗, 푟푗) ∈ {1, … , ℎ푗} × {ℎ푗 + 1, … , 푘푗} = 퐽ℎ,푗}. (113) 푗 푗 ℓ푗 푟푗 푚 Clearly, ℇ푘,푗(ℎ)contains ∑푗=1 ℎ푗(푘푗 − ℎ푗) elements. These symbols ets define the corresponding commutative Toeplitz Banach algebras 풯휆(ℇ푘,푗(ℎ))and 풯휆(ℇ푘(ℎ)). The following result has been proved in [193]. 푚 Proposition (6.3.19)[302]:Let k = (k1, … , k푚) ∈ ℤ+ . For each pair of orthogonal multi- ∞ indices 푝 and 푞 with |푝(푗)| = |푞(푗)| for all 푗 = 1, … , 푚 and each 푎 = 푎(푟1, … , 푟푚) ∈ 퐿푘−푞푟 we have

푇푎푇훷푝,푞 = 푇훷푝,푞푇푎 = 푇푎훷푝,푞. ∞ Now formula (101), Proposition (6.3.19), the decomposition (109) with 퐷훾푗 ∈ 풯휆(퐿푘−푞푟), and the notation in Theorem (6.3.2) permit us to essentially reduce the set of generators of the algebra ℬ푘(ℎ).Namely, we have: Theorem (6.3.20)[302]:The following commutative Banach algebras coincide ∞ 풯휆 (퐿푘−푞푟 ∪ ℇ푘(ℎ)) = 퐵푘(ℎ) (114) The algebra on the left hand side of (114) is clearly generated by its two commutative ∞ ∗ subalgebras 풯휆(퐿푘−푞푟) and 풯휆(ℇ푘(ℎ)). Whereas the first one is a commutative C -algebra, the second algebra is just a commutative Banach algebra and is not in variant under the ∗- 2 푛 ∞ operationof ℒ(𝒜휆(𝔹 )). Recall that 풯휆(퐿푘−푞푟)was analyzed. Our next aim is to analyze the structure of the finitely generated algebra 풯휆(ℇ푘(ℎ)). As was already mentioned, the structure of the algebra풯휆(ℇ푘(ℎ))does not depend on the weight parameter 휆. Thus in what follows we will always assume that 휆 = 0, i.e., the operators will 2 푛 2 푛 act on the unweighted Bergman space 𝒜 (𝔹 ): = 𝒜0(𝔹 ) and clarify the structure of the algebra 풯(ℇ푘(ℎ): = 풯0(ℇ푘(ℎ).

By Lemma (6.3.16)the Toeplitz operator with symbol 휓ℓ푗,푟푗 (푧) = 휉푗,ℓ푗휉푗,푟푗 , defined in (105), 푛 acts on the orthonormal basis [푒훼: 훼 ∈ ℤ+]as follows

205

0, if 훼푗 = 0, 푟푗

푇휓 푒훼 = √(푎푗ℓ + 1)푎푗푟 (115) ℓ푗,푟푗 푗 푗 . 푒훼 + 푒̃ℓ푗 − 푒̃푟푗otherwise. { 푘푗 + |훼(푗)| 푛 Let 퐻 bean abstract Hilbert space whose orthonormal basis is enumerated by ℤ+. If we consider set of operators 푇ℓ푗,푟푗on 퐻 which act on the basis elements according to (115), then the unital Banach algebra generated by such operators would be isomorphic and isometric to the algebra 푇(ℇ푘(ℎ)Now, we will choose a particular realization of 퐻 which is different from our present setting of weighted Bergman spaces over the unit ball together with a set of operators 푇ℓ푗,푟푗. We introduce the Segal–Bargmann space and certain Toeplitz operators acting on it. The main simplification we achieve in this way lies in the additional tensor product structure of the multi-dimensional Segal–Bargmann space. This feature will allow us tore present the corresponding Toeplitz operator algebra(and hence the algebra 풯 (ℇ푘(ℎ)))in the form of a tensor product, cf. (130). Westar with ℂn equipped with the standard Gaussian measure −푛 −|푧|2 푑휇푛(푧): = 휋 푒 푑푣(푧), where dv denotes the Lebesgue measure on ℂ푛 ≃ ℝ2푛. Denote by 퐻(ℂ푛) the space of entire functions on ℂ푛. The Segal–Bargmann space (or Fock space) ℱ2(ℂ푛) is defined as 2 푛 푛 푛 ℱ (ℂ ) ∶= 퐻(ℂ ) ∩ 퐿2(ℂ , 푑휇푛). 푛 2 푛 푛 Denote by 푷 the orthogonal projection from 퐿2(ℂ , 푑휇푛)ontoℱ (ℂ ). with 푔 ∈ 퐿∞(ℂ ) the 2 푛 Toeplitz operator 푻푔 with symbol g acts on ℱ (ℂ ) in standard way 2 푛 2 푛 푻푔: 푓 ∈ ℱ (ℂ ) → 푷(푔푓) ∈ ℱ (ℂ ). 푚 푛 Given 푘 = (푘1, … , 푘푚) ∈ ℤ+ with |푘| = 푛, we interpret ℂ as a product space ℂ푛 = ℂ푘1 × … × ℂ푘푚 푛 푘푗 and we write 푧 = (푧(1), … , 푧(푚)) ∈ ℂ , where 푧(푗): = (푧푗, 1, … , 푧푗, 푘푗) ∈ ℂ . With respect to polar coordinates we express 푧(푗) ≠ 0 in the form 푧(푗) = 푟푗휉(푗), where 푧(푗) 2푘푗−1 휉(푗) = ∈ 푆 and 푟푖 ≔ |푧(푗)| ∈ ℝ+ |푧(푗)| Let (ℓ푗, 푟푗) ∈ {1, … , ℎ푗} × {ℎ푗+1, … , 푘푗}. We interpret the elementary 푘-quasi-homogeneous ̅ n functions 훹ℓ푗,푟푗: = 휉ℓ푗,푟푗휉ℓ푗,푟푗 as elements in L∞(ℂ ). It can be checked by an easy calculation (see [104]) that 0, if훼푗 = 0, 푟푗

푇휓 푓훼 == √(푎푗ℓ + 1)푎푗푟 ℓ푗,푟푗 푗 푗 . 푓훼 + 푒̃ℓ푗 + 푒̃푟푗 otherwise. { 푘푗 + |훼(푗)| Here the monomials 1 훼 푛 푓훼 = (푧) 푧 with 훼 ∈ ℤ+ √훼! form the standard orthonormal basis in ℱ2(ℂ푛). 206

That is the set of Toeplitz operators 푇휓 , where 푇휓 ∈ ℇ푘(ℎ), obeys the relations (115). ℓ푗,푟푗 ℓ푗,푟푗 Denote by 픗(ℇ푘(ℎ)) the unital Banach algebra generated by 푇휓 with 푇휓 ∈ ℇ푘(ℎ). ℓ푗,푟푗 ℓ푗,푟푗 According to the above remarks we have Lemma (6.3.21)[302]:The assignment 푇휓 → 푇휓 extends to an isometric isomorphism ℓ푗,푟푗 ℓ푗,푟푗 between the Banach algebras 풯 (ℇ푘 (ℎ)) and 픗(ℇ푘(ℎ)). We start with a classical result on Toeplitz operators with continuous symbols, see [162,291]. Denote by 퐶(𝔹̅푛) the algebra of all functions continuous on the closed unit ball 𝔹̅푛, and let 풯(퐶(𝔹̅푛))be the C∗-algebra generated by all Toeplitz operators Ta act in g on the Bergman space 𝒜2(𝔹̅푛)and having symbols푎 ∈ 퐶(𝔹̅푛). Theorem (6.3.22)[302]:(See [162,291].) The algebra 풯(퐶(𝔹̅푛)) is irreducible and contains the ideal 퐾 of all compact operators on 𝒜2(𝔹푛). Each operator 풯 ∈ 풯(퐶(𝔹̅푛)) has the form 푛 푇 = 푇푎 + 퐾, where 푎 ∈ 퐶(𝔹 ) and 퐾 ∈ 풦. (116) The quotient algebra 풯̂(퐶(𝔹̅푛)) = 풯(퐶(𝔹̅푛))/풦 is isomorphic and isometric to 퐶(푆2푛−1), and under their identification the homomorphism 휋: 풯(퐶(𝔹̅푛)) → 풯̂(퐶(𝔹̅푛)) ≅ 퐶(푆2푛−1) (117) is given by 휋: 푇 = 푇푎 + 퐾 → 푎|푆2푛−1 . we note that the representation (116) is not unique. An ambiguity comes from the fact that for 푛 2푛−1 any two functions 푎, 푎1 ∈ 퐶(𝔹 ) with (푎 − 푎1)|푆 ≡ 0 the difference 푇푎 − 푇푎1 is compact, and thus 푇푎 + K = 푇푎1 + 퐾1, for 퐾1 = 퐾 + (푇푎 − 푇푎1) ∈ 풦. In order to make the representation (116) unique (and in a sense canonical) we proceed as follows. We introduce the C∗-algebra 퐻(퐶(푆2푛−1)) consisting of all functions that are homogeneous of order zero on 𝔹n and continuous on 푆2푛−1 ≅ 휕𝔹푛.Let 풯(퐻(퐶(푆2푛−1))) be the C∗-algebra generated by all Toeplitz operators acting on the Bergman space 𝒜2(𝔹푛) having symbols in 퐻(퐶(푆2푛−1)). With any pair of functions 푎 ∈ 퐶(𝔹̅푛) and 푎̂ ∈ 퐻(퐶(푆2푛−1)) such 푛 that (푎 − 푎̂)|푆2푛−1 ≡ 0 we have 푇푎 − 푇푎̂ ∈ 풦. Moreover, the algebras 풯(퐶(𝔹 )) and 풯(퐻(퐶(푆2푛−1)))consist of the same operators, in spite of the fact that they have different systems of generators. Each operator 푇 ∈ 풯(퐶(𝔹푛)) = 풯(퐻(퐶(푆2푛−1))) admits the (unique) canonical representation 2푛−1 푇 = 푇푎̂ + 퐾, where 푎̂ ∈ 퐻(퐶(푆 )) and 퐾 ∈ 풦. (118) we note that none of the above operator 푇푎̂is compact (unless 푇푎̂ ≡ 0 and thus 푇푎̂ = 0), and the essential spectrum of 푇â is given by 2푛−1 푛 푒푠푠 − 푠푝 푇푎̂ = 푎̂(푆 ) = 푎̂(𝔹 ). ‖ ‖ This implies the estimate 푟(푇푎̂) ≥ 푎̂ 퐿∞ for the spectral radius r(Tâ ) of Tâ which, together ‖ ‖ ‖ ‖ with 푇푎̂ ≤ 푎̂ 퐿∞, shows that ‖ ‖ ( ) ‖ ‖ ( 2푛−1) 푇푎̂ = 푟 푇푎̂ = 푎̂ 퐿∞ for all 푎̂ ∈ (퐻(퐶 푆 ). (119) the above observations permit us to give another equivalent description of the quotient algebra 풯̂(퐶(𝔹̅푛)) = 풯̂(퐻(퐶(푆2푛−1))) ≅ 퐶(푆2푛−1). Indeed, the assignment ̂ 푇̂ = 푇푎̂ + 푘 ↦ 푇푎̂ 207 gives a Banach space isometric isomorphism 푛 2푛−1 2푛−1 풯̂(퐶(𝔹̅ )) → 푇퐻(퐶(푆 )): = {푇푎̂: 푎̂ ∈ (퐻(퐶(푆 ))}. (120) 2푛−1 This isomorphism becomes algebraic after introducing the multiplication law in 푇퐻(퐶(푆 )) as 푇푎̂1 ⊙ 푇푎̂2 = 푇푎̂1푎̂2. With our previous notation let 휆 = 0, 푛 ∈ ℕ and 푚 = 1, so that 푘 = (푛), and ℎ = (ℎ), with 1 < ℎ < 푛 − 1. Let the tuple 휓 = (휓1, . . . , 휓훾) (121) be the (somehow) ordered set of 훾: = ℎ(푛 − ℎ) elementary quasi-homogeneous symbols in 푧푗푧푙̅ ℇ (ℎ) = {휓 : = ∶ 푗 = 1, . . . , ℎ, 푙 = ℎ + 1, . . . , 푛}, 푘 푗,푙(푧) |푧|2 ( ) And let 푇 휓 = (푇휓1, . . . , 푇휓훾 ) be the ordered set of the corresponding Toeplitz operators. We 2푛−1 introduce the Banach algebra ℬ(ℇ푘(ℎ)), asubalgebra of 퐻(퐶(푆 )), being the unital algebra enerated by all elementary quasi-homogeneous functions from ℇ푘(ℎ), as well as the unital 2푛−1 Banach algebra 풯(ℇ푘(ℎ)), a subalgebra of 풯(퐻(퐶(푆 ))), which is generated by the Toeplitz operators in 푇(휓). Note that in our particular case (푚 = 1) the functions 휓푗 ∈ 휓 continuously extend to the sphere 푆2푛−1 ≅ 휕𝔹푛, and thus we are in the framework of the previous subsection. The general case of 푚 > 1 where such a continuity on the boundary is not fulfilled will be treated in the section. We define the following tuples of multi-indexes 푛 푛 푃: = {(푝, 푞) ∈ ℤ+ × ℤ+: 푝 = (푝1, . . . , 푝ℎ, 0, . . . , 0), 푞 = (0, . . . ,0, 푞ℎ+1 , . . . , 푞푛), |푝| = |푞|}. 훾 If (푝, 푞) ∈ 푷, then there is 훼 = 훼(푝, 푞) ∈ ℤ+ such that 푝 푞 푧 푧̅ 훼 = 휓훼1 … 휓 훾 . (122) |푧|2|푝| 1 훾 푝 푞 Consider the space of polynomials ℱ푷: = {휑(푝,푞)(푧): = 푧 푧̅ : (푝, 푞) ∈ 푷}. It is easy to see that 2푛 푛 all the elements of ℱ푷 are harmonic polynomials on ℝ ≅ ℂ . Moreover, we have: n Lemma (6.3.23)[302]: The functions in ℱ푷 are orthogonal in L2(𝔹 ). If were strict them to a 2푛−1 2푛−1 sphere 푟푆 of radius 푟 ∈ (0,1), then they define orthogonal functions in 퐿2(푟푆 , 휎) where 휎 denotes the usual surface measure on 푟푆2푛−1.

Proof: With (푝, 푞), (푟, 푠) ∈ 푃 and suitable numbers 퐶(푝, 푞) > 0 it holds ∫ 푧푝푧̃푞 푧̅̅푟̅̅푧̅̃̅푠 푑푣(푧) = ∫ 푧푝+푠푧̃푞+푟 푑푣(푧)

𝔹푛 𝔹푛

푝1 푝ℎ 푠ℎ+1 푠푛 푟1 푟ℎ 푞ℎ+1 푞푛 = ∫ 푧1 … 푧ℎ 푧ℎ+1 … 푧푛 푧1̃ … 푧̃ℎ 푧ℎ+1 … 푧̃푛 푑푣(푧) = C(푝, 푞)훿푝,푟훿푠,푞. 𝔹푛 The second assertion follows by the same argument. 2 푛 푛 Let 퐴 ∈ ℒ(퐴 (𝔹 )), then we write 퐵[퐴](푧) ∈ 퐿∞(𝔹 ) for the Berezin transform of A. More precisely, 퐵[퐴](푧) is defined by 1 퐵[퐴](푧) ≔ 2 퐴〈퐾(·, 푧), 퐾(·, 푧)〉0, (123) ‖퐾(·, 푧)‖0 Where K: 𝔹n × 𝔹n → C is the reproducing kernel of the unweighted Bergman space 퐴2(𝔹푛)

208

1 퐾(푢, 푧) = . (1 − 〈푢, 푧〉)푛+1 2 푛 휔 푛 휔 푛 Recall that 퐵: ℒ(퐴 (𝔹 )) → 퐶푏 (𝔹 ) is linear and injective. Here we write 퐶푏 (𝔹 ) for the space of bounded real analytic functions on 𝔹n. 훾 Lemma (6.3.24)[302]: Let 훼 ∈ ℤ+ and (푝, 푞) ∈ 푷 be related to 훼 via (122), then 훼 퐵 [푇 훼1 … 푇 훾] (푧) = 푧푝푧̃푞퐻 (|푧|), (124) 휓1 휓훾 |푝| Where the function H|p| fulfills lim H|p| (ρ) = 1 and it has the explicit form ρ↑1 2 (푛 + |푝|)! 퐻 (|푧|) = (1 − |푧|2)푛+1 . |푝| (|푝| − 1)! 푛! ∞ ( | |) 2 푒 − 푛 + 푝 푠 2|푝|−1 × ∫ 푛+|푝|+1 푠 푑푠 (125) 2 −푠2 0 (1 − |푧| 푒 ) 훼 Proof: According to Lemma (6.3.17) the operator product 푇 훼1 … 푇 훾 acts on the orthonormal 휓1 휓훾 n 2 n basis [eβ: β ∈ ℤ+] of A (𝔹 ) in the form 훼 푇 훼1 … 푇 훾e = 푚(훽)푒 휓1 휓훾 β 훽+푝−푞, Where m(β) is defined by 1 (훽 + 푝)! √ , 푖푓 훽 − 푞 ≥ 0 (componentwise), 푚(훽): = {(푛 + |훽|)|푝|! (훽 − 푞)! 0, otherwise Hence it follows 훼 훼 〈푇 훼1 … 푇 훾푒 퐾(·, 푧), 퐾(·, 푧)〉 = ∑ 〈푇 훼1 … 푇 훾푒 푒 , 푒 〉 푒̅̅̅̅(̅푧̅̅)푒 (푧) 휓1 휓훾 훽 0 휓1 휓훾 훽 훽 휂 0 훽 휂 푛 훽,휂∈ℤ+ ̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅ = ∑ 푚(훽)푒훽(푧)푒훽+푝−푞 (푧) = ∑ 푚(훽 + 푞)푒훽(푧)푒훽+푝(푧) = (∗). 푛 훽푗 ≥푞푗 훽∈ℤ+ Using the explicit form of 푚(훽 + 푞) above and the expression for eβ(z) in (86) together with|푝| = |푞|we obtain ∞ 푧푝푧̃푞 훤(푛 + |훽| + |푝| + 1) |푧훽|2 푧푝푧̃푞 훤(푛 + ℓ + |푝| + 1) |푧|2ℓ (∗) = ∑ |푝| = ∑ |푝| . 푛! 푛 (푛 + |훽| + |푝|) 훽! 푛! (푛 + ℓ + |푝|) ℓ! 훽∈ℤ+ ℓ=0 In the last equation we have applied the multinomial theorem. In order to obtain an integral representation of (∗) we use the well-known relations 1 1 2 = ∫ 푒−(푛+ℓ+|푝|)|푥| 푑푥, (푛 + ℓ + |푝|)|푝| 휋|푝| 𝔹2|푝| ∞ 푢ℓ 훤 (푡) ∑ 훤(푡 + ℓ) = , ℓ! (1 − 푢)푡 ℓ=0 Which hold true for 푢 ∈ [0.1]and 푡 ≥ 0. Interchanging summation and integration implies 209

∞ 2 ℓ 푧푝푧̃푞 1 (푒−|푥| |푧|2) ( ) −(푛+|푝|)|푥|2 ( | | ) ∗ = |푝| ∫ 푒 ∑ Γ 푛 + ℓ + 푝 + 1 푑푥 푛! 휋 2|푝| ℓ! ℝ ℓ=0 2 푧푝푧̃푞 (푛 + |푝|)! 푒−(푛+|푝|)|푥| = ∫ 푑푥 |푝| 2 푛+|푝|+1 푛! 휋 ℝ2|푝| (1 − 푒−|푥| |푧|2) Finally, by changing to polar coordinates in the last integral, we obtain the expression (125). The limit behavior 푙푖푚 퐻|푝| (휌) = 1 can be directly checked from (125). However, we can also 휌↑1 give a more abstract argument. By (118) we can write the product of Toeplitz operators in the form 훼1 훼훾 푇 … 푇 = 푇 푧푝푧̃푞 + 퐾, (126) 휓1 휓훾 |푧|2|푝| Where 퐾 is a compact operator. If 푓 is a continuous function in a neighborhood of 푆2푛−1 and 푛 2푛−1 bounded on 𝔹 then it is well-known that 푙푖푚 퐵[푇푓](휌푧) = 푓(푧) for all 푧 ∈ 푆 (cf.[21]). 휌↑1 Moreover, it holds 푙푖푚 퐵[퐾](푧) = 0. From (126) we obtain |푧|↑1 훼 푙푖푚 퐵 [푇 훼1 … 푇 훾] (휌푧) = 푧푝푧̃푞 휌↑1 휓1 휓훾 2푛−1 For all푧 ∈ 푆 .Together with (124) it follows that 푙푖푚 퐻|푝| (휌) = 1. 휌↑1 Recall (see (117)) that the Banach algebra homomorphism π: T(C(𝔹n)) → 퐶(푆2푛−1) is given by 휋(푇푎 + 퐾) = 푎|푆2푛−1 . Here 퐾 is compact and a ∈ C(𝔹̅n) is continuous up to the boundary. Proposition (6.3.25)[302]:The restriction of 휋 to the algebra 풯(ℰ푘(ℎ)) is injective. Proof: Let 푇 ∈ 풯(ℰ푘(ℎ)) with 휋( 푇) = 0, then we want to show that 푇 = 0. Choose sequence 훼 푇 = ∑ 푎 (ℓ)푇 훼1 … 푇 훾 ∈ 풯(ℰ (ℎ)) ℓ 훼 휓1 휓훾 푘 푛 훼∈ℤ+ Such that 푙푖푚 푇ℓ = 푇 in the norm topology. Here for each ℓ ∈ ℤ+only finitely many ℓ→∞ 훼 coefficients푎 (ℓ)are non-zero. As was already mentioned the operator product 푇 훼1 … 푇 훾 훼 휓1 휓훾 admits the decomposition 훼 훼 푇 훼1 … 푇 훾 = 푇 훼1 … 푇 훾 + 퐾 , 퐾 ∈ 풦, 휓1 휓훾 휓1 휓훾 훼 훼 훼 And we have 휋(푇 훼1 … 푇 훾) = 푧푝푧̃푞|푧|−2|푝| = 푧푝푧̃푞, where (p, q) and α are related sin (122). 휓1 휓훾 Since π is continuous we have 푝 푞 0 = 휋(푇) = 푙푖푚 휋(푇ℓ) = 푙푖푚 ∑ 푎훼(푝, 푞)(ℓ)푧 푧̃ , ℓ→∞ ℓ→∞ 푝,푞 Where the convergence on the right hand side is in 퐶(푆2푛−1). In particular, the convergence 2푛−1 takes place in 퐿2(푆 , 휎). Due to the orthogonality result in Lemma (6.3.23) it follows

210

푝 푞 2 푝 푞 2 ‖∑ 푎 (푝, 푞)(ℓ)푧 푧̃ ‖ = ∑|푎 (푝, 푞)(ℓ)| ‖푧 푧̃ ‖ 2푛−1 → 0(ℓ → ∞). 훼 훼 퐿2(푆 ,휎 ) 푃,푞 2푛−1 푝,푞 퐿2(푆 ,휎 ) as a consequence we have that 푙푖푚 푎훼(푝, 푞)(ℓ) = 0. ℓ→∞ Now we consider the sequence of Berezin transforms 퐵[푇ℓ]. According to Lemma (6.3.24) we have for all ℓ ∈ ℤ+ ∞ 푝 푞 푝 푞 퐵[푇ℓ](푧) = ∑ 푎훼(푝, 푞)(ℓ)푧 푧̃ 퐻|푝|(|푧|) = ∑ 퐻푙(|푧|) ∑ 푎훼(푝, 푞) (ℓ)푧 푧̃ . 푝,푞 푙=1 (푝,푞) |푝|=|푞|=푙 by continuity of the Berezin transform, it follows that푙푖푚 퐵[푇ℓ] (푧) = 퐵[푇](푧)uniformly for ℓ→∞ 푛 2푛−1 푧 ∈ 𝔹 . In particular, if we fix 푟 ∈ (0,1) and restrict 퐵[푇ℓ] to 푟푆 , then we obtain a 2푛−1 p q 2푛−1 convergent sequence in 퐿2(푟푆 ) Since z z̃ are orthogonal in 퐿2(푟푆 ) and by applying 푙푖푚 푎훼(푝, 푞)(ℓ) = 0, we obtain ℓ→∞ 푙푖푚 퐻|푝|(|푧|) 푎훼(푝,푞)(ℓ) = 0 ℓ→∞ 2푛−1 for all (푝, 푞) and therefore 퐵[푇ℓ](푧) converges to zero in 퐿2(푟푆 ). Since r was arbitrary it follows that 푙푖푚 퐵 [푇ℓ](푧) = 퐵[푇](푧) = 0 a.e. on Bnand from the continuity of 퐵[푇] we see ℓ→∞ that 퐵[푇] identically vanishes on 𝔹푛. Since the Berezin transform B is injective on bounded operators we have 푇 = 0. Corollary (6.3.26)[302]:The algebra 풯(ℰk(h)) does not contain any non-zero compact operator. Each element 푇 ∈ 풯(풯(ℰ푘(ℎ))) admits a unique representation 푇 = 푇휓 + 퐾휓, (127) ∞ Where 휓 ∈ 퐵(ℰ푘(ℎ)) and 퐾휓is the compact operator from 퐾 ∩ 풯0(퐿푘−푞푟 ∪ ℰ푘(ℎ)) uniquely determined by휓;i.e.,if both operators T1 = Tψ + K1 and 푇2 = 푇휓 + 퐾2 belong to 푇(ℰ푘(ℎ)), then 퐾1 = 퐾2,and thus 푇1 = 푇2. Proof: According to Proposition (6.3.24)we know that the homomorphism π is injective on 풯(ℰ푘(ℎ)); as it vanishes on compact operators, we have that 퐾 ∩ 풯(ℰ푘(ℎ)) = {0}. Representation (127) follows from(118). Assuming that both 푇1 = 푇휓 + 퐾1 and푇2 = 푇휓 + 퐾2 belong to 풯(ℰ푘(ℎ)), we have 푇1 − 푇2 = 퐾1 − 퐾2 ∈ 퐾 ∩ 풯(ℰ푘(ℎ)) = {0}. Thus 퐾1 = 퐾2 is uniquely determined by 휓. We mention that the latter result remains true for the algebras 풯휆(ℰ푘(ℎ))with the only ∞ difference that 퐾휓 ∈ 퐾 ∩ 풯휆(퐿푘−푞푟 ∪ ℰ푘(ℎ)). We mention that the exact form of the compact operator in (127) can be easily figured out. Indeed, let ℱ be a dense subset of 풯(ℰ푘(ℎ))consisting of finite sums of finite products of its generators. For elements of ℱ the concrete form of the compact operator in (127) can be obtained using (108) and (109). Let now {푇푝}푝∈ℕ, where 푇푝 = 푇휓푝 + 퐾휓푝 ∈ ℱbe a sequence that uniformly converges to 푇 = 푇휓 + 퐾 ∈ 풯(ℰ푘(ℎ))\ℱ. Then the sequence {푇푝}푝∈ℕ converges to 풯̂ in the quotient algebra 풯̂(퐻(퐶(푆2푛−1))). The isomorphism (120) implies that 푇휓 → 푇휓 uniformly, and thus 퐾 = 푙푖푚 퐾휓 . 푝 푝→∞ 푝 211

We have then 풯(ℰ푘(ℎ)) = 풯(ℰ푘(ℎ))/(풯(ℰ푘(ℎ)) ∩ 풦) ≅ (풯(ℰ푘(ℎ)) + 풦)/풦 ⊂ 풯 (퐻(퐶(푆2푛−1)))/풦 ≅ 퐶(푆2푛−1). (128)

Lemma (6.3.27)[302]: For each휓푗,푙 ∈ ℰk(h), the spectrum of 푇휓푗,푙, is given by sp푇휓푗,푙 = 1 𝔻̅ (0, ). 2 1 Proof:Follows from two facts: 푒푠푠 − 푠푝 푇 =Range 휓 | 2푛−1 = 𝔻̅ (0, ), and the spectral 휓푗,푙 푗,푙 푆 2 1 radius of 푇 is equal to which follows from (119). 휓푗,푙 2 We recall the notion of the joint spectrum (see [285] or [27]). Let A be commutative Banach algebra with identity and let 푥1, … , 푥푛 ∈ 퐴. The joint spectrum of푥1, … , 푥푛 is the subset 푛 휎퐴(푥1, … , 푥푛)of ℂ defined by 휎퐴(푥1, . . . , 푥푛) = {(휑(푥1), . . . , 휑(푥푛)): 휑 ∈ 푀(퐴)}, Where 푀(퐴) is the compact set of maximal ideals (≡multiplicative functionals) of 퐴. 2n−1 By (128), the algebra 풯(ℰ푘(ℎ)) is isomorphic to the unital subalgebra of C(S ) generated 2푛−1 by the elements of 휓 in (101), with the following assignment: 푇휓푗 ⟼ 휓푗|푆 . Identifying them we calculate the joint spectrum of elements of 푇(휓), relative to 퐶(푆2푛−1), as the joint spectrum of 휓 in the algebra 퐶(푆2푛−1). Lemma (6.3.28)[302]: The joint spectrum of the Toeplitz operators with symbols in 휓 is given by 휎(푇 (휓)) = 휓(푆2푛−1) ⊂ ℂℎ(푛−ℎ). Proof: As 푆2푛−1 is the compact set of maximal ideals of 퐶(푆2푛−1),we have 2푛−1 휎(푇 (휓)) = 휎퐶(푆2푛−1)(휓 ) = 휓(푆 ). At the same time the unital Banach algebra 풯(ℰ푘(ℎ))it self, considered as a finitely generated algebra by elements of 푇(휓) is isomorphic to the “polynomial” algebra 푃(휎(푇(휓))), and, by [285], its compact set of maximal ideals coincides with the poly nominally convex hull휎̂(푇(휓)) of σ(푇(휓)), i. e. 푀(풯(ℰ푘(ℎ)) = 휎̂(푇(휓)). From [285] yields Theorem (6.3.29)[302]:The Banach algebra 풯(ℰ푘(ℎ)) is isomorphic to the algebra 푃(휎̂(푇(휓))). We proceed now with the description of the set 휎(푇(휓)).With 푟 ∈ ℕ and 푠 > 0 let 𝔹̅푟(0, 푠) be the closed ball in ℝr of radius s centered at the origin. Example(6.3.30)[302]:Given 푛 > 1, consider 푚 = 1, so that 푘 = (푛), and ℎ = (1). In this case we have with our former notation 푧 푧̅ 푧 푧̅ 푧 푧̅ ℰ (ℎ) = {휓 (푧) = 1 2 , 휓 (푧) = 1 3 , … , 휓 (푧) = 1 푛} . 푘 1,2 |푧|2 1,3 |푧|2 1,푛 |푧|2 푖휃푗 By changing to polar coordinates 푧푗 = 푟푗푒 , with r푗 ∈ ℝ+ for 푗 = 1, … , 푛 and writing 푟 = (푟1, 푟2, … , 푟푛) we obtain:

212

2푛−1 푖(휃1−휃2) 푖(휃1−휃푛) 휓(푆 ) = {(푟1푟2푒 , . . . , 푟1푟푛푒 ): |푟| = 1, 휃푗 ∈ [0, 2휋), 푗 = 1, … , 푛} 1 푛−1 ⊂ 𝔻̅ (0, ) . 2 푖(휃1−휃푗+1) Setting 휔푗 = |휔푗|푡푗 = 푟1푟푗+1푒 for푗 = 1,2, . . . , 푛 − 1gives 푛−1 1 |휔|2 = ∑ |휔 |2 = 푟2 (1 − 푟2) ≤ , 푗 1 1 4 푗=1 Which implies that 1 1 휓(푆2푛−1) = {(휔 , … , 휔 ) ∈ ℂ푛−1: |푤| ≤ } = 𝔹̅푛−1 (0, ). 1 푛−1 2 2 Notethatthecase푛 > 1, 푚 = 1, 푘 = (푛),andℎ = (푛 − 1)givesthesame result:for 푧 푧̅ 푧 푧̅ 푧 푧̅ 휓 = (휓 (푧) = 1 푛 휓 (푧) = 2 푛 . . . , 휓 (푧) = 푛−1 푛), 1,푛 |푧|2, 2,푛 |푧|2, 푛−1,푛 |푧|2 We have that 1 1 휓(푆2푛−1) = {(휔 , . . . , 휔 ) ∈ ℂ푛−1: |휔| ≤ } = 𝔹̅푛−1 (0, ). 1 푛−1 2 2 Example(6.3.31)[302]:Consider now the case: 푛 > 3, 푚 = 1, 푘 = (푛), and ℎ = (ℎ), with 1 < ℎ < 푛 − 1. In this case we have ℎ(푛 − ℎ) elementary quasi-homogeneous symbols, 푧푗푧푙̅ ℰ (ℎ) = {휓 (푧) = : 푗 = 1, . . . , ℎ, 푙 = ℎ + 1, . . . , 푛}. 푘 푗,푙 |푧|2, iθj Passing to the polar coordinates zj = rje , for j = 1, . . . , n, and with |z| = 1 we have 푖(휃푗−휃푙 ) 휓푗,푙 = 푟푗푟푙푒 , where 푗 = 1, . . . , ℎ, 푙 = ℎ + 1, . . . , 푛. 2푛−1 The range of 휓 = (휓1,ℎ+1, . . . , 휓1,푛, . . . , 휓ℎ,ℎ+1, . . . , 휓ℎ,푛) on 푆 is calculated as 2푛−1 푖(휃푗−휃푙 ) 푖(휃푗−휃푙 ) 푖(휃ℎ−휃푛 ) 휓(푆 ) = {(푟1푟ℎ+1푒 , … , 푟푗푟푙푒 , … , 푟ℎ푟푛푒 ) |푟| = 1, 휃푗 − 휃푙 ∈ [0, 2휋), 푗 = 1, . . . , ℎ, 1 (푛−ℎ) 푙 = ℎ + 1, . . . , 푛 ⊂ 𝔻̅ (0. ) . 2 Let 푖(휃푗−휃푙 ) 푟1푟ℎ+1푒 = 휔1,ℎ+1 = 푎1,ℎ+1푡1,ℎ+1, . . . , 푖(휃푗−휃푙 ) 푟푗푟푙푒 = 휔푗,푙 = 푎푗,푙 푡푗,푙, . . . , 푖(휃ℎ−휃푛) 푟ℎ푟푛푒 = 휔ℎ,푛 = 푎ℎ,푛푡ℎ,푛, 1 here 푎푗,푙 ∈ ℝ+, 푡푗,푙 ∈ 푆 , 푗 = 1, . . . , ℎand 푙 = ℎ + 1, . . . , 푛. Moreover, the above components tj,l obey the relations

푇 = {푡푗1,푙1푡푗1,푙2푡푗2,푙1푡푗2,푙2 = 1: forall 푗1, 푗2 ∈ 1, . . . , ℎ and 푙1, 푙2 ∈ ℎ + 1, . . . , 푛}. note that not all relations in T are independent. An equivalent reformulation of the relations ̅̅̅̅̅̅̅̅̅̅̅̅ Tisa follows. The equation 푡푗1,푙1푡𝑗1,푙2푡𝑗2,푙1 푡푗2,푙2 = 1 is equivalent to 푡푗2,푙2 = ̅̅̅̅̅̅ 푡𝑗1,푙1푡푗1,푙2푡푗2,푙1 showing that only 푛 − 1 of the ℎ(푛 − ℎ) variables t푗.1 are actually independent (e.g. taket1,h+1, . . . , t1,n,t2,h+1, . . . , th,h+1) which yields 푇 = {푡푗,푙 = 푡̅̅1̅,ℎ̅̅+̅1̅푡1,푙푡푗,ℎ+1: 푗 = 2, . . . , ℎ, 푙 = ℎ + 2, . . . , 푛}. Then we have 213

ℎ 푛 1 |휔|2 = ∑ ∑ |휔 |2 = (푟2 +··· +푟2)[1 − (푟2 +··· +푟2)] ≤ , 푗,푙 1 ℎ 1 ℎ 4 푗=1 푙=ℎ+1 Which implies that 1 1 휓(푆2푛−1) = {(휔 , . . . , 휔 ) ∈ ℂℎ(푛−ℎ): |휔| ≤ subject to T} ⊂ 𝔹̅ℎ(푛−ℎ) (0, ), 1,ℎ+1 ℎ,푛 2 2 Where 휔푗,푙 = 푎푗,푙푡푗,푙, with 푗 = 1, . . . , ℎand 푙 = ℎ + 1, . . . , 푛, as above. We unify the results of the above examples in the next lemma. Lemma (6.3.32)[302]:Let 푛 > 1, 푘 = (푛), and ℎ = (ℎ), with1 ≤ h ≤ n − 1.Then the vector 휓 in (4.21) has h(n − h) components, and 1 휓(푆2푛−1) = 훬(푛, ℎ) ⊆ 𝔹̅ℎ(푛−ℎ) (0, ), 2 Where 1 𝔹̅푛−1(0, ), if ℎ = 1, 표푟 ℎ = 푛 − 1, 2 1 훬(푛, ℎ) = {휔 = (휔 , . . . , 휔 ) ∈ 𝔹̅ℎ(푛−ℎ) (0, ): (129) 1,ℎ+1 ℎ,푛 2 { 휔 subject to 푇 }, otherwise. Corollary (6.3.33)[302]:The Banach algebra 풯(ℰ푘(ℎ)) is isomorphic to the polynomial algebra 푃(훬̂(푛, ℎ)), where 훬̂(푛, ℎ)is the polynomially convex hull of 훬(푛, ℎ). Proof: Follows from the above lemma and Theorem (6.3.29) 1 Note that, if ℎ = 1 or ℎ = 푛 − 1, then the set 훬(푛, ℎ) = 𝔹̅ℎ(푛−ℎ) (0, ) is convex and thus 2 coincides with its polynomially convex hull. At the same time, in the case 푛 > 3 and 1 < ℎ < 푛 − 1, an explicit description of the polynomially convex hull 훬̂(푛, ℎ): of 훬(푛, ℎ) seems to be quite anon-trivial task. We do not know the answer, and leave it as a problem (v) in the section. The following discussion provides a (rough) upper bound for 훬̂(푛, ℎ): 1 It is easy to see that 훬̂(푛, ℎ): is a subset of 𝔹̅h(n−h) (0, ). We will show here that it is 2 1 even proper subset. With푟 ∈ (0, ) consider the sets 2 훬푟 (푛, ℎ): = {휔 ∈ 훬(푛, ℎ): |휔| = 푟} ℎ(푛−ℎ) −2 And for each fixed 푧 ∈ ℂ define the holomorphic polynomial 푃푧(휔) ≔ 푟 〈휔, 푧〉. Let 푐 2ℎ(푛−ℎ)−1 훬푟(푛, ℎ) be the (open) complement of 훬푟(푛, ℎ) in the 푟-sphere 푆푟 . For any 푧 ∈ 푐 훬푟(푛, ℎ) there is 0 < 훾 < 1 such that 2 |〈휔, 푧〉| ≤ 훾 푟 , for all 휔 ∈ 훬푟(푛, ℎ). Hence it follows for 휔 ∈ 훬푟(푛, ℎ) and by using the maximum principle in the last equality: −1 −1 2푟 1 = |푃푧(푧)| ≥ 훾 푠푢푝 |푃푧(휔)| ≥ 훾 푠푢푝 |푃푧(2푟휔)| = 푠푢푝 푃푧(휔) . 휔∈훬푟 (푛,ℎ) 휔∈훬1 (푛,ℎ) 훾 휔∈훬(푛,ℎ) ⁄2 푐 푐 Thus, no point in 훬푟(푛, ℎ) with 훾 ≤ 2푟 be longs to the polynomial convex hull of 훬푟(푛, ℎ). 1 In particular this holds for all points in 훬푐 (푛, ℎ) = 휕𝔹̅ℎ(푛−ℎ) (0, ) \훬(푛, ℎ) ≠ ∅. 1/2 2 Now we list several properties of 훬(푛, ℎ) f or the case 푛 ≥ 3and 1 < ℎ < 푛 − 1.

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1 (i) 훬̂(푛, ℎ) is a compact subset of 𝔹̅ℎ(푛−ℎ) (0, ). 2 (ii) None of the points of 훬(푛, ℎ) is interior, i.e., this set has an empty interior. (iii)The set 훬(푛, ℎ) is a contractible star-like set, i.e., together with each of its point the set 1 훬(푛, ℎ) contains the radius of 𝔹̅ℎ(푛−ℎ) (0, ) passing through this point. 2 (iv).Both 훬(푛, ℎ) and its complement ℂℎ(푛−ℎ)\훬(푛, ℎ) are connected sets. (v)The set 훬(푛, ℎ) is invariant under the following action of the (푛 − 1)-dimensional torus 푛−1 푛−1 핋 . For 휏 = (휏1,ℎ+1, . . . , 휏1,푛, 휏2,ℎ+1, . . . , 휏ℎ,ℎ+1) ∈ 핋 and each point 휔 = (휔1,ℎ+1, . . . , 휔ℎ,푛) ∈ 훬(푛, ℎ) the coordinates of 푢 = 휏 · 휔 ∈ 훬(푛, ℎ) are of the form 휏푗,푙휔푗,푙 , if 푗 = 1 and 푙 = ℎ + 1, . . . , 푛, . 표푟 푗 = 2, . . . , ℎ 푎푛푑 푙 = ℎ + 1, 푢푗,푙 = { 휏̅̅1̅,̅ℎ̅̅+̅1̅휏1,푙휏푗,ℎ+1휔푗,푙 , otherwise. Consider now the general case of 푛 > 1, with 푚 > 1, that islet 푘 = (푘1, . . . , 푘푚) and ℎ = (ℎ1, . . . , ℎ푚). In this case, by (112), 푚

ℰ푘(ℎ) ∶= ⋃ ℰ푘,푗(ℎ ), 푗=1 Where for each j ∈ {1, . . . , m} we have used the notation in (113). We consider as well the ordered sets 휓 and 휓[푗] formed by elements of ℰk(h) and ℰ푘,푗(ℎ), 푗 = 1, . . . , 푚, respectively, together with the corresponding unital Banach Toeplitz operator n algebras: 풯(ℰ푘(ℎ)), generated by operators acting on 𝒜2(𝔹 ), and 풯(ℰk,j(h)), generated by 푘푗 operators acting on 𝒜2(𝔹 ), where 푗 = 1, . . . , 푚. 푚 푛 With the above multi- index 푘 = (푘1, . . . , 푘푚) ∈ ℤ+ , we interpret ℂ as a product space ℂ푛 = ℂ푘1 ×···× ℂ푘푚. As is well known, in this situation the standard Hilbert space tensor product decomposition holds ℱ2(ℂ푛) = ℱ2(ℂ푘1) ⊗···⊗ ℱ2(ℂ푘푚). Similarly we have 푘1 푘푛 푘1 푘푚 𝒜푘 ∶= 𝒜2(𝔹 ) ×···× (𝔹 ) = 𝒜2(𝔹 ) ⊗···⊗ 𝒜2(𝔹 ). We introduce as well the unital Banach algebras: 픗(ℰk(h)), generated by Toeplitz operators 2 푛 2 kj on ℱ (ℂ ) with symbols in ℰ푘(ℎ), and 풯(ℰ푘,푗(ℎ)), generated by Toeplitz operators onℱ (ℂ ) with symbols in ℰk,j(h), where 푗 = 1, . . . , 푚. By Lemma (6.3.21) we have 푚 푚

풯(ℰ푘,푗(ℎ) ≅ 픗(ℰ푘(ℎ)) = ⊗ 픗ℰ푘,푗(ℎ) ≅ ⊗ 풯(ℰ푘,푗(ℎ)). (130) 푗 = 1 푗 = 1 since the choice of the tensor product norm is somehow tricky, some comments to the above formula have to be added. In the setting of C∗-algebras the task is simpler and therefore we first ∗ ∗ extend the algebras 풯(ℰ푘,푗(ℎ))and 풯(ℰ푘,푗(ℎ)) to the corresponding C -algebras 풯 (ℰ푘,푗(ℎ)) ∗ ∗ and 풯 (ℰ푘,푗(ℎ)) where 푗 = 1, . . . , 푚. Note that all these C -algebras are of type I. A simple ∗ method of proving that 풯 (ℰ푘,푗(ℎ)) is of type I uses the observation that it is asubalgebra of 풯(퐶(𝔹̅푘푚푗푘)),which by its description in Theorem (6.3.23) is a 퐺퐶푅-algebra. Thus these C∗-

215 algebras are nuclear (see, for example, [44,182]). This in turn implies that the C∗-cross-norms on the tensor products 푚 푚 ∗ ∗ ⊗ 픗 (ℰ푘,푗(ℎ)) and ⊗ 풯 (ℰ푘,푗(ℎ)) 푗 = 1 푗 = 1 are uniquely defined and coincide, in particular, with the spatial cross-norm [80,44], being the 2 푛 standard norm of operators acting on the Hilbert spaces ℱ (ℂ )) and Ak, respectively. Finally the tensor products of Banach algebras 푚 푚

⊗ 픗(ℰ푘,푗(ℎ)) and ⊗ 풯(ℰ푘,푗(ℎ)) 푗 = 1 푗 = 1 In her it the operator norm from their C∗-algebra extensions. Theorem (6.3.34)[302]:The compact set 푀(풯휆(ℰ푘(ℎ))) of maximal ideals of the algebra 풯휆(ℰ푘(ℎ))is given by 푀풯휆(ℰ푘(ℎ)) = 훬̂(푘1, ℎ1) ×···× 훬̂(푘푚, ℎ푚). (131) Proof: As was stated before, the description of the algebra 풯λ(ℰk(h)) does not depend on the weight parameter λ. That is all algebras 풯휆(ℰ푘(ℎ)) , where 휆 ∈ (−1, ∞), are isomorphic and thus have the same compact set of maximal ideals. By (110) the compact set of maximal ideals of the algebra 풯(ℰ푘(ℎ)) (the unweighted case휆 = 0) coincides with the one of the algebra' 푚 ⊗푗=1 풯(ℰ푘,푗(ℎ)). Then by the terminology of [136], the norm on 퐴푘 is uniform, and the corresponding operator (spatial) norm is ordinary. By [136], 푚

푀 ( ⊗ 풯(ℰ푘,푗(ℎ))) = 푀 (풯(ℰ푘(ℎ)) ×···× 푀(풯(ℰ푘,푚(ℎ)), 푗 = 1 which, together with Corollary (6.3.34) and there marks after Lemma (6.3.29) finishes the proof. We now describe the space of maximal ideals in (131) in a different form. As before let 푘 = n (푘1, . . . , 푘푚)and consider the following compact subset of the boundary ∂𝔹 푛 1 퐷푐표푚푝: = {(푧(1), . . . , 푧(푚)) ∈ ℂ : |푧(푗 )| = 푓표푟 푗 = 1, . . . , 푚} √푚 2푘1−1 2푘푚−1 ≅ 푆 1 ×···× 푆 1 . √푚 √푚 We interpret the tuple 휓 = (휓[1], . . . , 휓[푚]) of ordered elementary k-quasi-homogeneous symbols on ℂn as a vector valued function 훾 휓 ∶ 퐷푐표푚푝 → ℂ , 푚 Here 훾 = ∑푗=1 ℎ푗(푘푗 − ℎ푗). As a consequence of Theorem (6.3.35) we have: Corollary (6.3.35)[302]: The compact set 푀(푇휆(퐸푘(ℎ)))in(131) coincides with the polynomial convex hull 휓̂ (퐷푐표푚푝) of the range 휓(퐷푐표푚푝). Proof: It is easy to check that 휓(퐷푐표푚푝) = 훬(푘1, ℎ1) ×···× 훬(푘푚, ℎ푚). Note that the relation 푋̂ × 푌̂ = 푋̂× 푌 holds for compact subsets 푋 ⊂ ℂ푛1 and 푌 ⊂ ℂ푛2. Hence we have

휓̂ (퐷푐표푚푝) = [훬(푘1, ℎ1) ×···× 훬(푘푚, ℎ푚)] = 훬̂(푘1, ℎ1) ×···× (푘푚, ℎ푚). Now, the assertion follows from Theorem (6.3.35). 216

We recall some standard notation (see [285]). Given acompact (polynomially convex) set 푀 ⊂ ℂ푞, we denote by 푃(푀) the closed subalgebra of 퐶(푀) consisting of all functions that uniformly on M can be approximated by analytic polynomials. The algebra 퐴(푀) is the subalgebra of C(M) consisting of all functions that are analytic on the interior int(M) of 푀. Note that 퐴(푀) = 퐶(푀) in the case where 푖푛푡(푀) = ∅. Recall as well that the inclusion 푃(푀) ⊂ 퐴(푀) holds. Although many partial results (both positive andcounterexamples)areknown,thequestionwhetherthe algebras 푃(푀)and 퐴(푀) coincide still remains open for general subsets 푀 ⊂ ℂ푞. Theorem (6.3.36)[302]: The Gelf and trans form is generated by the following mapping of generators of the algebra 풯(ℰ푘(ℎ)) 푇휓 ⟼ 휔푗,ℓ ,푟 (132) ℓ푗 ,푟푗 푗 푗

Where 휓ℓ푗 ,푟푗 is given in (93) and 휔 = (휔1,1,ℎ1+1, . . . , 휔푗,ℓ푗,푟푗, . . . , 휔푚, ℎ푚, 푘푚) ∈ 푀(풯휆(ℰ푘(ℎ))). (i)The Gelf and image of the algebra 풯λ(ℰk(h)) coincides with P(M),where 푀 = 푀(풯휆(ℰ푘(ℎ))) = 훬̂(푘1, ℎ1) ×···× 훬̂(푘푚, ℎ푚). (ii)The isomorphism 풯λ(ℰk(h))) → P(M) is generated by the mapping (132) of generators of the algebra (풯λ(ℰk(h))). (iii)In the case where either ℎ푗 = 1orℎ푗 = 푘푗 − 1 for all 푗 = 1, . . . , 푚 we have that 1 1 푀 = 𝔹̅푘1−1 (0, ) ×···× 𝔹̅푘푚−1 (0, ) and 푃(푀) = 퐴(푀). 2 2

217

List of Symbols

Symbol Page 푑푖푎푔 diagonal 1 ⊝ Direct difference 3 푚푖푛 minimum 14 푚푎푥 maximum 15 푑푒푡 Determinant 21 퐻∞ essential Hardy space 22 퐻푝 Hardy space 22 푑푖푚 dimension 23 Ker kernel 23 퐿1 Lebesgue space of the real line 24 퐿2 Hibert space 25 푠푢푝푝 support 28 퐺푊 Gleason – Whitny 31 푖푛푓 infimum 33 퐿푞 Dual Lebesgue space 34 sup supremum 34 퐻1 Hardy space 34 dom domain 35 퐻2 Hardy space 41 퐿∞ essential Lebesgue space 43` WOT weak operator topology 47 퐴푢푡 Automorphism 48 ⊕ orthogonal sum 48

218

⊗ tensor product 48 푙푎푡 lattice 50 푅푎푛 range 53 ℓ2 Hilbert space of sequences 56 푅푒푝 representation 56 퐴푙푔 Algebra 60 푞 − 퐼푛푛 quasi-inner 63 푑푖푠푡 distance 65 푖푚 imaginary 80 푡푟 trace 86 푙푒푥 lexicographic 93 푅푒 real 96 푝푟표푗 projection 112 푎푟푔 argument 131

퐻푢 Hankel operator 137 2 𝒜휆 Bergman spaces 163 푟푎푑 radial 163 푐푙표푠 closure 175

⨂̂ 휀 Injective tensor product 182 푠푝 spectrum 185 푒푠푠 essential 207 ⨀ multiplication law 208 푐표푚푝 compact 216

219

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