Sudan University of Science and Technology College of Graduate Studies Subdiagonal Subalgebras and Commutative Banach Algebras by Toeplitz Operators 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 By Siddig Ibrahim HassabAlla Gomer Supervisor Prof. Dr. Shawgy Hussein Abdalla 12017 Dedication To my Family I 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. II 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 الخﻻصة حددنا متباينات القيم الذاتية والمجاميع الهيرميتية والمصفوفات الناظمة وحسبان شيبورت ومبرهنة ويﻻنت مع الفئات الطيفية وجبر باناخ. تم ايضاح المصفوفات الجزئية اﻻساسية مع نظرية الدالة غير التبديلية والتمديدات الوحيدة. اعطينا تطبيقات لمحددة فيقليد−كادسون ومبرهنة التحليل الى عوامل نوع ريس−سيزيقو ﻷجل 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−متﻻشية القوى والمولدة بواسطة مؤثرات تبوليتز مع الرموز شبه−المتجانسة شبه−نصف القطرية. IV 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 non- commutative. These result were extended then to the classes of symbols, subordinated to the 2 n quasi-hyperbolic and quasi-parabolic groups. Let λ ( ) denote the standard weighted Bergman space over the unit ball n in ℂn. New classes of commutative Banach 2 n algebras 풯(λ) which are generated by Toeplitz operators on λ ( ) have been recently discovered in Vasilevski. ). These algebras are induced by the action of the quasi-elliptic group of biholomorphisms of n. 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 푛 and in term of their generators they were described. VI 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.