Diagonals of operators Majorization, a Schur–Horn theorem and zero-diagonal idempotents
A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy
Department of Mathematical Sciences McMicken College of Arts and Sciences University of Cincinnati, June 2016
Author: Jireh Loreaux Chair: Gary Weiss, Ph.D. Degrees: B.A. Mathematics, 2010, Committee: Jonathan Brown, Ph.D. University of Cincinnati Victor Kaftal, Ph.D. Costel Peligrad, Ph.D. Shuang Zhang, Ph.D. Abstract
This dissertation investigates two related but mostly independent topics. A diagonal of an operator is the sequence which appears on the diagonal of the matrix representation with respect to a prescribed orthonormal basis.
The classical Schur–Horn Theorem provides a characterization of the set of diagonals of a
n n selfadjoint operator on C (or R ) in terms of its eigenvalues. The relation which provides this description is called majorization. New kinds of majorization (p-majorization and approximate p-majorization) are defined herein, and it is shown that they characterize the diagonals of positive compact operators with infinite dimensional kernel. When the kernel is finite dimensional the diagonals lie between two closely related classes of sequences: those whose zero set differs in cardinality from the zero set of the eigenvalues by p and are also p-majorized (or in the case of the second set, approximately p-majorized) by the eigenvalue sequence. These results extend the work of Kaftal and Weiss in 2010 which describes precisely the diagonals of those positive compact operators with kernel zero.
Zero-diagonal operators are those which have a constant zero diagonal in some or- thonormal basis. Several equivalences are obtained herein for an idempotent operator to be zero-diagonal including one which is basis independent, answering a 2013 question of Jasper about the existence of nonzero zero-diagonal idempotents. These techniques are extended to prove that any bounded sequence of complex numbers appears as the diagonal of some idempotent, and that the diagonals of finite rank idempotent operators are those complex-valued absolutely summable sequences whose sum is a positive integer.
ii c 2016 by Jireh Loreaux. All rights reserved.
Acknowledgments
The author was supported by the University of Cincinnati Department of Mathematical
Sciences, the Charles Phelps Taft Research Center, the Choose Ohio First Scholarship
Program, as well as the following conferences: Non-Commutative Geometry and Operator
Algebras; Southeastern Analysis Meeting; Great Plains Operator Theory Symposium;
International Conference on Operator Theory; East Coast Operator Algebras Symposium;
Wabash Modern Analysis Seminar; and Singular Traces: Theory and Applications hosted by the Centre International de Rencontres Math´ematiques.
I would like to thank my advisor, Gary Weiss, for his continual prodding, thorough feedback and valuable insights, without which I might never have completed this disserta- tion. I am appreciative of the candor with which he communicates both commendation and exhortation. I am also indebted to him for our many disagreements over writing style and mathematical reasoning which resulted in more economical, precise and lucid exposition. Moreover, his support is a primary reason I remained at the University of
Cincinnati for graduate school.
I would like to acknowledge Victor Kaftal for devoting his time and energy to instruct me on the subject of von Neumann algebras even when no formal course could be offered within the department. I must also thank Shuang Zhang for siphoning time from his busy schedule as Department Head to teach me the basics of C∗-algebras.
iv I owe my gratitude to my committee for bothering to read, edit and critique this dissertation. I thank the constituents of the functional analysis seminar for listening to me ramble seemingly unceasingly concerning these results.
I am grateful to Daniel Beltit¸˘afor introducing me to the work of Fan, Fong and Herrero, without which my research on diagonals of idempotents may never have flourished. I am indebted to John Jasper for suggesting to me at GPOTS 2013 several of the major problems of study in this dissertation. I thank Jean-Christophe Bourin for communicating that Theorem 4.2.7 is a corollary of his pinching theorem [Bou03, Theorem 2.1].
My friends in the department, especially Sasmita Patnaik, Marcos Lopez, Michael
Snook, Dewey Estep, Seth Ball and Catalin Dragan deserve special mention for listening to my complaints and providing ample distraction from work when the need arose. Their laughter and companionship were essential to slogging through coursework and research daily. I owe Michael Snook a great deal for the many fruitful discussions concerning libre software, GNU/Linux, cryptography, and of course, LATEX. I would not be where I am today were it not for my parents, Richard and Debra Loreaux, who instilled in me a passion for mathematics and made great sacrifices to provide me with the best opportunities to pursue this career. Their love, support and encouragement have been mainstays during my journey long before I entered the University of Cincinnati.
And finally, the most influential people in my life: my wife, Katie, and my daughters
Dara and Anastacia. Words cannot adequately express Kate’s role in this arena. She pushed me onward when I felt as though I wanted to quit; she never doubted me for a second and encouraged me daily; most importantly, she has never given up on me in spite of my many failings. As for my daughters, they have inspired me and made sure
I always had my hands full. Their smiles and laughter provide me with an unending source of joy and happiness.
“Great are the works of the Lord; They are studied by all who delight in them.”
Psalm 111:2
v This dissertation is dedicated in memoriam to my father
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Introduction 1 1.1. What are diagonals of operators? ...... 1 1.2. Information exchange between operators and diagonals ...... 4 1.3. Description of results ...... 9 1.4. Notation ...... 11 2. Background 13 2.1. Majorization and extensions of the Schur–Horn Theorem ...... 13 2.1.1. Recent History ...... 15 2.1.2. Main contribution ...... 22 vii 2.2. Diagonals of idempotents and frames ...... 23 2.2.1. Projections and idempotents ...... 23 2.2.2. General techniques for diagonals of non-selfadjoint operators . . . 26 2.2.3. Jasper’s problem and the relation to frames ...... 29 3. Schur–Horn theorems 31 3.1. Preliminaries on stochastic matrices ...... 33 3.2. p-majorization (sufficiency) ...... 36 3.2.1. Discussion on majorizations ...... 47 3.3. Approximate p-majorization (necessity) ...... 54 3.4. E( (A)) convexity ...... 62 U 4. Diagonals of idempotents 66 4.1. Zero-diagonal idempotents ...... 67 4.2. Diagonals of the class of idempotents and applications ...... 84 4.3. Diagonals of the class of finite rank idempotents ...... 91 Bibliography 96 A. Prerequisite material 102 A.1. Hilbert spaces ...... 102 A.2. Bounded operators on Hilbert spaces ...... 104 A.3. Trace-class and Hilbert–Schmidt operators ...... 107 B. A primer on zero-diagonal operators 108 B.1. Zero-diagonal operators are “trace-zero” operators ...... 109 B.2. The shape of the set of traces ...... 118 Index 125 viii Notation Functions R Real numbers Tr The standard trace on B( ), Nonnegative real numbers H R+ page 4 C Complex numbers diag The normal -isomorphism h ∗ from `∞ to , pages 2, 3 Ah Operators EA Conditional expectation onto , A Mn(K) n n matrices over a field K page 2 × W (T ) Numerical range of the operator B( ) Bounded operators on a Hilbert H T , page 6 space, page 1 W (T ) Essential numerical range of the Ideal of compact operators e K operator T , page 111 + Positive elements of σ(T ) Spectrum of the operator T , K K page 98 1 Ideal of trace-class operators, L page 100 R Tr T Set of traces of an operators T , { } page 27 2 Ideal of Hilbert–Schmidt L Number classes operators, page 100 A masa (maximal abelian Positive integers A N -subalgebra) in B( ), page 2 ∗ H Z Integers The diagonal masa in B( ) Ah H with respect to the basis h, Z+ Nonnegative integers page 2 ix (A) Unitary orbit of the operator A, `2 Complex-valued U page 19 square-summable sequences, page 97 (A) Partial isometry orbit of the V operator A, page 19 `∞ Bounded complex-valued sequences RA Range projection of the operator A, page 12 ξ, η Sequences related by majorization Re T Real part of T , page 98 ξ∗ Decreasing rearrangement of a Im T Imaginary part of T , page 98 sequence ξ c+, page 11 ∈ 0 [S, T ] Commutator ST TS, page 5 s(A) Singular value sequence sn(A) − h i of the compact operator A, Relations page 11 Majorization, pages 7, 14, 16 Set operations ≺ p p-majorization, page 37 ≺ co(S) Convex hull of a set S in a vector space -p Approximate p-majorization, page 54 S◦ Interior of a set S Strong majorization, page 15 4 S Closure of a set S Weak majorization, page 15 ≺≺ Vectors Sequences Hilbert space, pages 1, 95 H dn A sequence h i e, f, g, h Orthonormal sets in H c0 Complex-valued sequences converging to zero, page 11 (x, y) Inner product of x, y , page ∈ H 95 + c0 Nonnegative sequences converging to zero, page 11 span x closed linear span of x ∗ c0 Nonnegative nonincreasing [x, y] The line segment joining x, y sequences converging to zero, paramaterized by tx + (1 t)y − page 11 for 0 t 1 ≤ ≤ `1 Complex-valued absolutely x y The rank-one operator ⊗ summable sequences z (z, y)x 7→ x Chapter 1 Introduction 1.1. What are diagonals of operators? Consider a separable complex Hilbert space and the -algebra B( ) of bounded H ∗ H (continuous) linear operators. Later we will be concerned primarily with the case when is infinite dimensional, but for now we will allow both finite and infinite dimensional H . For convenience, denote by N := dim the Hilbert dimension of (as opposed to H H H the algebraic dimension). Let ( , ) : C denote the inner product on . If the · · H × H → H reader is unfamiliar with these topics please consult Appendix A before continuing for a brief overview of relevant concepts. N Given an operator T B( ) and an orthonormal basis h = hi , we can define ∈ H { }i=1 the matrix representation [T ]h with respect to the basis h whose ij-th entry is given by (T hj, hi). Then the diagonal of T with respect to the basis h is the (finite or infinite) ∞ bounded sequence (T hi, hi) ` . This is a natural way to view the diagonal sequences h i ∈ of an operator. In fact, when N < this is precisely the standard matrix representation ∞ from linear algebra. There is another, perhaps more elegant, way to view the diagonals (i.e., diagonal sequences) of an operator T . Fix an orthonormal basis h for , and consider the H 1 operators of the form[1] N X diag ai := ai(hi hi), h h i ⊗ i=1 ∞ for scalar sequences ai ` . The matrix representation of diag ai with respect to h i ∈ h h i the basis h is a1 0 0 ··· 0 a2 0 ··· 0 0 a3 ··· . . . . . . . .. and for this reason such operators are called diagonal operators. It is clear that they form an abelian -subalgebra of B( ), and in fact is maximal with respect to these ∗ Ah H Ah properties, and hence it is called a masa[2] (maximal abelian selfadjoint subalgebra) of B( ). H There is a natural map EA : B( ) h defined as h H → A N X EA (T ) := (T hi, hi)(hi hi). h ⊗ i=1 The map EA is the unique trace-preserving conditional expectation onto . Moreover, h Ah this map is faithful (i.e., T 0 and E(T ) = 0 imply T = 0) and normal (i.e., Tn T ≥ ↑ in the strong operator topology implies E(Tn) E(T )). With respect to the matrix ↑ representation [T ]h, the conditional expectation EAh consists of changing the off-diagonal entries of T to zero. Thus we may also refer to the collection of EAh (T ) as h varies over all orthonormal bases of as the diagonals of T . This paradigm has the added benefit H (over sequences) that we never leave the setting of B( ). Another benefit to viewing the H [1] For vectors x, y ∈ H, the tensor product x ⊗ y is the rank-one operator on B(H) which acts on a vector z ∈ H by (x ⊗ y)z = (z, x)y. Thus hi ⊗ hi is the rank-one projection onto the vector hi. [2] The cautious reader may note that there are masas A in B(H) which cannot be obtained as Ah for some orthonormal basis h. However, all atomic masas (i.e., those which are type I as von Neumann algebras) do arise in this manner, and thus these are natural masas to consider because B(H) is a type I factor. Moreover, these are the only masas of B(H) which admit trace-preserving conditional expectations. 2 diagonals of T in this manner is that this concept can be naturally extended to other von Neumann algebras, but such a discussion is outside our scope. We may refine this view as follows: Given an orthonormal basis h and an operator U B( ), the collection Uh forms an orthonormal basis if and only if U is unitary (i.e., ∈ H UU ∗ = U ∗U = 1). Furthermore, if h0 is any other orthonormal basis, there is a unitary U satisfying h = Uh0. In this context, for each unitary U we find N ∗ X ∗ EA (UTU ) = (UTU hi, hi)(hi hi) h ⊗ i=1 N X ∗ ∗ 0 0 = (TU hi,U hi)(Uh Uh ) i ⊗ i i=1 (1.1.1) N X = (T h0 , h0 )U(h0 h0 )U ∗ i i i ⊗ i i=1 ∗ = UEAh0 (T )U . Letting (T ) denote the unitary orbit of T under conjugation (i.e., UTU ∗ U unitary ), U { | } we find that every diagonal of T is unitarily equivalent to a member of EA ( (T )). Thus h U we may view the diagonals of T instead as elements in this latter set. Moreover, the -isomorphism diag : `∞ given by ∗ h → Ah N X diag ai = ai(hi hi) h h i ⊗ i=1 takes the diagonal sequences of T coming from the matrix representations of T to the diagonals viewed as elements of EA ( (T )). Furthermore, (1.1.1) allows us to conclude h U ∗ ∗ UEA ( (T ))U = EA (U (T )U ) = EA ( (T )). h0 U h U h U Therefore the diagonals of T in one basis are unitarily equivalent to the diagonals in any other basis, so there is no reason for preferential treatment of bases. 3 We now have several equivalent ways of viewing diagonals of operators: diagonal sequences arising from matrix representations; • conditional expectations onto diagonal masas ; • Ah elements of the expectation of the unitary orbit. • We will choose whichever description is most convenient for the given situation. In particular, in Chapter 3 we will view diagonals as EA ( (T )), the image under the h U conditional expectation of the unitary orbit. In Chapter 4 we will view diagonals in terms of matrix representations with respect to the relevant orthonormal basis. 1.2. Information exchange between operators and diagonals In this section we describe several facts concerning diagonals of an operator T B( ) ∈ H which are suggestive of their importance. Details are often omitted for the sake of clarity of exposition. For now, let be finite dimensional. H (i) The trace of T is the linear functional on B( ) defined as the sum of the diagonal H entries N X Tr T := (T hi, hi), i=1 which is independent of the choice of orthonormal basis h. Indeed, if e is any other orthonormal basis then expanding h in terms of e and applying properties of the inner product we have N N N N ! X X X X (T hi, hi) = T (hi, ej)ej , (hi, ek)ek i=1 i=1 j=1 k=1 N N N X X X = (hi, ej)(hi, ek)(T ej, ek), i=1 j=1 k=1 4 and switching the order of summation N N N N X X X X (T hi, hi) = ((ek, hi)hi, ej)(T ej, ek) i=1 j=1 k=1 i=1 N N X X = (ek, ej)(T ej, ek) (1.2.1) j=1 k=1 N N N X X X = δj,k(T ej, ek) = (T ei, ei). j=1 k=1 i=1 This independence of basis is equivalent to the statement Tr[T,U] = 0 for any unitary operator U, where [T,U] := TU UT . Indeed, if Ue = h, then − N N N N X X X X (UT hi, hi) = (UT ei, ei) = (UT Uhi, Uhi) = (T Uhi, hi), i=1 i=1 i=1 i=1 by (1.2.1) and U ∗U = I. Moreover, this implies Tr[T,S] = 0 for any other operator S[3], since S may be written as a linear combination of four unitary operators. It is a straightforward exercise to show that Tr X = 0 if and only if 0,..., 0 is a h i diagonal of X (see Lemma 4.1.9). If an operator X is a commutator (i.e., X = [T,S]), then by the above remarks Tr X = 0. Sh¯oda’sLemma [Sh¯o37]proves the converse, that X is a commutator if Tr X = 0. The proof proceeds as follows: If Tr X = 0, then there is a basis e in which X has zero diagonal. Choose S to be a diagonal operator in the basis e with distinct diagonal entries s1, . . . , sn. Then let T be the operator whose entries in xij the basis e are given by tij := when i = j and tii = 0. Then a matrix computation si−sj 6 shows X = ST TS. − Fact 1.2.1. An operator X has 0,..., 0 as a diagonal in some basis if and only if it h i is a commutator, i.e., X = [S, T ]. [3] In fact, the trace is the unique (up to a scalar multiple) faithful positive linear functional which vanishes on commutators. It is often this property which is used to define traces in more general settings. 5 Now remove the restriction that be finite dimensional. H (ii) The diagonal entries of T (with respect to any given orthonormal basis h) are also elements of the numerical range W (T ) = (T h, h) (h, h) = 1 . A nonobvious fact { | } concerning the numerical range is the Toeplitz–Hausdorff Theorem [Hau19; Toe18], which states that W (T ) is a convex subset of C. It is also true that W (T ) σ(T ), which ⊇ is obvious when is finite dimensional since the spectrum consists of the eigenvalues H and these are always elements of the numerical range. Hence the convex hull of the spectrum σ(T ) is also included in the closure of the numerical range. When T is normal (i.e., when [T,T ∗] = 0 so T commutes with its adjoint) this can be strengthened to ∗ W (T ) = co σ(T ). In particular, if T 0 (i.e., T = T is selfadjoint and σ(T ) R+), ≥ ⊆ then W (T ) = co σ(T ) R+. The more interesting fact is that the converse is true. That ⊆ is, if W (T ) R+ then T 0. Because every element of the numerical range appears ⊆ ≥ on the diagonal of T in some basis, and the diagonal entries are always members of the numerical range, we have: Fact 1.2.2. The diagonal of T is a nonnegative sequence for every choice of orthonormal basis h if and only if T 0. ≥ (iii) The ideal of compact operators is the closure in the operator norm of the finite K rank operators. It is the only proper norm-closed ideal of B( ). A standard exercise in H operator theory is to show that for any orthonormal basis h and any compact K , ∈ K one has Khi 0. This implies that the diagonal entries of K in any basis converge k k → to zero. The converse seems to the author to be less well-known. That is, if K is an operator whose diagonal entries in any basis converge to zero, then K . ∈ K A similar result holds for the ideal of trace-class operators 1 which consists of those L compact operators K such that K := (K∗K)1/2 has summable eigenvalues when ∈ K | | repeated according to multiplicity. Precisely, if K 1 then for any orthonormal basis h, ∈ L 6 the diagonal (Khi, hi) is absolutely summable. Conversely, if in any basis the diagonal h i of K is absolutely summable, then K is trace-class. Both of these results are subsumed by the generic statements: Fact 1.2.3. If is a two-sided ideal (not necessarily closed) of B( ) then I H (i)If E (K) for every basis h, then K . h ∈ I ∈ I (ii)If is arithmetic-mean closed[4] and K , then E (K) for any basis h (that I ∈ I h ∈ I is, is diagonally invariant). I In other words, if is arithmetic-mean closed then an operator lies in if and only if I I every diagonal lies in . This subsumes the previous results since and 1 are arithmetic- I K L mean closed. The condition in (ii) that be arithmetic-mean closed is necessary in I general. Indeed, for the finite rank operators, witness any nonzero rank-one operator ξ ξ with ξ `2 of infinite support. ⊗ ∈ (iv) In light of the above examples, it is natural to ask which sequences are the diagonals of a specific operator. The vast majority of the work on this question in the literature is concerned solely with selfadjoint operators, with the original finite dimensional result being proved in two parts by Schur [Sch23] and Horn [Hor54], see the latter for details. For finite dimensional , they characterized the diagonals of a selfadjoint operator X B( ) H ∈ H in terms of a relation called majorization. N Definition 1.2.4. Given ξ, η R we say that ξ is majorized by η, denoted ξ η if ∈ ≺ n n N N X ∗ X ∗ X X ξ η , 1 n N, and ξi = ηi, i ≤ i ≤ ≤ i=1 i=1 i=1 i=1 where ξ∗, η∗ denote the monotone nonincreasing rearrangements of ξ, η. With this definition we can state the classical Schur–Horn Theorem. [4] For details see [KW11, Theorem 4.5]. Briefly, I is arithmetic-mean closed if I = a(Ia), where Ia and aI are the arithmetic mean and pre-arithmetic ideals generated respectively by: operators with s-numbers the arithmetic means of the s-numbers of operators from ideal I, and operators whose arithmetic means of their s-numbers are s-numbers of operators in I. 7 Theorem 1.2.5 (Schur–Horn, [Hor54; Sch23]). Let be finite dimensional and let H N X B( ) be a selfadjoint operator with eigenvalue sequence λ R . The diagonals of ∈ H ∈ X are characterized by N EAh ( (X)) = diag d d R , d λ . U { h | ∈ ≺ } Since the proof of the classical Schur–Horn Theorem in 1954 there have been several attempts to extend it to the infinite dimensional setting. We will cover these in detail in Section 2.1. (v) Instead of determining those sequences which are the diagonals of a specific operator, we can ask which are the diagonals of a class of operators, that is, the union of the diagonals of the individual operators in that class. There is a variety of material on this subject which we briefly review now. In the same paper [Hor54] in which he characterizes the diagonal sequences of a fixed selfadjoint matrix in Mn(C), Horn identifies the diagonals of the class of rotation matrices n as the convex hull of the points in R whose coordinates are either 1 or 1 with an even − number of negative entries. He then uses this to identify the diagonals of the classes of orthogonal matrices and of unitary matrices. The diagonals are those real-valued, respectively complex-valued, sequences whose sequence of absolute values lie in the convex n hull of the points in R whose coordinates are either 1 or 1. See [Hor54, Theorems − 8-11]. Fong shows in [Fon86] that any bounded sequence of complex numbers appears as the diagonal of a nilpotent operator in B( ) of order four (N 4 = 0), thus characterizing H diagonals of the broader classes of nilpotent (N k = 0 for some k) and also quasinilpotent k 1/k operators (limk→∞ N = 0). In this paper Fong remarks that a finite complex-valued k k sequence appears as the diagonal of a nilpotent matrix in Mn(C) if and only if its sum is zero. 8 More recently, Giol, Kovalev, Larson, Nguyen and Tener [Gio+11] classified the diagonals of idempotent matrices in Mn(C) as those whose sum is a positive integer less than n, along with the constant sequences 0,..., 0 and 1,..., 1 (see Theorem 2.2.2 h i h i below). In Section 2.2 we provide additional detail on the subject of diagonals of idempotents. 1.3. Description of results In Chapter 2 we explore the history and significance of the topics encountered in this dissertation, as well as providing a more detailed explanation of our main theorems. The remainder of the work is divided into several independent pieces. In Chapter 3, the results of which have been published in [LW15], we prove an infinite dimensional Schur–Horn theorem. As stated in Theorem 1.2.5, the classical Schur–Horn Theorem provides a characterization of the diagonals of a selfadjoint operator acting on a finite dimensional Hilbert space in terms of its eigenvalues. More specifically, it says that the diagonal sequences are precisely those which are majorized by the eigenvalue sequence. The work in Chapter 3 is concerned with extending the Schur–Horn Theorem to the infinite dimensional setting for positive compact operators. This extends recent work of Kaftal and Weiss [KW10] which provided the first exact characterization of the diagonals of a positive compact operator with kernel zero. One key aspect of Chapter 3 is the definition and investigation of new types of majorization, called p-majorization (Definition 3.2.2) and approximate p-majorization (Definition 3.3.1). In some sense, these types of majorization are -close, and in fact they coincide when p = 0 or p = . Our first main theorem (Theorem 3.2.4) asserts that ∞ sequences which are p-majorized by the eigenvalues of a positive compact operator and which have at most p fewer zeros than the eigenvalue sequence are diagonal sequences of that positive compact operator. Our second main theorem (Theorem 3.3.4) asserts a partial converse to the first by stating that a diagonal sequence of a positive compact 9 operator which has at least p fewer zeros than the eigenvalue sequence are approximately p-majorized by the eigenvalue sequence. Thus this bounds the diagonal sequences of a positive compact operator between two sets which are somehow -close when the kernel is finite dimensional, and provides a complete characterization of the diagonal sequences when the kernel is infinite dimensional (Corollary 3.3.6). Our last main theorem in this chapter (Corollary 3.4.1) proves that the diagonals of a positive compact operator with infinite dimensional kernel form a convex set. Chapter 4, the results of which have been published in [LW16], is devoted to the study of diagonals of idempotent operators. We begin by answering a question due to Jasper concerning the existence of nonzero idempotents with zero diagonal. We not only show that there exist nonzero idempotents with zero diagonal, we provide several equivalent conditions for an idempotent to have zero diagonal in some basis (Theorem 4.1.5). One of these conditions, that the idempotent cannot be written as the sum of a projection and a Hilbert–Schmidt operator, is especially useful because it is basis independent. Since the proof of Theorem 4.1.5 is essentially nonconstructive, we also provide a more hands-on approach to constructing a basis in which an idempotent has zero diagonal (Algorithm 4.1.11). These techniques then generalize to allow us to prove that any bounded complex-valued sequence appears as the diagonal of some idempotent (Theorem 4.2.7). Because Theorem 4.2.7 bears no particular resemblance to the finite dimensional case (Theorem 2.2.2 due to [Gio+11]), we also investigated the diagonals of the class of finite rank idempotents. Ultimately, we prove that the diagonals of finite rank idempotents are precisely those absolutely summable complex-valued sequences whose sum is a positive integer (Theorem 4.3.3), which is a natural analogue of the finite dimensional case. 10 1.4. Notation For a set S, let S denote its cardinality. will always denote a separable complex | | H Hilbert space of either finite or infinite dimension. Lowercase Fraktur letters (e.g. h, e, b) will always denote either orthonormal sets or orthonormal bases for the Hilbert space in question, and the corresponding lowercase roman glyph (e.g., h, e, b) will represent an element of this orthonormal basis. An atomic masa (i.e., arising as diagonal operators with respect to a fixed orthonormal basis) will be denoted by , or more simply by when the basis h is understood. Ah A Uppercase roman letters such as T, S, X will generally denote operators in B( ). The H ideal of compact operators in B( ) will be denoted as . H K Let c0 denote the space of all complex-valued sequences converging to zero, and let + ∗ c0 ,c0 denote the cones of nonnegative sequences converging to zero and nonnegative nonincreasing sequences converging to zero. For a sequence ξ c+, let ξ∗ c∗ denote ∈ 0 ∈ 0 the monotonization of ξ, or rather the monotonization of ξ supp ξ when ξ is not finitely | supported. That is, ξ∗ := ξ∗ where ξ∗ denotes the j-th largest element of ξ. Notice h j i j that if ξ is finitely supported, then ξ∗ ends in zeros. However, if ξ has infinite support, then ξ∗ has no zeros, and in this case the monotonization ξ∗ reflects neither the zeros of ξ nor their multiplicity. This notion of monotonization coincides with the decreasing rearrangement[5] of a function from measure theory by taking the counting measure on N and restricting the resulting step function to be defined on N. The symbol d will often denote a diagonal sequence of some operator. For a compact [6] operator K , the singular value sequence s(K) = sn(K) is the sequence ∈ K h i eigenvalues of (K∗K)1/2 listed in decreasing order repeated according to multiplicity [5]For a measure space (X, µ) and a complex-valued measurable function f which is finite almost everywhere, the decreasing rearrangement f ∗ : [0, ∞) → [0, ∞) is given by ∗ −1 f (t) = inf{x ∈ R+ | µ(|f| (x, ∞)) ≤ t}. [6]also known as s-numbers 11 (with the zero eigenvalue necessarily omitted if K has infinite rank). Equivalently, the singular numbers can be defined as sn(K) := inf F K , F ∈Fnk − k where n denotes the set of finite rank operators whose rank is strictly less than n. F The range projection RA for an operator A B( ) is the orthogonal projection onto ∈ H ⊥ ∗ ⊥ ran A = ker A . Thus for a selfadjoint operator A, RA is the projection onto ker A and ⊥ hence Tr RA = dim ker A, and in general Tr RA = rank A. Throughout this dissertation ⊥ we opt for using Tr RA and Tr RA instead of rank A and dim ker A. 12 Chapter 2 Background 2.1. Majorization and extensions of the Schur–Horn Theorem The classical Schur–Horn Theorem (Theorem 1.2.5) characterizes the diagonals of a selfadjoint operator on a finite dimensional Hilbert space in terms of the majorization preorder (Definition 1.2.4) on finite real-valued sequences. This suggests the more general problem: Problem 2.1.1 (Schur–Horn Problem). Characterize the diagonals (or an appropriate substitute thereof) of selfadjoint operators (on an infinite dimensional space, in a von Neumann algebra, etc.) in terms of majorization (or a suitable modification thereof). Since the advent of the Schur–Horn theorem, there has been substantial progress towards developing infinite dimensional analogues. This was perhaps started by the work of Markus [Mar64] and Gohberg and Markus [GM64], and more recently the topic was revived by A. Neumann in [Neu99]. However, Neumann studied an approximate Schur–Horn phenomenon, i.e., the operator-norm closure of the diagonals of bounded selfadjoint operators (equivalently, the `∞-norm closure of the diagonal sequences), which Arveson and Kadison deemed too coarse a closure [AK06, Introduction paragraph 3]. Instead, they studied the expectation of the trace-norm closure of the unitary orbit of a 13 trace-class operator and then proved approximate Schur–Horn analogues for trace-class operators in B( ) (the type I∞ factor). They also formulated a Schur–Horn conjecture H [1] for type II1 factors, but discussion of this topic is outside the scope of this paper . Kaftal and Weiss in [KW10] extended Arveson and Kadison’s work by proving exact Schur–Horn theorems for positive compact operators and pinpointing the diagonals of those with either finite rank or zero kernel. Majorization plays an essential role in Schur–Horn phenomena, but also has given rise to new kinds of majorization inside and outside this arena. The Riemann and Lebesgue majorizations of Bownik–Jasper are applied to Schur–Horn phenomena in [BJ15a; BJ15b; Jas13b]. Similarly, the uniform Hardy–Littlewood majorization of Kalton–Sukochev [KS08] is used ubiquitously in Lord–Sukochev–Zanin [LSZ13] as an essential tool to study traces and commutators for symmetrically normed ideals. In this paper we also develop new kinds of majorization essential to our work (see Figure 2.1 and Figure 2.2 and the accompanying description). + The following Definition 2.1.2 of majorization on c0 from [LW15] agrees with most of the literature, but it is a departure from that of [KW10] which did not include this equality condition. When Kaftal–Weiss needed an equality-like condition, they used instead the more restrictive Definition 2.1.3 of strong majorization. Definition 2.1.2 ([LW15, Definition 1.2]). Let ξ, η c+. One says that ξ is majorized ∈ 0 by η, denoted ξ η, if for all n N, ≺ ∈ n n ∞ ∞ X ∗ X ∗ X X ξ η and ξj = ηj. j ≤ j j=1 j=1 j=1 j=1 [1] For work on II1 and II∞ factors, see the work of Argerami and Massey [AM07; AM08; AM13], Bhat and Ravichandran [BR14] and a recent unpublished work of Ravichandran [Rav14]. 14 Definition 2.1.3 ([KW10, Definition 1.2]). Let ξ, η c+. One says that ξ is strongly ∈ 0 majorized by η, denoted ξ η, if for all n N, 4 ∈ n n n X X X ξ∗ η∗ and lim inf (η∗ ξ∗) = 0. j ≤ j n j − j j=1 j=1 j=1 Note that when ξ η, one has ξ `1 if and only if η `1. In the η-summable case, ≺ ∈ ∈ majorization as defined above in Definition 2.1.2 is equivalent to strong majorization in Definition 2.1.3. However, in the nonsummable case, the latter is clearly a stronger constraint than the former. Strong majorization is not an essential tool in our main theorems, but we thought it important to emphasize the distinction between these various definitions of majorization. The reasons for our Definition 2.1.2 departure from that of Kaftal–Weiss are convenience, efficiency of notation and unification of cases. This notation allows us to state in a more unified way the results for both trace-class and non-trace-class operators simultaneously without splitting the conclusions into cases (compare Theorem 3.1.3 to the two cases in [KW10, Corollary 5.4]). 2.1.1. Recent History Gohberg and Markus In [GM64], Gohberg–Markus almost characterized the diagonals of a selfadjoint trace- class operator K. To state their result we need some additional notions of majorization. Definition 2.1.4 ([GM64, pages 202–203]). Let ξ, η c+. One says that ξ is weakly ∈ 0 majorized by η, denoted ξ η, if for all n N ≺≺ ∈ n n X X ξ∗ η∗. j ≤ j j=1 j=1 15 This allows us to naturally define majorization of real-valued sequences in `1. For a real-valued sequence ξ, let ξ+ := max ξj, 0 and ξ− := ( ξ)+. h { }i − Definition 2.1.5 ([GM64, pages 202–203]). Let ξ, η `1 be real-valued. One says ∈ that ξ is majorized by η, denoted ξ η, if both ξ+ η+ and ξ− η−, and also ≺ ≺≺ ≺≺ P∞ P∞ 1 ξ = 1 η. Note that Definition 2.1.2 and Definition 2.1.5 agree on their common domain of pairs of nonnegative sequences in `1. Thus, while we may have overloaded the notation , ≺ there can be no misunderstanding of its meaning in a given context. In this context Gohberg and Markus prove the following theorem. Theorem 2.1.6 ([GM64, Theorem 1]). Let T 1 be a selfadjoint trace-class operator ∈ L in B( ) with eigenvalue sequence λ c0 repeated according to multiplicity. Then any H ∈ diagonal d of T satisfies d c0 and d λ, and conversely, if d c0 is majorized by λ, ∈ ≺ ∈ then d 0 is a diagonal of T for some size of 0. ⊕ The reason this is almost a characterization is that the size of the zero 0 is not specified, and it is easy to show that not all d majorized by λ with arbitrarily sized zeros appear as diagonals. Indeed, if λ0 is a sequence consisting only of the nonzero values of a sequence [2] λ of infinite support, then λ0 λ and λ0 0 is a diagonal of T if and only if the size ≺ ⊕ ⊥ of 0 equals Tr RT . Theorem 2.1.6 can, however, be easily enhanced to provide an exact characterization of the diagonals of positive trace-class operators with kernel zero through the use of the upcoming elementary Proposition 3.2.1. The result is then a special case of the forthcoming Theorem 2.1.14. [2] To prove this statement, it suffices to show that if λ0 ⊕ 0 is a diagonal of T then the unit vectors producing the values of λ0 are, in fact, eigenvectors for T . This can be established by induction on λ0 (proceeding in decreasing order) and using the fact that if |T x, x| = kT k then x is an eigenvector of T . 16 Arveson and Kadison In [AK06], Arveson and Kadison consider positive trace-class operators and prove an approximate Schur–Horn Theorem for this class. It is approximate in the sense that instead of characterizing the diagonals of a positive trace-class operator, i.e., the expectation of the unitary orbit, they characterize the expectation of the trace-norm closure of the unitary orbit. They first provide an alternative way to view the trace-norm closure of the unitary orbit of a positive trace-class operator. Proposition 2.1.7 ([AK06, Proposition 3.1]). Let A 1 be a positive trace-class ∈ L+ operator with s(A) the singular values of A. k·k1 (i) (A) = S(A) := B + s(B) = s(A) . U { ∈ K | } k·k (ii) B (A) 1 if and only if A 0 and B 0 are unitarily equivalent for an infinite ∈ U ⊕ ⊕ dimensional 0. k·k (iii) If A is finite rank, (A) 1 = (A). U U Their approximate Schur–Horn Theorem for positive trace-class operators is: Theorem 2.1.8 ([AK06, Theorem 4.1]). Let be a masa in B( ) of diagonal operators A H with respect to an orthonormal basis h and let A 1 be a positive trace-class operator. ∈ L+ Then k·k1 + EA( (A) ) = diag d d c , d s(A) . U { h | ∈ 0 ≺ } For finite rank positive operators, this provides a complete description of their diagonal sequences as encapsulated in the following corollary. Corollary 2.1.9. Let be a masa in B( ) of diagonal operators with respect to an A H orthonormal basis h and let A B( )+ be a positive finite rank operator. Then ∈ H + EA( (A)) = diag d d c , d s(A) . U { h | ∈ 0 ≺ } 17 Proof. Apply Proposition 2.1.7 (iii) and Theorem 2.1.8. They then prove one inclusion of the Schur–Horn Theorem for positive compact operators: Theorem 2.1.10 ([AK06, Theorem 4.2]). Let be a masa in B( ) of diagonal A H operators with respect to an orthonormal basis h and let A + be a positive compact ∈ K operator. Then ∗ + EA(L AL) L 1 = diag d d c , d s(A) . { | k k ≤ } { h | ∈ 0 ≺≺ } This indeed has part of the Schur–Horn Theorem as a corollary because it implies that diagonals of a positive compact operator A are weakly majorized by the singular value sequence s(A). In addition, the sum of the diagonal sequence is always equal to the sum of the singular values since A is positive and the trace is preserved under conjugation by unitaries (i.e., change of bases). Therefore, the diagonals of A are majorized by the singular values s(A). Symbolically, this is written as + EA( (A)) diag d d c , d s(A) . (2.1.1) U ⊆ { h | ∈ 0 ≺ } When A + has finite rank, Corollary 2.1.9 shows the sets in (2.1.1) are equal, thus ∈ K providing an exact Schur–Horn theorem for finite rank positive operators on an infinite dimensional Hilbert space[3]. It not hard to see that when A + has infinite rank and finite dimensional kernel, ∈ K then the inclusion is proper (reference Proposition 3.2.1). The inclusion is also proper when A has instead infinite dimensional kernel (see Theorem 3.3.4). Thus, the Schur–Horn [3]While similar, this result is inherently different from the classical Schur–Horn Theorem about selfadjoint operators on a finite dimensional space since the diagonal are infinite sequences instead of finite sequences. 18 Problem for positive compact operators is reduced to describing which proper subset of the sequences majorized by the singular values corresponds to the diagonal sequences. Kaftal and Weiss In [KW10], Kaftal and Weiss provided an exact[4] extension of the Schur–Horn Theorem to a certain subset of positive compact operators. That is, in terms of majorization they characterize precisely the expectation of the unitary orbit of strictly positive compact operators and the expectation of the partial isometry orbit (Definition 2.1.11 below) for all positive compact operators. They then ask for, but leave as an open question, a characterization of the expectation of the unitary orbit of positive compact operators with infinite rank and nonzero kernel. Definition 2.1.11 ([KW10, Page 3152] Partial Isometry Orbit). Given an operator A B( ), the partial isometry orbit of A is the set ∈ H ∗ ∗ (A) = V AV V B( ),V V = RA RA∗ . V { | ∈ H ∨ } Notice this extends to partial isometries the standard notion of unitary orbits (A) = U UAU ∗ unitary U ( ) . { | ∈ U H } At first, the partial isometry orbit may seem like a strange collection to consider, but our next result shows that for A + it is just another description for more familiar ∈ K objects. Recall that for A +, S(A) := B + s(B) = s(A) . ∈ K { ∈ K | } Proposition 2.1.12. If A + is a positive compact operator, then ∈ K k·k (A) = S(A) = (A) . V U [4]i.e., precise characterizations without taking closures of any kind 19 k·k If in addition A 1 , then these sets are also equal to (A) 1 . Furthermore, if A in ∈ L+ U fact has finite rank, then all these sets coincide with the unitary orbit (A). U Proof. We begin by establishing the first equality: (A) = S(A). Let V AV ∗ (A). V ∈ V By the conditions on V we know that V ∗V AV ∗V = A. Since V ∗V = 1, by standard k k inequalities concerning the singular value sequences [GK69, Page 27] we obtain s(A) = s(V ∗V AV ∗V ) s(V AV ∗) s(A). ≤ ≤ This proves (A) S(A). V ⊆ To prove the reverse inclusion, take B S(A). Since A, and therefore also B, are ∈ positive and compact, they have respective orthonormal eigenvectors e = en , f = fn { } { } corresponding to the eigenvalues sn(A) = sn(B), taking only those n for which these are positive. Define V to be the partial isometry which maps span e isometrically onto ∗ span f via V en = fn for all n N. Then V V = RA since it is the projection onto ∈ ∗ ∗ span en n∈ , and so V AV (A) and a computation guarantees V AV = B. This { } N ∈ V proves the inclusion S(A) (A). ⊆ V k·k We now prove the second equality: S(A) = (A) . This proof mimics the proof of U Proposition 2.1.7 (i) from [AK06] with only slight modifications. The singular value sequences are known to satisfy sn(A) sn(B) A B (see, for example, [GK69, | − | ≤ k − k Corollary 2.3]), and therefore S(A) is norm closed. Along with the inclusion (A) U ⊆ k·k (A) = S(A), it follows trivially that (A) S(A). For the reverse inclusion, let V U ⊆ B S(A). For each n N we may write ∈ ∈ 0 0 A = An + An and B = Bn + Bn, 20 0 0 where s(An) = s(Bn) = s1(A), . . . , sn(A), 0,... and A = B = sn+1(A). It is h i k nk k nk ∗ clear that there are unitaries Un for which UnAnUn = Bn. Thus B UnAU = k − nk 0 0 ∗ k·k k·k B UnA U 2sn+1(A) 0, and hence B (A) . This proves S(A) (A) . k n − n nk ≤ → ∈ U ⊆ U The additional result when A 1 is precisely Proposition 2.1.7 (i), and when A has ∈ L+ finite rank we apply Proposition 2.1.7 (iii). In light of Proposition 2.1.12, the following Schur–Horn type result of Kaftal–Weiss is a natural extension of Theorem 2.1.8 by Arveson–Kadison to positive compact operators. Here we present a reformulation of their theorem using our notation for majorization. Theorem 2.1.13 ([KW10, Proposition 6.4]). Let A +. Then ∈ K E( (A)) = B + s(B) s(A) . V { ∈ D ∩ K | ≺ } The common underpinning between Theorem 2.1.8 and Theorem 2.1.13 is S(A). Since S(A) does not encode the dimension of the kernel of A both theorems sidestep the effect of this information on the diagonal sequences of A. The techniques of Kaftal–Weiss, however, allow for the analysis of special cases, as evidenced by the following theorem. Theorem 2.1.14 ([KW10, Proposition 6.6]). Let A + with RA = I. Then ∈ K E( (A)) = E( (A)) B RB = I . U V ∩ { ∈ D | } The question of precisely what is E( (A)) for all A + has only been partially U ∈ K answered. In particular, it was answered when A has finite rank by Corollary 2.1.9 or when RA = I (that is, when A is strictly positive) by the previous theorem. We should note that Corollary 2.1.9 can also be deduced from Theorem 2.1.13 and the finite rank fact that (A) = (A), which itself can be found in Proposition 2.1.12 (or see U V Remark 3.2.5 for a direct approach). 21 2.1.2. Main contribution Corollary 2.1.9 and Theorem 2.1.14 left open the case when A has infinite rank and nonzero kernel. We attempt here to narrow this gap. Indeed, we characterize E( (A)) U when A has infinite dimensional kernel. When A has finite dimensional kernel, we give a necessary condition for membership in E( (A)) (which we conjecture is also sufficient), U and we give a sufficient condition for membership in E( (A)) (which we know not to be U necessary when 0 < Tr R⊥ < , see Example 3.2.7 below which also appears in [KW10, A ∞ Proposition 6.10, Example 6.11]). Moreover, these two conditions are somehow “-close” in an admittedly vague sense[5], thereby significantly restricting the range of sequences for which it is unknown whether or not they are diagonals. These main results are embodied in Theorem 3.2.4, Theorem 3.3.4 and Corollary 3.3.6[6]. Both of these membership conditions involve new kinds of majorization, which here we call p-majorization and herein we introduce approximate p-majorization (for 0 p , Definition 3.2.2 and ≤ ≤ ∞ Definition 3.3.1 below). There is a natural hierarchy of these new types of majorization which the diagram in Figure 2.1 describes succinctly. All of these implications are natural (see the discussions following Definition 3.2.2 and Definition 3.3.1) except the two corresponding to the dashed arrows, which are proved in Proposition 3.2.8 and are only applicable when both sequences in question are in c+ `1. A linear interpretation of 0 \ the diagram in Figure 2.1 is presented below it in Figure 2.2. -1 -2 - ··· ∞ and ≺ ≺ 64 1 2 ≺ ≺ ··· ≺∞ Figure 2.1.: Hierarchy of majorization [5]Compare Definition 3.2.2 to Definition 3.3.1. [6]We refrain from stating these results in their entirety in the introduction because doing so would require definitions which do not appear until Chapter 3, namely p-majorization (Definition 3.2.2) and approximate p-majorization (Definition 3.3.1). 22 -1 1 -2 - ( and 4) ≺ ≺ ··· ∞ ≺∞ ≺ 6 Figure 2.2.: Linear hierarchy of majorization 2.2. Diagonals of idempotents and frames 2.2.1. Projections and idempotents As mentioned in the previous section, a large portion of the work concerning diagonal sequences of operators deals with the selfadjoint case. In Chapter 4 we study diagonal sequences of idempotents, so diagonals of projections (selfadjoint idempotents) are of particular relevance to us. These were characterized by Kadison in Theorem 2.2.1 [Kad02a; Kad02b]. We find this theorem especially interesting because it straddles the fence between a characterization of diagonals of a single operator in a specified class (Section 1.2 item (iv)) and of a class of operators (Section 1.2 item (v)). Indeed, although it is stated as a characterization of the diagonals of the class of projections, it can easily be adapted to identify the diagonals of any fixed projection. This is because two projections P,P 0 B( ) are unitarily equivalent if and only if Tr P = Tr P 0 and ∈ H 0 Tr(1 P ) = Tr(1 P ) and thus share the same diagonals. Moreover, for dk a diagonal − − h i sequence for P , these quantities are precisely the sum of the diagonal entries dk and the sum of 1 dk, respectively. Then one can apply the four finite/infinite cases in the next − theorem. Theorem 2.2.1 ([Kad02a; Kad02b]). Given an infinite sequence dk [0, 1]N with h i ∈ X X a = dk and b = (1 dk), − dk<1/2 dk≥1/2 2 ∗ then there is a projection P B( ) (i.e., P = P = P ) with diagonal dn if and only ∈ H h i if one of the following mutually exclusive conditions holds: (i) either a or b is infinite; 23 (ii) a, b < and a b Z. ∞ − ∈ The requirement that 0 dk 1 for all k N is clearly necessary since 0 P I and ≤ ≤ ∈ ≤ ≤ the diagonal entries of P are elements of its numerical range. The second condition, that a b Z, is less obvious but can be viewed as a kind of index obstruction to an arbitrary − ∈ sequence in [0, 1]N appearing as the diagonal of a projection. Indeed, in [Arv07] Arveson provided details on this index obstruction and showed that it applies more generally to any normal operator with finite spectrum (each element of infinite multiplicity) that consists of the vertices of a convex polygon. Since we study diagonals of idempotents in B( ), which when not projections are H non-selfadjoint, we are especially interested in diagonals of non-selfadjoint operators. One particularly relevant result in this direction is the aforementioned characterization of diagonals of idempotent matrices in Mn(C) by Giol, Kovalev, Larson, Nguyen and Tener [Gio+11]. n Theorem 2.2.2 ([Gio+11, Theorem 5.1]). A finite sequence dk C appears as the h i ∈ diagonal of an idempotent D Mn(C) if and only if one of the following three mutually ∈ exclusive conditions holds. (i) dk = 0 for all k (in which case D = 0); (ii) dk = 1 for all k (in which case D = I); P (iii) dk 1, . . . , n 1 . ∈ { − } Since Tr D 1, . . . , n 1 for any nonzero, non-identity idempotent matrix (as ∈ { − } is well-known, see for instance Lemma 4.1.1), this theorem says that this is the only requirement for a sequence to appear as the diagonal of some idempotent. We prove a natural extension of this result to finite rank idempotents on an infinite dimensional space in the next theorem. H 24 Theorem 4.3.3 ([LW16, Theorem 4.3]). The diagonals of the class of nonzero finite rank idempotents consist precisely of those absolutely summable sequences whose sum is a positive integer. Giol, Kovalev, Larson, Nguyen and Tener proved Theorem 2.2.2 because of its relevance to frame theory. Because of a similar frame-theoretic question (characterizing inner prod- ucts of dual frame pairs), Jasper asked for a characterization of diagonals of idempotents in B( ). For a key test case, Jasper posed the following two operator-theoretic questions H (private communication, May 2013 [Jas13a]): Question 2.2.3. If an idempotent has a basis in which its diagonal is absolutely summable, is it finite rank? Question 2.2.4. If an idempotent has a basis in which its diagonal consists solely of zeros (i.e., is a zero-diagonal operator in the terminology of Fan [Fan84]), is it finite rank? If we restrict the idempotents to be selfadjoint (i.e., projections), then they are positive operators and the answer to each question is certainly affirmative since the trace is preserved under conjugation by a unitary operator (i.e., a change of basis). In fact, for projections, having an absolutely summable (or even summable) diagonal is a characterization of those projections with finite rank since rank P = Tr P . Moreover, the only projection with a zero diagonal is the zero operator for this same reason. Hence, a negative answer to either of these questions for the entire class of idempotents would be a notable departure from the case of projections and would therefore suggest that the classification of their diagonals is potentially harder than one might na¨ıvely expect. As it turns out, Larson constructed a nonzero (and necessarily infinite rank) idempotent that lies in a continuous nest algebra which has zero diagonal with respect to this nest[7] [7] An operator T has zero diagonal with respect to the nest if PλTPλ = 0 for some linearly ordered set of projections {Pλ}λ∈Λ inside the nest such that with respect to the decomposition of the identity P I = Λ Pλ every element of the nest is block upper-triangular. 25 [Lar85, Proof of Theorem 3.7]. However, the existence of an idempotent with zero diagonal with respect to a nest algebra certainly depends on the order type of the nest to some extent. Indeed, the nest algebra consisting of the upper triangular matrices with respect to some basis en n∈ for has order type ω (the first infinite ordinal), and a { } N H simple computation shows that the only idempotent with zero diagonal inside this nest algebra is the zero operator (proceed by induction on the superdiagonals). Leaving the realm of nest algebras, we can ask two questions: Which idempotents are zero-diagonal? • Which idempotents have an absolutely summable diagonal? • As it turns out, both of these questions have the same answer, which we provide in Theorem 4.1.5, the statement of which can be found below on page 28. 2.2.2. General techniques for diagonals of non-selfadjoint operators The techniques for analyzing diagonals of non-selfadjoint operators seem to differ greatly from those used for selfadjoint operators. For example, the techniques used in determining diagonals of selfadjoint operators often rely heavily on majorization and keeping track of the explicit changes of the basis (or equivalently, the unitary operators) involved in the construction. In contrast, the Toeplitz–Hausdorff Theorem, that the numerical range W (T ) of a bounded operator T is convex, is one of the central tools in the work of Fan, Fong and Herrero [Fan84], [FFH87], [FF94] to determine diagonals of non-selfadjoint operators. Indeed, they frequently use the nonconstructive version of the Toeplitz– Hausdorff Theorem despite the existence of constructive versions in which a formula is specified for the vector yielding the prescribed value of the quadratic form (T f, f). The Fan, Fong and Herrero results relevant to us here are restated below. The first is an infinite dimensional analogue of the finite dimensional result that an n n matrix × has trace zero if and only if it is zero-diagonal. 26 Theorem 2.2.5 ([Fan84, Theorem 1]). If T B( ) and there exists some basis ∈ H ∞ ej for for which the partial sums { }j=1 H n X sn := (T ej, ej) j=1 have a subsequence converging to zero, then T is zero-diagonal. Fan’s proof of [Fan84, Theorem] has a flaw (see Appendix B for details), but it can be circumvented with a slight modification. Appendix B is devoted to reproving Theorem 2.2.5 in order both to fix the flaw and to provide more detail than [Fan84]. ∞ Definition 2.2.6 ([FFH87]). Let T B( ) and let e = ej be a basis for . ∈ H { }j=1 H Pn Suppose the partial sums sn = (T ej, ej) converge to some value s C. Then we j=1 ∈ say that Tre T := s is the trace of T with respect to the basis e. The set of traces of T , denoted R Tr T , is then the set of all such traces Tr T as e ranges over all orthonormal { } e bases for which Tre T is defined (i.e. lim sn exists and is finite). Observe that in order to make sense of this definition, it is essential both that these trace values are finite and that we must order e by N. A curious fact about the set R Tr T from Definition 2.2.6 is that it may take on only { } four different shapes: the plane, a line, a point or the empty set. It is no coincidence that these shapes coincide with those obtainable as the limits of convergent rearrangements of a series of complex numbers (i.e., the L´evy–SteinitzTheorem extending the Riemann Rearrangement Theorem to complex numbers [L´ev05;Ste13]). Theorem 2.2.7 ([FFH87, Theorem 4]). Suppose T B( ). Then there are four ∈ H possible shapes that R Tr T can acquire. More specifically, R Tr T is: { } { } iθ 1 (i) the plane C if and only if for all θ R, (Re e T )+ / (the trace-class); ∈ ∈ L iθ 1 iθ 1 (ii) a line if and only if for some θ R, both (Re e T )± / but (Im e T ) ; ∈ ∈ L ∈ L (iii) a point if and only if T 1; ∈ L 27 iθ 1 iθ 1 (iv) the empty set if and only if for some θ R, (Re e T )+ / but (Re e T )− . ∅ ∈ ∈ L ∈ L In fact, their proof of Theorem 2.2.7 shows that given T B( ) there exists a basis ∈ H e for , e ordered by N, for which every element of R Tr T can be obtained from a H { } basis which is a permutation of e. For the next theorem Fan–Fong utilize the previous two theorems to provide intrinsic (i.e., basis independent) criteria for when a bounded operator is zero-diagonal. Theorem 2.2.8 ([FF94]). An operator T is zero-diagonal if and only if for all θ R, ∈ iθ iθ Tr(Re e T )+ = Tr(Re e T )−. We neither use nor cite this theorem elsewhere, however, it seems interesting to include it because it shares its intrinsic nature with our Theorem 4.1.5 (i). Theorem 4.1.5 ([LW16, Theorem 2.5]). For D B( ) an infinite rank idempotent ∈ H the following are equivalent: (i) D is not a Hilbert–Schmidt perturbation of a projection; (ii) the nilpotent part[8]of D is not Hilbert–Schmidt; (iii) R Tr D = C; { } (iv) D is zero-diagonal; (v) D has an absolutely summable diagonal; (vi) D has a summable diagonal (i.e., R Tr D = ). { } 6 ∅ An interesting general fact is that (v) and (vi) of Theorem 4.1.5 are actually equivalent for any bounded operator, not merely idempotents (see Proposition 4.1.12). [8]We have not yet defined the nilpotent part of an idempotent D, but it is a natural object defined in Lemma 4.1.1 that gives a canonical decomposition for idempotents. In particular, we can write I 0 D = ( T 0 ) and we call T the nilpotent part of D. 28 2.2.3. Jasper’s problem and the relation to frames The tools developed in Chapter 4 for the proof of Theorem 4.1.5 led to a complete answer to Jasper’s original problem: Problem 2.2.9. Characterize the diagonals of the class of idempotent operators. In short, there are no restrictions on the diagonals of the class of idempotent operators. ∞ Theorem 4.2.7 ([LW16, Theorem 3.7]). Every dn ` admits an idempotent D h i ∈ ∈ B( ) whose diagonal is dn with respect to some orthonormal basis. H h i We mentioned briefly that Jasper’s problem had its origins in frame theory. Below we give a short description of the frame-theoretic equivalent[9] of Problem 2.2.9. We begin with the definition of a frame. Definition 2.2.10. A collection x = xj j∈J of vectors in is said to be a frame if the { } H P so-called frame operator S = xj xj satisfies aI S bI for some positive scalars j∈J ⊗ ≤ ≤ a, b. The analysis operator (or frame transform) is the operator θ : B( ) `2(J) defined x H → P 2 by θ = ej xj where e = ej j∈J is the standard basis on ` (J). The synthesis x j∈J ⊗ { } ∗ operator is just the adjoint θx of the frame transform. Then the frame operator is simply ∗ ∗ S = θx θx and the so-called Gram matrix (or Grammian) G = θxθx is the operator on 2 ` (J) whose ij-th matrix coefficient (with respect to e) is (xj, xi)`2(J). ∗ Definition 2.2.11. Given a frame x, a frame y is said to be dual to x if θy θx = I, i.e., if the synthesis operator of y is a left inverse for the analysis operator of x. The canonical ∗ −1 dual frame is the frame associated to the analysis operator θx(θx θx) . [9]The equivalence of these two problems was originally described to us by Jasper, but a fairly concise explanation can be found on this MathOverflow post [10113]. 29 Another way to view dual frames is via idempotents. Consider that y is dual to x if and ∗ ∗ ∗ ∗ ∗ ∗ only if θxθy is idempotent. Indeed, if they are dual then (θxθy )(θxθy ) = θx(θy θx)θy = θxθy , ∗ ∗ ∗ ∗ and conversely, if θxθy is idempotent, then θxθy = θxθy θxθy and multiplying on the right ∗ ∗ by θx and on the left by θy we obtain I = θxθy . Furthermore, if Px denotes the projection ∗ −1 ∗ θx(θx θx) θx , then Px is the range projection of θx. Moreover, it is the range projection of any idempotent θ θ∗ where y is dual to x. This guarantees that the map y θ θ∗ is a x y → x y bijection between the dual frames of x and the set of idempotents with range projection Px. ∗ One can notice that the diagonal of the idempotent θxθy is precisely the sequence of inner products (xj, yj) . Therefore a characterization of diagonals of idempotents h i provides a characterization of inner products of dual frame pairs, thereby establishing the correspondence. 30 Chapter 3 Schur–Horn theorems Most of the results of this chapter have been published in [LW15]. In this chapter we pursue an exact Schur–Horn theorem for positive compact operators with infinite rank and infinite dimensional kernel. We also provide some information in the case when the operator has infinite rank and finite dimensional kernel. We begin in Section 3.1 with a brief primer on the closely related subject of stochastic matrices, which are ubiquitous throughout the work of [KW10] from whence flow many of our results. In Section 3.2 we start by providing a necessary condition on the kernel dimension of diagonals of a positive operator. However, the main thrust of the section is the definition of p-majorization (for p Z+ , Definition 3.2.2) and the proof that this ∈ ∪ {∞} type of majorization is sufficient to guarantee membership in the diagonal sequences of an operator when the difference in the dimension of the kernels is less than or equal to p (Theorem 3.2.4). We then follow a short tangent on the relationship between p-majorization and strong majorization (Definition 2.1.3), and how this may be exploited in the determination of diagonals. As a partial converse of Theorem 3.2.4, in Section 3.3 we define approximate p- majorization (for p Z+ , Definition 3.3.1) and show that this type of majorization ∈ ∪ {∞} is necessary for membership in the diagonal sequences of an operator when the difference 31 in the dimension of the kernels is at least p (Theorem 3.3.4). Since -majorization and ∞ approximate -majorization are equivalent, these theorems establish a characterization ∞ of diagonals of positive compact operators with infinite rank and infinite dimensional kernel (Corollary 3.3.6). We conclude by proving that in the setting of Corollary 3.3.6 the diagonals form a convex set. Convexity is closely related to the Schur–Horn Theorem (see discussion in Section 3.4) and in all the cases in which the diagonals of a positive compact operator are known, they form a convex set. However, it is in general false that the diagonals of an arbitrary operator (even a normal operator on a three-dimensional space) constitute a convex set. In fact, given Kadison’s characterization of diagonals of projections, one can see that the diagonals of an infinite rank projection with infinite dimensional kernel are not convex as the next example shows. Example 3.0.1. In the notation of Theorem 2.2.1, note that there is a projection with diagonal d such that a + b < and a b = 1. Moreover, the sequence d0 = d0 given by ∞ − h ji 0 if dj < 1/2, 0 dj := 1 if dj 1/2 ≥ is obviously the diagonal of a projection. If we consider the convex combination ϕ = λd + (1 λ)d0 for 0 < λ < 1, and − X X aλ = ϕk and bλ = (1 ϕk), − ϕk<1/2 ϕk≥1/2 then we compute X X aλ = ϕk = λdk = λa, ϕk<1/2 dk<1/2 32 and X X bλ = (1 ϕk) = (1 λdk 1 + λ) = λb. − − − ϕk≥1/2 dk≥1/2 Thus aλ bλ = λ(a b) = λ / Z, and hence ϕ is not the diagonal of a projection by − − ∈ Theorem 2.2.1. 3.1. Preliminaries on stochastic matrices Stochastic matrices play a central role in investigating Schur–Horn theorems due to the following definition and lemma. Definition 3.1.1 (Stochasticity). A matrix P with positive entries is called substochastic if its row and column sums are bounded by 1; • column-stochastic if it is substochastic and its column sums equal 1; • row-stochastic if it is substochastic and its row sums equal 1; • doubly stochastic if it is both row- and column-stochastic; • unistochastic if it is the Schur-product of a unitary matrix with its complex • conjugate (the Schur-product of two matrices A = (aij) and B = (bij) is the matrix (aijbij), that is, it is the entrywise product of A, B); orthostochastic if it is the Schur-square of an orthogonal matrix, i.e., of a unitary • with real entries. And the connection between expectations of orbits and stochastic matrices is: Lemma 3.1.2 ([KW10, Lemmas 2.3, 2.4]). Let ξ, η `∞ and for any contraction ∈ 2 L = (Lij) B( ), let Qij := Lij for all i, j. Then ∈ H | | ξ = Qη if and only if diag ξ = E(L diag η L∗). 33 Furthermore, (i) Q is substochastic; (ii) L is an isometry if and only if Q is column-stochastic; (iii) L is a co-isometry (L∗ is an isometry) if and only if Q is row-stochastic; (iv) L is unitary if and only if Q is unistochastic; (v) L is orthogonal if and only if Q is orthostochastic. For completeness we repeat the straightforward short proof. ∞ Proof. Given ξ, η ` , notice that for any n N, ∈ ∈ ∗ ∗ (E(L diag η L )en, en) = (L diag η L en, en) ∞ ∞ X X = diag η) L¯njej, L¯nkek j=1 k=1 ∞ ∞ X X = ηjL¯njej, L¯nkek j=1 k=1 ∞ X 2 = Lnj ηj = (Qη)n. | | j=1 Notice now that ∞ ∞ X X ∗ ∗ ∗ 2 Qij = LijL = (LL ei, ei) = L ei 1 for every i, (3.1.1) ji k k ≤ j=1 j=1 and similarly ∞ X 2 Qij = Lej 1 for every j. (3.1.2) k k ≤ i=1 (i) Immediate from (3.1.1) and (3.1.2). (ii) If L is an isometry, then it is immediate from (i) and the equality cases of (3.1.2) that Q is column-stochastic. Conversely assume that Q is column-stochastic and hence 34 ∗ Lej = 1 for all j by (3.1.2). Then L Lej, ej = 1 for all j and thus it follows that k k h i E(I L∗L) = 0. Since E is faithful and I L∗L 0 because L is a contraction by − − ≥ hypothesis, it follows that L∗L = I. (iii) Apply (ii) to L∗. (iv) Immediate from (ii) and (iii). (v) Immediate from (iv) and the fact that L has real entries. Lemma 3.1.2 provides a dictionary which translates between the realms of sequences and stochastic matrices and of diagonals of operators (expectations of orbits). Many of the results in [KW10] are stated and proved in terms of these stochastic matrices. We reformulate here one of their theorems more relevant to our study using our notation for majorization. Theorem 3.1.3 ([KW10, Corollary 5.4]). If ξ, η c∗, then ∈ 0 ξ = Qη for some orthostochastic matrix Q ξ η. ⇐⇒ ≺ Note that via Lemma 3.1.2, the above Theorem 3.1.3 along with Proposition 3.2.1 imply Theorem 2.1.14 which characterizes diagonals of positive compact operators with zero kernel. In fact, because Theorem 3.1.3 is about orthostochastic matrices as opposed to unistochastic matrices, we can conclude that Theorem 2.1.14 is valid even when the underlying Hilbert space is real instead of complex. Moreover, all our theorems in this H chapter are applicable to real Hilbert spaces since Theorem 3.2.4 is the only one for which this fact is not easily verified, but its extension to real Hilbert spaces is handled in Remark 3.2.6. 35 3.2. p-majorization (sufficiency) A result which was known to Kaftal and Weiss [KW10, Proposition 6.6 (i)], and almost certainly to others, is a necessary condition for membership in the expectation of the unitary orbit of a positive operator. Namely, the dimensions of the kernels of operators in the range of the expectation of the unitary orbit of a positive operator cannot increase from the dimension of the kernel of the operator itself. Proposition 3.2.1 ([LW15, Proposition 2.1]). If A B( )+ and B E( (A)), then ∈ H ∈ U ker B ker(UAU ∗) for some U ( ), and hence also Tr R⊥ Tr R⊥. ⊆ ∈ U H B ≤ A Vector proof. Since B E( (A)), B = E(UAU ∗) for some unitary U ( ), and since ∈ U ∈ U H ⊥ ∗ ⊥ Tr RA = dim ker A = dim ker UAU = Tr RUAU ∗ , the required trace inequality follows from the inclusion ker B ker UAU ∗, which itself follows from A 0 and ⊆ ≥ ker B = ker E(UAU ∗) ∗ = span en E(UAU )en, en = 0 { | h i } ∗ = span en UAU en, en = 0 { | h i } ∗ 1/2 2 = span en (UAU ) en = 0 { | k k } ker(UAU ∗)1/2 = ker UAU ∗ ⊆ Projection proof. Since A 0, R⊥ is the largest projection P so that P AP = 0. Since ≥ A B E( (A)), there is some unitary U ( ) so that B = E(UAU ∗). Notice also that ∈ U ∈ U H R⊥ since B . Therefore, B ∈ D ∈ D ∗ ⊥ ∗ ⊥ ∗ ⊥ ∗ ⊥ E(U(U RBU)A(U RBU)U ) = E(RBUAU RB) ⊥ ∗ ⊥ ⊥ ⊥ = RBE(UAU )RB = RBBRB = 0. 36 ∗ ⊥ ∗ ⊥ ∗ Then since E is faithful and U(U RBU)A(U RBU)U is positive, it is zero, and thus conjugating by U ∗ shows (U ∗R⊥U)A(U ∗R⊥U) = 0. By the maximality of R⊥, U ∗R⊥U B B A B ≤ ⊥ RA and therefore Tr R⊥ = Tr(U ∗R⊥U) Tr R⊥. B B ≤ A Before we proceed with our analysis, we need to introduce next a concept similar to [KW10, Definition 6.8(ii)] which here we call p-majorization[1]. Roughly speaking, it is majorization along with eventual p-expanded majorization. And this leads us to the definition below of -majorization which is both new and fruitful. ∞ Definition 3.2.2. Given ξ, η c+ and 0 p < , we say that ξ is p-majorized by η, ∈ 0 ≤ ∞ denoted ξ p η, if ξ η and there exists an Np N such that for all n Np, we have ≺ ≺ ∈ ≥ the inequality n+p n X X ξ∗ η∗. k ≤ k k=1 k=1 And -majorization, denoted ξ ∞ η, means ξ p η for all p N. ∞ ≺ ≺ ∈ Note that ξ 0 η is precisely the statement that ξ η (recall Definition 2.1.2, which ≺ ≺ 0 includes equality of the sums). One also observes that if ξ p η and p p, then ≺ ≤ ξ p0 η (we use often the special case that p-majorization implies 0-majorization, i.e., ≺ majorization). For this reason, ξ p η for infinitely many p is equivalent to ξ p η for all ≺ ≺ p < , in which case we say that ξ is -majorized by η and we write ξ ∞ η. Although ∞ ∞ ≺ not immediately obvious, it is in fact possible to have ξ ∞ η even when η does not have ≺ 1 finite support. For example, let ξ := η1, η1, η2, η2,... (hint: Np = p). 2 h i One should also take note that p-majorization is actually strictly stronger than p0- 0 + majorization when p < p. That is, there exist sequences ξ, η c for which ξ p0 η but ∈ 0 ≺ ξ p η. From the remarks of the previous paragraph, it suffices to exhibit ξ, η when 6≺ [1]The author, along with Gary Weiss, introduced the concept of p-majorization in [LW15, Definition 2.2]. The reader should note that the asterisks (indicating monotonization) which appear in the displayed inequality of Definition 3.2.2 are missing from [LW15, Definition 2.2]. This is a typo in [LW15]. An identical statement holds for [LW15, Definition 3.1] and Definition 3.3.1. 37 p = p0 + 1. To produce such sequences, start with any 0 < η c∗ and define ∈ 0 (p) ξ := η1/p,..., η1/p, η2, η3,... . h| {z } i p times (p) (p) Even though ξ is not necessarily monotone, it is not difficult to verify that ξ p0 η ≺ (p) but ξ p η. 6≺ Remark 3.2.3 ([LW15, Remark 2.3]). Because (A) (A), Theorem 2.1.13 for A V ⊇ U ∈ + implies that s(B) s(A) is a necessary condition for B E( (A)); Proposition 3.2.1 K ≺ ∈ U shows Tr R⊥ Tr R⊥ is another necessary condition for membership in E( (A)) and with B ≤ A U majorization is equivalent to membership in E( (A)) when Tr R⊥ = 0 by Theorem 2.1.14. U A It was natural in [KW10] to ask how the role of majorization is impacted by the dimension of these kernels. In particular here we enhance this program by asking, what role, if any, p = Tr R⊥ Tr R⊥ plays in relating membership in E( (A)) to majorization (with the A − B U convention that p = 0 when Tr R⊥ = = Tr R⊥. Equivalently, p is the minimal n for A ∞ B which Tr R⊥ Tr R⊥ + n. This is the strategy that guides our program. A ≤ B The result below appears in [KW10] as Lemma 6.9 for p < , the proof of which ∞ utilizes orthostochastic matrices. We provide a different proof which instead utilizes expectations of unitary orbits because it leads to a straightforward extension to the p = case. See Remark 3.2.6 for when an orthostochastic matrix can be produced to ∞ implement the construction. ⊥ ⊥ Theorem 3.2.4 ([LW15, Theorem 2.4]). Let A, B +, B and Tr R Tr R . ∈ K ∈ D B ≤ A ⊥ ⊥ If for some 0 p , s(B) p s(A) and Tr R Tr R + p, ≤ ≤ ∞ ≺ A ≤ B then B E( (A)). ∈ U Proof. Without loss of generality we may assume that A (i.e., A, B are simultaneously ∈ D diagonalized) because (A) = (UAU ∗) for every U ( ). We may further reduce U U ∈ U H ⊥ ⊥ to the case when Tr RB = 0 via a splitting argument. Because dim ker B = Tr RB < 38 Tr R⊥ = dim ker A, without loss of generality we can assume S := ker B ker A. Then A ⊆ ⊥ with respect to = S S , one has A = A1 A2, B = B1 B2, A1 = 0 = B1, and B2 H ⊕ ⊕ ⊕ ⊥ ∗ has zero kernel. Thus once we find a unitary U on S for which E(UA2U ) = B2, then I U satisfies E((I U)A(I U)∗) = B. ⊕ ⊕ ⊕ Case 1: A has finite rank. This is a direct consequence of Corollary 2.1.9. However, below we give an alternate proof which shows directly that (A) = (A) (even if A is not necessarily selfadjoint), U V and then applies Theorem 2.1.13. If A has finite rank, then (A) = (A), even if A is not necessarily selfadjoint. Indeed, U V the elements of (A), by Definition 2.1.11, have the form V AV ∗ for some partial isometry V ∗ ∗ ∗ for which V V = RA RA∗ (:= P ), so PA = AP = A. Then V AV = (VP )A(VP ) ∨ and VP is also a partial isometry with (VP )∗(VP ) = P . But this partial isometry VP is finite rank since RA and RA∗ , and hence also P , are finite rank, and so VP can be extended to a unitary U for which V AV ∗ = (VP )A(VP )∗ = UAU ∗. This shows (A) (A), and hence equality. That the conclusion of Theorem 3.2.4 holds in this case V ⊆ U is then covered by Theorem 2.1.13, since A 0, (A) = (A), and since s(B) p s(A) ≥ U V ≺ implies s(B) s(A). ≺ ⊥ ⊥ Case 2: A has infinite rank and Tr RB = Tr RA (the p = 0 case). ⊥ With the earlier reduction that Tr RB = 0 and using p = 0, Case 2 is a direct consequence of Theorem 2.1.14 and Theorem 2.1.13. ⊥ ⊥ Case 3: A has infinite rank and Tr RB < Tr RA (the most complicated case). Since Tr R⊥ Tr R⊥ + p, if necessary, by passing to a smaller p we may assume A ≤ B Tr R⊥ = Tr R⊥ + p, even if Tr R⊥ = (in which case p = ). Since Tr R⊥ = 0, one has A B A ∞ ∞ B Tr R⊥ = p, and since Tr R⊥ < Tr R⊥, one has 1 p . A B A ≤ ≤ ∞ ∞ We now employ another splitting. First let ej denote the basis that simultaneously { }j=1 p diagonalizes A, B, and then assign them different names, that is, let fj be the { }j=1 collection of ej’s such that (Aej, ej) = 0. Since A is diagonalized with respect to the 39 ∞ p ∞ basis ej the collection fj forms an orthonormal basis for ker A. Let gj { }j=1 { }j=1 { }j=1 ∞ ∞ ⊥ consist of the remainder of the set ej , which means gj is a basis for ker A. { }j=1 { }j=1 ⊥ Then = 1 2, where 1 = ker A and 2 = ker A. Let Np := n N n p and H H ⊕ H H H { ∈ | ≤ } ∞ ∞ then define diagH ⊕H : ` (Np) ` (N) by 1 2 × → D ϕ1 0 0 0 ··· ··· ...... ...... 0 ϕp 0 0 diag ϕ, ρ = ··· ··· (3.2.1) H1⊕H2 h i 0 0 ρ1 0 ··· ··· . . . . . . .. . . ρ . 2 . 0 0 0 .. ··· ··· p ∞ ⊥ Then because of the way in which we chose fj and gj and because Tr R = 0, { }j=1 { }j=1 B if we let η0 := s(A) and ξ = s(B) then A, B are 0 A = diagH1⊕H2 0, η and B = diag ξ, ∞ 0 where 0 ` (Np) is the zero sequence and by hypothesis ξ p η . ∈ ≺ The heuristic idea of the proof is the following in descriptive informal language. First construct a sequence ξ0 which is a sparsely compressed version of ξ but sufficient to retain majorization by η0, i.e., ξ0 η0. Next apply Case 2 to obtain a special unitary U for ≺ which E(U(diag 0, η0 )U ∗) = diag 0, ξ0 . Finally, apply another unitary to decompress h i h i the diagonal 0, ξ0 to the diagonal ξ. h i p 0 p Now inductively choose sequences of nonnegative integers Nm and N { }m=1 { m}m=0 with the following properties: 0 0 0 0 0 (i) 0 = N < N + 1 < N1 < N < N + 1 < N2 < N < if p = , 0 0 1 1 2 ··· ∞ 0 (or < Np < N if p < ). ··· p ∞ 40 (ii) For each m Np, whenever n Nm (m 1), one has ∈ ≥ − − n+m n X X 0 ξk η . ≤ k k=1 k=1 0 (iii) ξNm + ξN ξNm−1 for each m Np. m ≤ ∈ For transparency and brevity, we only loosely describe the construction of the pair of p 0 p 0 sequences Nm and N . The construction proceeds in pairs, Nm,N . We { }m=1 { m}m=0 m 0 0 may choose Nm to satisfy property (ii) since ξ p η and hence ξ m η because m p. ≺ ≺ ≤ For this we use the fact that property (ii) is an eventual property in the sense that if it holds for some Nm it holds for any larger Nm. Moreover, because ξ > 0 and ξ 0, ξ has ↓ infinitely many strictly decreasing jumps, i.e., for infinitely many j one has ξj ξj+1. If necessary, increase Nm so that it satisfies this condition. Then since ξ 0, we may choose → 0 Nm to satisfy property (iii). To construct the entire pair of sequences, simply iterate this 0 process while simultaneously ensuring Nm+1 > Nm + 1, which we can guarantee because property (ii) is an eventual property. 0 0 0 Next since N < Nm < N < Nm+1 < N for all 1 m < p, one has for p = 0 m m+1 ≤ ∞ 0 0 0 0 N + 1 = 1 N1 < N N2 1 < N 1 N3 2 < N 2 0 ≤ 1 ≤ − 2 − ≤ − 3 − ≤ · · · 0 0 Nm (m 1) < N (m 1) Nm+1 m < N m · · · ≤ − − m − − ≤ − m+1 − ≤ · · · 0 Np (p 1) < N (p 1) for p < . · · · ≤ − − p − − ∞ When p < , if we set N 0 = for convenience of notation, then, regardless of whether ∞ p+1 ∞ p < or p = , these inequalities partition N as ∞ ∞ p G 0 0 N = Nm (m 1),Nm+1 m − − − m=0 41 0 0 with each Nm (m 1) Nm−1 (m 2),Nm (m 1) and m Np. − − ∈ − − − − ∈ Next define the sequence ξ0 which shifts and alters ξ at one point in each interval 0 0 Nm−1 (m 2),Nm (m 1) : for each m Np (or for each m Np+1 if p < , in − − − − ∈ ∈ ∞ which case the last interval is [N 0 (p 1), )), set p − − ∞ ξN + ξN 0 if k = Nm (m 1); m m − − 0 ξk = 0 0 ξk+m−1 if N(m−1) (m 2) k < Nm (m 1) but − − ≤ − − k = Nm (m 1). 6 − − 0 0 This partition of N ensures that ξ is well-defined. Property (iii) guarantees that ξ is monotone decreasing. And property (ii) allows us to conclude that ξ0 η0 which ≺ will follow from equations (3.2.2)–(3.2.4). We omit their straightforward natural proofs for the sake of clarity of exposition. Equations (3.2.2)-(3.2.3) have natural proofs by induction and (3.2.4) is merely one case for when p < . ∞ For all m Np, ∈ k k+m−1 X 0 X 0 ξ = ξj for N (m 2) k < Nm (m 1), (3.2.2) j m−1 − − ≤ − − j=1 j=1 and k k+m−1 X 0 X 0 ξ = ξN 0 + ξj for Nm (m 1) k < N (m 1). (3.2.3) j m − − ≤ m − − j=1 j=1 When p < , the last interval requires separate consideration. That is, if N 0 (p 1) ∞ p − − ≤ 0 0 0 k < N p = and N (p 1) j k one has ξ = ξj+p and so p+1 − ∞ p − − ≤ ≤ j k k+p X 0 X 0 ξ = ξj for k [N (p 1), ) (3.2.4) j ∈ p − − ∞ j=1 j=1 42 It is now simple to prove that ξ0 η0 using condition (ii) and equations (3.2.2)–(3.2.4). ≺ 0 Indeed, notice that for m Np, if Nm−1 (m 2) k < Nm (m 1) and since also ∈ − − ≤ − − 0 m 1 Np 0 and ξ η , and considering separately the cases m = 1 and m > 1, − ∈ ∪ { } ≺ one has k k+m−1 k X 0 X X 0 ξj = ξj ηj. (3.2.5) (3.2.2) ξ≺≤η0 j=1 j=1 or (ii) j=1 0 And for m Np, if Nm (m 1) k < Nm (m 1), one has ∈ − − ≤ − − k k+m−1 k+m k X 0 X X X 0 ξj = ξN 0 + ξj ξj ηj. (3.2.6) (3.2.3) m ≤ ≤ j=1 j=1 j=1 (ii) j=1 Finally, if p < and N 0 (p 1) k < N 0 p = , one has ∞ p − − ≤ p+1 − ∞ k k+p k X 0 X X 0 ξj = ξj ηj. (3.2.7) (3.2.4) ≤ j=1 j=1 (ii) j=1 0 0 Recalling that N is the disjoint union of N (m 2),Nm (m 1) for m Np, (m−1) − − − − ∈ (3.2.2–3.2.4) and (3.2.5–3.2.7) imply that for all k N ∈ k k k X X 0 X 0 ξj ξ η . ≤ j ≤ j j=1 (3.2.2–3.2.4) j=1 (3.2.5–3.2.7) j=1 P∞ P∞ 0 P∞ 0 0 Passing to the limit as k yields ξj = ξ = η since ξ η . Hence → ∞ j=1 j=1 j j=1 j ≺ ξ0 η0. ≺ By Case 2 and our earlier reduction (first paragraph of proof) applied to A = 0 0 0 diag 0, η and B = diag 0, ξ , we obtain a unitary U of the form IH W H1⊕H2 h i H1⊕H2 h i 1 ⊕ for which ∗ 0 0 E(UAU ) = B = diagH1⊕H2 0, ξ . 43 0 p ∞ Since U has this form, for all f, f fj and g gj one has ∈ { }j=1 ∈ { }j=1 0 = UAU ∗f, f 0 = UAU ∗f, g = UAU ∗g, f . h i h i Then for all m Np define Vm on span fm, gNm−(m−1) given by the unitary 2 2 matrix ∈ { } × a b m m Vm = (3.2.8) bm am − where s s 0 ξNm ξNm am = and bm = . 0 0 ξNm + ξNm ξNm + ξNm 0 0 ∗ Then Cm = 0 ξ0 is the compression of UAU to span fm, gN −(m−1) and if Nm−(m−1) { m } one interprets Vm as acting on this same subspace, one computes ξN ∗ m ∗ VmCmVm = (3.2.9) ξN 0 ∗ m because ∗ am bm 0 0 am bm VmCmVm = − 0 bm am 0 ξ bm am − Nm−(m−1) 0 bm(ξN + ξN 0 ) am bm m m − = 0 0 am(ξNm + ξNm ) bm am 2 0 0 bm(ξNm + ξN ) ambm(ξNm + ξN ) = m m 2 0 0 ambm(ξNm + ξNm ) am(ξNm + ξNm ) ξN m ∗ = . ξN 0 ∗ m 44 0 p Next let = span fm, g and let V be the unitary defined by H H { Nm−(m−1)}m=1 p M V = IH0 Vm. ⊕ m=1 From the above computations, one can see that the diagonal operator E(V UAU ∗V ∗) = Π∗(diag ξ)Π, for an appropriate permutation Π of the basis en n∈ . But conjugation by operators { } N which permute the basis corresponding to E, in particular Π, commutes with E, and ∗ ΠVU is unitary, so E((ΠVU)A(ΠVU) ) = diag ξ. Remark 3.2.5. In the proof of Case 1, we only used the facts that A had finite rank and s(B) s(A). Although not needed in this case, the other hypotheses hold automatically ≺ and for edification we explain why. Even though s(B) p s(A) for some p 0 implies ≺ ≥ s(B) s(A), one has the stronger converse: when s(A) has finite support, s(B) s(A) ≺ ≺ implies s(B) p s(A) for every p 0, i.e., s(B) ∞ s(A). Indeed, let Np be the largest ≺ ≥ ≺ index for which s(A) has a nonzero value. Then for all k Np, we have ≥ k+p k+p k X X X sj(B) sj(A) = sj(A) ≤ j=1 j=1 j=1 and therefore s(B) p s(A). Since p is arbitrary, s(B) ∞ s(A). This shows that the ≺ ≺ second inequality in the hypotheses is satisfied for p = since its right-hand side is ∞ infinite. The first inequality is satisfied since Tr R⊥ = because A is finite rank. A ∞ Remark 3.2.6 ([LW15, Remark 2.5] Orthostochasticity). In the above proof, if A is already diagonalized with respect to the basis en n∈ , then all the unitary operators { } N either are orthogonal with respect to this basis, or can be chosen as such. Indeed, Π and V are orthogonal, and U = IH W where W , coming from Theorem 2.1.14, can 1 ⊕ 45 be chosen to be orthogonal by [KW10, Corollary 6.1, NS(ii0) and S(ii0)]. Thus if is a H real Hilbert space and A is a positive compact operators, then it is diagonalizable via an orthogonal matrix, hence Theorem 3.2.4 is valid even when is real instead of complex. H + A consequence of this combined with Lemma 3.1.2 is that if ξ, η c , ξ p η and if ∈ 0 ≺ −1 −1 −1 ξ (0) η (0) p + ξ (0) , ≤ ≤ then there exists an orthostochastic matrix Q for which ξ = Qη. Example 3.2.7 ([KW10, Proposition 6.10, Example 6.11]). The following example, in conjunction with Lemma 1.6, will show that the condition s(B) p s(A) is not necessary ≺ for B E( (A)) and 0 < p := Tr R⊥ Tr R⊥ < . In particular, the converse of ∈ U A − B ∞ Theorem 3.2.4 fails for this case, but is true when Tr R⊥ Tr R⊥ is either infinite or A − B undefined as proved by Corollary 3.3.6. A counterexample is the following. Let η˜ = 0, η where 0 < η c∗. Let Q be an h i ∈ 0 orthostochastic matrix with the property that Qij = 0 if and only if i > j > 1 (so that otherwise Qij > 0 and rows and columns sum to one), e.g., [KW10, Example 6.11]. Then choosing ξ := Qη˜ we claim that ξ 1 η. One can see this from the calculation 6≺ n n n ∞ X ∗ X X X ξ ξi = Qijη˜j i ≥ i=1 i=1 i=1 j=1 n ∞ X X = Qijηj−1 i=1 j=2 n n ∞ n X X X X = Qijηj−1 + Qijηj−1 j=2 i=1 j=n+1 i=1 n−1 ∞ n n−1 X X X X = ηj + Qijηj−1 > ηj, (3.2.10) j=1 j=n+1 i=1 j=1 where the latter inequality follows since η > 0. 46 This example can be easily extended to create similar orthostochastic examples (e.g., Lp−1 1/2 1/2 Q˜ = Q ) for when 1 < p < and 0 < ξ p η, but ξ = Q˜η˜ for 1 1/2 1/2 ⊕ ∞ 6≺ η˜ = 0, η1, 0, η2,..., 0, ηp, ηp+1,... with p zeros. Then since Q˜ is orthostochastic, by h i Lemma 3.1.2 one has that diag ξ = U(diagη ˜)U ∗ for some orthogonal matrix U with 2 Q˜ij = Uij . Therefore p-majorization for any 1 p < is not necessary to characterize | | ≤ ∞ E( (A)). U To verify ξ p η observe that for sufficiently large n, one has 6≺ n p−1 n−2p+2 X ∗ X X ∗ ξ = ηi + (Q 0, ηp, ηp+1,... )i i h i i=1 i=1 i=1 p−1 n−2p+1 X X > ηi + ηi+p−1 (3.2.10) i=1 i=1 p−1 n−p n−p X X X = ηi + ηi = ηi. i=1 i=p i=1 3.2.1. Discussion on majorizations The Figures 2.1–2.2 at the end of the introduction show the interconnections between various types of majorization. Here through Proposition 3.2.10 we begin a discussion of some of these interconnections. Next we exhibit a relationship between strong majorization (recall Definition 2.1.3) and -majorization. As stated, Proposition 3.2.8 may seem 4 ∞ to apply to summable sequences, but in fact the hypotheses, majorization and not strong majorization, negate that possibility as addressed just after the proof. Proposition 3.2.8 ([LW15, Proposition 2.7]). If ξ, η c+ and ξ η, then ∈ 0 ≺ ξ η = ξ ∞ η. 64 ⇒ ≺ 47 Proof. It suffices to show that if ξ η, then the lim inf condition implies ξ p η for every ≺ ≺ p N. Indeed, suppose that ∈ n X lim inf (η∗ ξ∗) = > 0. n→∞ k − k k=1 + ∗ Then since ξ c one can choose N N for which ξ < for all k N. Then for all n ∈ 0 ∈ k 2p ≥ sufficiently large for which both n N and Pn (η∗ ξ∗) > one has that ≥ k=1 k − k 2 n+p n n X X X ξ∗ < ξ∗ + p < η∗. k k · 2p k k=1 k=1 k=1 Note that Proposition 3.2.8 applies only to ξ, η / `1, since if either ξ or η is summable, ∈ majorization implies both are summable, and in this case majorization and strong majorization are equivalent (see Definition 2.1.3, succeeding comment). Furthermore, the converse of Proposition 3.2.8 fails because there exist sequences ξ, η c∗ `1 for which ∈ 0 \ ξ ∞ η hence also ξ η, and yet ξ η, as exhibited in the following example. ≺ ≺ 4 Example 3.2.9 ([LW15, Example 2.8]). In this exposition so far we have not consid- ered any hands-on examples of -majorization. In refuting this possible converse of ∞ Proposition 3.2.8 we provide one, but first we explain our natural motivation for it. ⊥ Motivation. Suppose A + has Tr RA = = Tr R . Then there is some basis with ∈ K ∞ A respect to which A = diag s1(A), 0, s2(A), 0,... . It is a natural question to ask if there h i exists some B E( (A)) with Tr R⊥ = 0. The answer is yes by Theorem 3.2.4, but ∈ U B there is a more straightforward way to see this in this case. Indeed, if we let Rj denote L the 2 2 rotation by π/4 acting on the subspace span e2j−1, e2j , and U = Rj, then × { } ∗ 1 ∞ B := E(UAU ) = diag D2s(A), where D2ϕ denotes the 2-ampliation of ϕ ` , i.e., 2 ∈ D2ϕ = ϕ1, ϕ1, ϕ2, ϕ2,... . h i 48 ∗ 1 Example. Fix any η c and choose ξ := D2η ∞ η, where -majorization is easily ∈ 0 2 ≺ ∞ verified by taking Np = p. Then k P2n X j=n+1 ηj if k = 2n, n N; (ηj ξj) = ∈ − j=1 P2n−1 1 ηj + ηn if k = 2n 1, n N. j=n+1 2 − ∈ From this it follows that k k X k ηk (ηj ξj) η k . (3.2.11) 2 ≤ − ≤ 2 d /2e j=1 1 From the second inequality in (3.2.11) it follows that if lim inf kηk = 0, then ξ = 2 D2η 4 η −1 (e.g. for η = ((k + 1) log(k + 1)) , lim kηk = 0). However, the first inequality of 1 −1 (3.2.11) shows the inverse, that if lim inf kηk > 0, then ξ = D2η η (e.g. for η = k , 2 64 1 lim kη2k = 2 ). This example provides the failure of the converse of Proposition 3.2.8. ∗ 1 Moreover one has, for η c , D2η η if and only if lim inf kηk = 0. ∈ 0 2 4 The previous example shows that for appropriate η c∗, there exist ξ c∗ which are ∈ 0 ∈ 0 counterexamples to the converse of Proposition 3.2.8. However, with more work one can + ∗ show for every sequence η c , there is some ξ c with ξ ∞ η and ξ η, as the next ∈ 0 ∈ 0 ≺ 4 proposition shows. Though it will not be used later, we present it here for completeness. Proposition 3.2.10 ([LW15, Proposition 2.9]). For every η c+ there exists some ∈ 0 ∗ ξ c with ξ ∞ η and ξ η. ∈ 0 ≺ 4 Proof. Without loss of generality, we may assume η c∗ since the definitions of - ∈ 0 ∞ majorization and strong majorization depend only on its monotonization η∗. 1 If lim inf kηk = 0, set ξ = 2 D2η. By the example preceding this proposition, we have ξ ∞ η and ξ η. ≺ 4 It therefore suffices to assume lim inf kηk > 0. For this we adopt the following conventions. Let juxtaposition denote concatenation of finite sequences. Let `(ϕ) denote 49 P P`(ϕ) Pb the length of a finite sequence ϕ. Let ϕ be an abbreviation of j=1 ϕj and a ϕ Pb an abbreviation for j=a ϕj. Finally, if ϕ is a sequence (not necessarily finite) and b a, b dom ϕ with a b, let ϕ denote the finite subsequence ϕa, ϕa+1, . . . , ϕb . ∈ ≤ |a h i ∗ If lim inf kηk > 0 and η c , then η > 0. To construct ξ we proceed inductively. Let ∈ 0 0 k be any positive sequence converging to zero and let N = 1. Then let N1 be the h i 0 η 0 N0 smallest positive integer for which 0 < ηN . Let p1 be the largest positive integer 1 ≤ 2 j ηN0 k 0 0 0 for which p1ηN1 ηN , i.e., p1 := η 2 and (p1 + 1)ηN1 > ηN = η1. Then setting ≤ 0 N1 ≥ 0 p1 (1) N1 z }| { ξ = η ηN , . . . , ηN , |2 h 1 1 i this choice of p1 guarantees that N1 X X (1) 0 η ξ = η1 p1ηN < ηN . (3.2.12) ≤ − − 1 1 1 (1) N1 Denote the difference between the lengths of ξ and ηj by h i1 (1) M1 := `(ξ ) N1 = p1 1 1 − − ≥ 0 0 (1) P N1−1 and then exploiting η 0, choose N1 > `(ξ ) for which η N 0 −M < 1. Then one has → 1 1 0 0 N1−1 N1−1 0 X X (1) N1−1 0 η ξ η < ηN + 1 (3.2.13) ≤ − N1+1 1 1 1 50 0 (1) N1−1 0 because, noting that the length of ξ η is greater by M1 than N1 1, one has N1+1 − 0 0 0 N1−1 N1−1 N1−1 0 0 0 X X (1) N1−1 X X (1) N1−1 X N1−1 0 η ξ η N +1 = η ξ η N +1 + η N 0 −M ≤ − 1 − 1 1 1 1 1 1 N1 0 X X (1) X N1−1 (3.2.14) = η ξ + η N 0 −M − 1 1 1 < ηN1 + 1. Continuing with the induction, suppose that for some k N as in the previous k = 1 ∈ (k) (k) 0 case we are given a finite decreasing sequence ξ and Nk < Nk + Mk = `(ξ ) < Nk, (k) (k) with Mk k and with the last term of ξ being equal to ηN and ξ satisfying ≥ k Nk 0 X X (k) X Nk−1 0 η ξ < ηNk and η N 0 −M < k. (3.2.15) ≤ − k k 1 As in the previous k = 1 case, by an argument identical to that of (3.2.14), together these (3.2.15) inequalities imply 0 0 Nk−1 Nk−1 0 X X (k) Nk−1 0 η ξ η < ηN + k. (3.2.16) ≤ − Nk+1 k 1 1 (k+1) (k) 0 0 We then construct an extension ξ of ξ and Nk+1,Mk+1,Nk+1 with Nk < Nk+1 satisfying all of the properties in the preceding paragraph for k + 1 replacing k in all instances. The procedure mimics the base case as follows. Let Nk+1 be the smallest ηN0 0 k integer greater than N for which 0 < ηN . Along with (3.2.15) this inequality k k+1 ≤ 2 implies that Nk X (k) X ξ + 2ηN η + ηN 0 . k+1 ≤ k 1 51 Hence letting pk+1 be the largest positive integer for which Nk X (k) X ξ + pk+1ηN η + ηN 0 , (3.2.17) k+1 ≤ k 1 one has pk+1 2. And by the maximality of pk+1 one also has ≥ Nk X (k) X ξ + (p + 1)ηN > η + η 0 . (3.2.18) k+1 k+1 Nk 1 Adding to both sides of (3.2.17) and (3.2.18) the η-terms from Nk + 1 to Nk+1 excluding 0 (k+1) Nk, one defines ξ as denoted and obtains pk+1 times Nk+1 0 z }| { X (k) Nk−1 Nk+1 X ξ η N +1η N 0 +1 ηNk+1 , . . . , ηNk+1 η, (3.2.19) k k ≤ | {z } 1 ξ(k+1) and Nk+1 X (k+1) X ξ + ηNk+1 > η. (3.2.20) 1 So from (3.2.19)–(3.2.20) the difference of the (3.2.20) sums is nonnegative and less (k+1) than ηNk+1 . This shows that ξ , as defined in (3.2.19), satisfies the first inequalities of (3.2.15). Note further that ξ(k+1) is decreasing since ξ(k) and η are decreasing and (k) because the last term of ξ is ηNk . As for M1, and recalling that pk+1 2, set ≥ (k+1) (k) Mk+1:= `(ξ ) Nk+1 = (`(ξ ) + Nk+1 (Nk + 1) + pk+1) Nk+1 − − − = Mk 1 + pk+1 > Mk k, − ≥ ∗ 0 (k+1) and hence Mk+1 k + 1. Next, since η c we may choose some N > `(ξ ) ≥ ∈ 0 k+1 satisfying the last inequality of (3.2.15), for k + 1 replacing k. These facts again imply (3.2.16) for k + 1 replacing k by an argument identical to (3.2.14). 52 (k) 0 By induction, we have constructed ξ ,Nk,Mk,Nk with the desired properties (i.e., paragraph containing inequalities (3.2.15) and (3.2.16)). Furthermore, by construction each ξ(k) is an extension of the preceding ones. Thus, the infinite sequence ξ given by (k) (k) ξj := ξ when 1 j `(ξ ) is well-defined. Finally it suffices to show that ξ ∞ η j ≤ ≤ ≺ and ξ 4 η. In order to prove ξ ∞ η it suffices to observe the following two facts. Firstly, if ≺ 1 m N1 then ≤ ≤ m m m X X N1 X ξ = η ηN η. (3.2.21) 2 h 1 i ≤ 1 1 1 Secondly, for each k N, if Nk < m Nk+1 then ∈ ≤ m+M m X k X ξ η (3.2.22) ≤ 1 1 because m+Mk m+Mk 0 X X (k) Nk−1 Nk+1 ξ = ξ η η 0 ηNk+1 Nk+1 Nk+1 1 1 (k) `(ξ ) m+Mk 0 X (k) X (k) Nk−1 Nk+1 = ξ + ξ η η 0 ηNk+1 Nk+1 Nk+1 1 `(ξ(k))+1 (k) `(ξ ) m−Nk 0 X (k) X Nk−1 Nk+1 = ξ + η η 0 ηNk+1 Nk+1 Nk+1 1 1 Nk m−Nk m X X Nk+1 X η + η = η. ≤ Nk+1 1 1 1 Indeed, since Mk , these inequalities (3.2.21) and (3.2.22) imply ξ k η for infinitely ↑ ∞ ≺ many k N, i.e., ξ ∞ η. ∈ ≺ Finally, (3.2.16) implies ξ 4 η since 0 k N −1 X Xk lim inf (ηj ξj) lim inf (ηj ξj) lim inf(ηNk + k) = 0. k→∞ − ≤ k→∞ − ≤ k→∞ j=1 j=1 53 Operator consequences The following corollary of Theorem 3.2.4 and Proposition 3.2.8 gives a method of ensuring membership in E( (A)), for A +. The purpose of this U ∈ K corollary is to provide a more easily computable way to make this determination in special + cases. For if one is given sequences ξ, η c , establishing ξ p η or its negation seems ∈ 0 ≺ more difficult than verifying ξ η, which requires only ξ η and the strict positivity of 64 ≺ the associated lim inf condition. ⊥ Corollary 3.2.11 ([LW15, Corollary 2.10]). Suppose A, B +, B , Tr R ∈ K ∈ D B ≤ Tr R⊥, and s(B) s(A) but s(B) s(A), then B E( (A)). A ≺ 64 ∈ U Proof. By Proposition 3.2.8 one has s(B) ∞ s(A), and then using Theorem 3.2.4 one ≺ obtains B E( (A)). ∈ U 3.3. Approximate p-majorization (necessity) Theorem 3.2.4 of the last section shows that if p := Tr R⊥ Tr R⊥ 0 or when undefined A − B ≥ we set p = 0, then p-majorization (s(B) p s(A)) is a sufficient condition for membership ≺ in E( (A)), but Example 3.2.7 shows it is not necessary for 0 < p < . In our quest U ∞ to characterize E( (A)) in terms of sequence majorization, we introduce a new type of U majorization called approximate p-majorization[2], which is a necessary condition for membership in E( (A)). U Definition 3.3.1. Given ξ, η c+ and 0 p < , we say that ξ is approximately ∈ 0 ≤ ∞ p-majorized by η, denoted ξ p η, if ξ η and for every > 0, there exists an Np, N - ≺ ∈ such that for all n Np,, ≥ n+p n X X ξ∗ η∗ + η∗ . k ≤ k n+1 k=1 k=1 [2]The author, along with Gary Weiss, introduced the concept of approximate p-majorization in [LW15, Definition 3.1]. Cross-reference the footnote on page 37. The reader should note that the asterisks (indicating monotonization) which appear in the displayed inequality of Definition 3.2.2 are missing from [LW15, Definition 3.1]. This is a typo in [LW15]. 54 Furthermore, if ξ p η for infinitely many p N (equivalently obviously, for all p N), - ∈ ∈ this we call approximate -majorization and denote it by ξ ∞ η. ∞ - Remark 3.3.2 ([LW15, Remark 3.2]). Notice from the above definition that if ξ is p-majorized by η, then ξ is trivially approximately p-majorized by η. However, there is a partial converse with a small loss in that approximate p-majorization implies (p 1)- − majorization. That is, if p > 0 and ξ is approximately p-majorized by η, then by choosing < 1, from the above display ξ is (p 1)-majorized by η. Combining these two facts − yields that ξ ∞ η if and only if ξ ∞ η, which we will exploit later. Furthermore, by ≺ - Remark 3.2.5, if η has only finitely many nonzero terms, then any ξ which is majorized by η is -majorized by η and so also approximately -majorized by η. ∞ ∞ Example 3.3.3 ([LW15, Example 3.3]). It is important to note that p-majorization is distinct from approximate p-majorization. That is, for each 0 < p < we exhibit ∞ + sequences ξ, η c with ξ p η but ξ p η. When p = 1, it suffices to consider the ∈ 0 - 6≺ k+1 −k ∗ sequences ξ = (2 −3)/22k and η = 2 . Elementary calculations verify that ξ, η c ∈ 0 ∗ ∗ (that is, ξ = ξ and η = η ), and ξ 1 η but ξ 1 η. To produce analogous sequences for - 6≺ any p > 1, define p − 1 times (p) z }| { (p) ξ := 1,..., 1 , ξ1, ξ2,... and η := p 1, η1, η2,... . h i h − i (p) (p) (p) (p) Then ξ p η but ξ p η , which proves that p-majorization and approximate - 6≺ p-majorization are distinct. However, it should be noted that these examples were not nearly as easy for us to come by as those for p-majorization. In particular, the examples immediately preceding Remark 3.2.3 came naturally, but the single example above took some effort. For further discussion on pairs of sequences ξ p η but ξ p η see unifying - 6≺ Remark 3.3.9. 55 Our main theorem on necessity for membership in E( (A)) depends on approximate U p-majorization. Theorem 3.3.4 ([LW15, Theorem 3.4]). Suppose A + and B E( (A)). If ∈ K ∈ U ⊥ ⊥ p = min n N 0, Tr RA Tr RB + n , { ∈ ∪ { ∞} | ≤ } then s(B) -p s(A). Proof. Suppose A, B and p are as in the hypotheses of this theorem. We may assume that A has infinite rank for otherwise the conclusion holds because of Remark 3.3.2 and Theorem 3.2.4 proof of Case 1. We may also assume that N := Tr R⊥ < . Otherwise, Tr R⊥ = = Tr R⊥ by B ∞ B ∞ A Proposition 3.2.1 and therefore p = 0. Thus we would need to prove s(B) -0 s(A) which is equivalent to s(B) s(A), and this holds by Theorem 2.1.13. ≺ Note that since Tr R⊥ < , p satisfies Tr R⊥ = Tr R⊥ + p = N + p. It suffices to show B ∞ A B ⊥ s(B) r s(A) for all r N with r p. Because Tr R = N + p N + r, without loss - ∈ ≤ A ≥ of generality via unitary equivalence for A and permutations for B, and the fact that conjugation by a permutation commutes with the expectation E, we may assume that A = diag 0,..., 0, η˜ = diag η0 and B = diag 0,..., 0, s(B) = diag ξ, h| {z } i h| {z } i N+r N where η˜ is s(A) interspersed with p r zeros. (To aid intuition, in case p < we may − ∞ choose r = p and hence also η˜ = s(A).) Since B E( (A)) by Lemma 3.1.2(iv), there ∈ U 0 0 exists a unistochastic matrix Q = (qij) for which Qη = ξ. However, only the double stochasticity of Q is used here. For all m N one has ∈ m m m ∞ ∞ m X X 0 X X 0 0 X 0 X 0 ξi = (Qη )i = qijηj = ηj qij. (3.3.1) i=1 i=1 i=1 j=1 j=1 i=1 56 0 For a doubly stochastic matrix Q = (qij), denote the last quantity in equation (3.3.1) as 0 fm(Q, η ). Now fix any 0 < < 1, and choose N + r < Nr, N for which ∈ Nr, N+r X X N + r < q0 . (3.3.2) − ij i=1 j=1 The existence of Nr, follows from column-stochasticity, which yields ∞ N+r X X 0 qij = N + r. i=1 j=1 Certainly inequality (3.3.2) holds with Nr, replaced by any m Nr,, since Q is a doubly ≥ stochastic matrix and so its entries are nonnegative. Consider a permutation matrix Πm which fixes the first N + r coordinates and has the property that 0 Πmη = η := 0,..., 0, s1(A), s2(A), . . . , sm−N−r+1(A), ηm+2, ηm+3,... , h| {z } | {z } i N+r m−N−r+1 where ηm+2, ηm+3,... are the remaining η˜-terms. (To aid intuition, when p < and we ∞ choose r = p and η˜ = s(A), η0 is already in this form without need of the permutation Πm.) Then notice that Q is both doubly stochastic and satisfies inequality (3.3.2) if and −1 only if QΠm does the same. Notice also that inequality (3.3.2) depends only on the first N + r columns of Q. Then direct computations show 0 −1 0 −1 0 −1 fm(Q, η ) = fm(QΠm Πm, η ) = fm(QΠm , Πmη ) = fm(QΠm , η). (3.3.3) −1 Denote the entries of QΠm by qij. From the definition of η above it is clear that −1 ηm+1 ηk whenever k m + 1. This yields an upper bound for fm(QΠ , η) when ≥ ≥ m 57 m Nr, ≥ ∞ m −1 X X fm(QΠm , η) = ηj qij j=1 i=1 m m ∞ m X X X X = ηj qij + ηj qij j=N+r+1 i=1 j=m+1 i=1 m m ∞ m X X X X ηj qij + sup ηk qij ≤ j=N+r+1 i=1 k>m j=m+1 i=1 m m m m X X X X = ηj qij + ηm+1 m qij − j=N+r+1 i=1 j=1 i=1 m m m m ! X X X X = ηj qij + ηm+1 1 qij − j=N+r+1 i=1 j=1 i=1 m m m m ! X X X X = ηj qij + ηm+1 1 qij − j=N+r+1 i=1 j=N+r+1 i=1 N+r m ! X X + ηm+1 1 qij − j=1 i=1 m m m m ! X X X X ηj qij + ηj 1 qij ≤ − j=N+r+1 i=1 j=N+r+1 i=1 N+r m X X + ηm+1 N + r qij − j=1 i=1 m X < ηj + ηm+1 by (3.3.2). j=N+r+1 The inequalities above along with equation (3.3.3) yield m−N m X X 0 −1 si(B) = ξi = fm(Q, η ) = fm(QΠm , η) i=1 i=1 m X ηj + ηm+1 ≤ j=N+r+1 m−N−r X = si(A) + sm−N−r+1(A). i=1 58 Since is arbitrary, s(B) r s(A), and since the positive integer r p is arbitrary, - ≤ s(B) -p s(A). The next corollary is an analogue of Theorem 3.1.3. + Corollary 3.3.5. Let ξ, η c be sequences such that p := η−1(0) ξ−1(0) 0 (with ∈ 0 − ≥ the convention that p = 0 when both sets are infinite). Then = ξ p η ξ = Qη for some orthostochastic matrix Q ⇐ ≺ = ξ p η ⇒ - Proof. Apply Remark 3.2.6 for one implication. For the other implication, apply Lemma 3.1.2 and Theorem 3.3.4. One of our main results is Corollary 3.3.6 which, in the rather general setting where A has infinite rank and infinite dimensional kernel, we obtain a precise characterization of E( (A)) in terms of majorization and -majorization. U ∞ Corollary 3.3.6 ([LW15, Corollary 3.5]). Suppose A + has infinite rank and infinite ∈ K ⊥ dimensional kernel (Tr RA = = Tr R ). Then ∞ A E( (A)) = E( (A))fk E( (A))ik, U U t U the members of E( (A)) with finite dimensional kernel and infinite dimensional kernel, U respectively, are characterized by ⊥ E( (A))fk = B + s(B) ∞ s(A) and Tr R < U { ∈ D ∩ K | ≺ B ∞} and ⊥ E( (A))ik = B + s(B) s(A) and Tr R = . U { ∈ D ∩ K | ≺ B ∞} 59 Proof. If B E( (A)), then B E( (A)) and we know that s(B) s(A) by Theo- ∈ U ∈ V ≺ ⊥ rem 2.1.13. But when Tr R < , from Theorem 3.3.4 we know that s(B) ∞ s(A) B ∞ - which is equivalent to s(B) ∞ s(A). Thus the left-hand set in Corollary 3.3.6 is ≺ contained in the right-hand set. ⊥ Next suppose that B + lies in the right-hand set. If Tr R = , then ∈ D ∩ K B ∞ Theorem 3.2.4 with p = 0 shows that B E( (A)). Similarly, if Tr R⊥ < , then ∈ U B ∞ s(B) ∞ s(A) and again, by Theorem 3.2.4 for p = , we find that B E( (A)). ≺ ∞ ∈ U Corollary 3.3.6 can be expressed as [ n ⊥ o E( (A)) = B + Tr R + p = and s(B) p s(A) . U ∈ D ∩ K B ∞ - 0≤p≤∞ This motivates the following conjectured characterization for E( (A)), which remains an U open problem. And if this conjecture should prove false, is there a proper majorization characterization of E( (A))? U Conjecture 3.3.7 ([LW15, Conjecture 3.6]). Let A +. Then ∈ K [ n ⊥ ⊥ ⊥ o E( (A)) = B + s(B) p s(A) and Tr R Tr R Tr R + p . U ∈ D ∩ K - B ≤ A ≤ B 0≤p≤∞ In general, we have the following squeeze theorem for diagonals of positive compact operators. Corollary 3.3.8. Let A +. Then ∈ K [ n ⊥ ⊥ ⊥ o B + s(B) p s(A) and Tr R Tr R Tr R + p ∈ D ∩ K ≺ B ≤ A ≤ B 0≤p≤∞ E( (A)) ⊆ U ⊆ [ n ⊥ ⊥ ⊥ o B + s(B) p s(A) and Tr R Tr R Tr R + p . ∈ D ∩ K - B ≤ A ≤ B 0≤p≤∞ 60 Moreover, these are equal when Tr R⊥ 0, , otherwise the first inclusion is proper. A ∈ { ∞} Comparing the definitions of p-majorization (Definition 3.2.2) and approximate p- majorization (Definition 3.3.1), we see that they are somehow -close in a vague sense. Thus, although Corollary 3.3.8 does not provide a complete characterization of diagonals of positive compact operators with nonzero finite dimensional kernel, it does squeeze them between these two -close sets. The following remark provides a method for producing pairs of sequences ξ c+ and ∈ 0 ∗ η c for which ξ p η but ξ p η. ∈ 0 - 6≺ Remark 3.3.9 ([LW15, Remark 3.7]). Recall that Example 3.2.7 provided an orthos- tochastic matrix Q such that for every η c∗, setting ξ = Qη˜, where η˜ = 0, η , yields ∈ 0 h i ξ 1 η. Using Lemma 3.1.2 one has diag ξ E( (diag η)). By Theorem 3.3.4, 6≺ ∈ U ∗ ∗ ξ = s(diag ξ) -1 s(diagη ˜) =η ˜ = η, hence ξ -1 η. In general, given two finite sequences, say ϕ, ζ, of lengths n and n + p Pn Pn+p (where n is arbitrary) and having the same sum (i.e., j=1 ϕj = j=1 ζj), one can 0 0 0 0 prepend ϕ to η and ζ to ξ, clearly obtaining new sequences ξ , η with ξ -p η but 0 0 ξ p η . 6≺ Although we did not mention it earlier, in Example 3.2.7 there is a substantial amount of freedom in choosing Q. In particular, examination of [KW10, Example 6.11] ensures that each sum-1 strictly positive column vector followed by a Gram–Schmidt process produces a distinct orthostochastic matrix Q that can be used in Example 3.2.7. Perhaps these orthostochastic Q can be exploited or modified to prove Conjecture 3.3.7. 61 3.4. E(U(A)) convexity Historically, convexity played a central role and is ubiquitous in majorization theory. For example, Horn [Hor54, Theorem 1] integrates theorems of Hardy, Littlewood and P´olya [HLP88] and Birkhoff [Bir46] to prove n n ξ R ξ η = co η˜ R η˜k = ηπ(k), π Πn , { ∈ | ≺ } { ∈ | ∈ } where Πn is the set of n n permutation matrices. For operators, in [Hor54], using × ∗ [Sch23], Horn proved that E( (X)) is convex whenever X = X Mn(C) by establishing U ∈ the characterization n E( (X)) = diag d d R , d λ , U { | ∈ ≺ } where λ is the eigenvalue sequence of X (Theorem 1.2.5). However, the verification that E( (X)) is convex is immediate from its majorization characterization even without the U theorem of Horn integrating Hardy, Littlewood, P´olya and Birkhoff mentioned above. Likewise it is straightforward to verify that if η c+, then ∈ 0 ξ c+ ξ η { ∈ 0 | ≺ } is convex. In particular, this leads to the results of Kaftal and Weiss on the convexity of the expectation of the partial isometry orbit of a positive compact operator. Corollary 3.4.1 ([KW10, Corollary 6.7]). Let A +. Then ∈ K E( (A)) is convex. • V If RA = I or A has finite rank, then E( (A)) is convex. • U Since we have a characterization of E( (A)) when A + has both infinite rank and U ∈ K infinite dimensional kernel, it seems natural to ask if E( (A)) is convex in this case. U 62 The answer is positive, however the verification is much less obvious to us (see below Corollary 3.4.3). But first, a lemma. Lemma 3.4.2 ([LW15, Lemma 4.2]). Suppose that ξ, ζ, η c+, 0 < λ < 1, 0 p, q ∈ 0 ≤ ≤ −1 −1 −1 −1 such that ξ p η, ζ q η. If r = min p + ξ (0) ζ (0) , q + ζ (0) ξ (0) , ∞ - - { \ \ } then λξ + (1 λ)ζ r η. − - Proof. Set ϕ := λξ + (1 λ)ζ. There are two cases: either η has finite support, or − not. If the former, then since ξ η and ζ η, one easily has ϕ η by the comment ≺ ≺ ≺ immediately preceding Corollary 3.4.1. Then since η has finite support, we can improve this to ϕ ∞ η (see Remark 3.3.2 or proof of Theorem 3.2.4 Case 1), which is equivalent ≺ to ϕ -∞ η. Thus ϕ -r η. The second case: η has infinite support. For now, suppose both p, q are finite. Let −1 ∗ −1 π : N N ϕ (0) be a bijection monotonizing ϕ: ϕ = ϕπ(k). Since ϕ (0) = → \ k ξ−1(0) ζ−1(0), one has ∩ −1 −1 −1 −1 −1 −1 −1 N ϕ (0) = N (ξ (0) ζ (0)) ξ (0) ζ (0) ζ (0) ξ (0) , (3.4.1) \ \ ∪ t \ t \ which immediately yields the disjoint union −1 −1 π([1, m]) = π([1, m]) N (ξ (0) ζ (0)) (cardinality km) ∩ \ ∪ −1 −1 π([1, m]) ξ (0) ζ (0) (cardinality sm) t ∩ \ −1 −1 π([1, m]) ζ (0) ξ (0) (cardinality tm) t ∩ \ for each m N. Therefore m = km + sm + tm, and each term increases to the cardinality ∈ of its corresponding set from equation (3.4.1). From this cardinality equation, it is clear that m sm = km + tm and m tm = km + sm are both increasing sequences. However, − − 63 we may further conclude that they increase without bound. Indeed, −1 −1 m sm = π([1, m]) N (ξ (0) ζ (0)) − ∩ \ ∪ −1 −1 + π([1, m]) ξ (0) ζ (0) ∩ \ −1 = π([1, m]) N ζ (0) , ∩ \ −1 which shows m sm N ζ (0) = supp ζ . Likewise m tm supp ξ . Moreover, ξ, ζ − ↑ \ | | − ↑ | | are both infinitely supported since they are majorized by η and η is infinitely supported (as shown simply in the proof of Theorem 3.2.4 Case 2, first paragraph). Thus we have verified m sm, m tm . Therefore, given 0 < < 1, once m is sufficiently large so − − ↑ ∞ that m sm Np, and m tm Nq,, one has − ≥ − ≥ m m m X X X ϕ∗ = λ ξ + (1 λ) ζ k π(k) − π(k) k=1 k=1 k=1 m−s m−t Xm Xm λ ξ∗ + (1 λ) ζ∗ ≤ k − k k=1 k=1 m−sm−p ! X ∗ λ η + ηm−s −p+1 ≤ k m k=1 m−tm−q ! X + (1 λ) η∗ + η∗ − k m−tm−q+1 k=1 m−r Xm η∗ + η∗ , ≤ k m−rm+1 k=1 where rm = min sm + p, tm + q . The above computation proves that ϕ r η. But { } - m since rm r, either rm , or eventually rm = r. In either case ϕ r η. ↑ ↑ ∞ - Finally, to remove the restriction that p, q be finite, observe that the above proof 0 0 0 0 actually showed that for any p p, q q with p , q < , one has ϕ r0 η, where ≤ ≤ ∞ - m 0 0 0 r = min sm + p , tm + q . The proof now splits into several subcases. m { } 64 If p < , q = and ξ−1(0) ζ−1(0) < , then if one chooses both p0 = p and ∞ ∞ \ ∞ q0 = p + ξ−1(0) ζ−1(0) , one has r0 = r = p + ξ−1(0) ζ−1(0) for m sufficiently large \ m \ −1 −1 so that sm = ξ (0) ζ (0) . \ If p < , q = and ξ−1(0) ζ−1(0) = , then for p0 = p and any q0 < , ∞ ∞ \ ∞ ∞ 0 0 0 eventually r reaches q since sm . Therefore ϕ q0 η, which means ϕ ∞ η since q m ↑ ∞ - - was arbitrary. The case where p = and q < holds by symmetric arguments. ∞ ∞ 0 0 0 If p = = q, then for p = q = k N, one has rm k and so ϕ k η, hence ∞ ∈ ≥ - ϕ -∞ η. Examination of the proof of Lemma 3.4.2 actually shows that we may replace approx- imate p-majorization with p-majorization everywhere in the statement of the lemma and the result remains valid. Indeed, the only difference in the proof is that the terms involving disappear when p-majorization is used. An operator E( (A)) consequence of this is: U ⊥ Corollary 3.4.3 ([LW15, Corollary 4.3]). If A + and Tr RA = = Tr R , then ∈ K ∞ A E( (A)) is convex. U Proof. Take B,C E( (A)) and 0 < λ < 1. Let D = λB + (1 λ)C and also let ∈ U − d = λb + (1 λ)c be their corresponding diagonal sequences in c+. If Tr R⊥ = , − 0 D ∞ that D E( (A)) follows immediately from Corollary 3.3.6 and from the fact that ∈ U convex combinations of elements majorized by η are themselves majorized by η. However, if Tr R⊥ < , we use the dichotomy of Corollary 3.3.6 for B,C. Observing that D ∞ Tr R⊥ = b−1(0) and similarly for C,D, simply notice that either b−1(0) d−1(0) = B \ ∞ −1 −1 ⊥ or b (0) d (0) < (so Tr R < ), in which case b ∞ a by Corollary 3.3.6. \ ∞ B ∞ ≺ −1 −1 Likewise, c (0) d (0) = or c ∞ a. Thus by Lemma 3.4.2 one has d ∞ a. So \ ∞ ≺ ≺ by Corollary 3.3.6 D E( (A)). ∈ U 65 Chapter 4 Diagonals of idempotents The vast majority of the results of this chapter have been published in [LW16]. This chapter is devoted to the investigation of diagonals of idempotent operators on B( ). The H case when is finite dimensional was already studied by Giol, Kovalev, Larson, Nguyen H and Tener in [Gio+11] which we stated as Theorem 2.2.2. Motivated by Questions 2.2.3 and 2.2.4 due to Jasper, in Section 4.1 we investigate idempotents with zero diagonal. In Theorem 4.1.5 we provide several equivalent conditions for an idempotent to have a zero diagonal, including one which is basis[1] independent. Algorithm 4.1.11 provides an algorithm to generate a basis with respect to which a suitable idempotent has zero diagonal. In Section 4.2 we characterize the diagonals of the class of idempotent operators, concluding in Theorem 4.2.7 that any bounded sequence appears as the diagonal of some idempotent. Since this result is such a drastic departure from the nature of Theorem 2.2.2 and to tie up loose ends, we describe in Section 4.3 the diagonals of the class of finite rank idempotent operators with Theorem 4.3.3. [1]A “basis” herein is always an orthonormal basis of the underlying Hilbert space. 66 4.1. Zero-diagonal idempotents We begin with a canonical decomposition of idempotents into 2 2 operator matrices. × Lemma 4.1.1 ([LW16, Lemma 2.1]). Let D2 = D B( ) be an idempotent. Then ∈ H with respect to the decomposition = ker⊥ D ker D, D has the following block matrix H ⊕ form: I 0 D = , T 0 where I B(ker⊥ D) is the identity operator and T B(ker⊥ D, ker D) is a bounded ∈ ∈ operator which we call the nilpotent part of the idempotent D, short for the corner of 0 0 the nilpotent operator T 0 . Note that the term ‘nilpotent part’ is a natural slight abuse of language in that T itself is not nilpotent; T 2 is not even defined. Proof. The only non-obvious fact we must prove is that the upper left-hand corner of D is the identity on the compression to ker⊥ D. To verify this let x 0 ker⊥ D be ⊕ ∈ arbitrary and let D(x 0) = y z. Then because D is idempotent one has ⊕ ⊕ D(x 0) = D2(x 0) = D(y z) = D(y 0). ⊕ ⊕ ⊕ ⊕ Since x 0, y 0 ker⊥ D on which D acts one-to-one, x = y. ⊕ ⊕ ∈ An important stepping stone to our first main theorem is the following proposition in which the idempotent acts on and its nilpotent part is normal. H ⊕ H 67 Proposition 4.1.2 ([LW16, Proposition 2.2]). Suppose is separable infinite dimen- H sional and the idempotent D B( ) has the respective block matrix form ∈ H ⊕ H I 0 D = T 0 where T B( ) is normal. Then ∈ H iπ/2 1 1 (i) both (Im D)± = (Re e D)± if and only if T ; ∈ L ∈ L iθ 1 (ii) (Re e D)+ / for −π/2 < θ < π/2; ∈ L iθ 1 2 (iii) (Re e D)− for −π/2 < θ < π/2 if and only if T . ∈ L ∈ L Proof. The core of the proof is an analysis of the 2 2 case followed by a straightforward × application of the Borel functional calculus to the operator case. For z C, let Az M2(C) be given by ∈ ∈ 1 0 Az := . z 0 Then fixing π/2 < θ π/2, − ≤ 2 cos θ e−iθz¯ iθ iθ −iθ ∗ 2(Re e Az) = e Az + e Az = , eiθz 0 iθ 2 2 which has characteristic polynomial det(λ 2(Re e Az)) = λ 2 cos θλ z . Hence − − − | | iθ the selfadjoint matrix 2(Re e Az) has eigenvalues which depend on z by p 2 2 λ±(z) = cos θ cos θ + z . (4.1.1) ± | | 68 When z = 0, normalized eigenvectors corresponding to these eigenvalues are 6 λ+(z) λ−(z) 2 2 2 2 √λ+(z)+|z| √λ−(z)+|z| x+(z) = and x−(z) = . (4.1.2) eiθz eiθz 2 2 2 2 √λ+(z)+|z| √λ−(z)+|z| On the other hand, when z = 0, the normalized eigenvectors are just the standard basis 1 0 x+(0) = ( 0 ) and x−(0) = ( 1 ). We now return to the operator case. Since T B( ) is normal, the Borel functional ∈ H calculus provides a -homomorphism Φ : (σ(T )) W ∗(T ) from the bounded Borel ∗ B → functions on the spectrum of T to the abelian von Neumann algebra generated by T Φ for which Id T , where the identity function on σ(T ) is Id (z) = z [KR97a, σ(T ) 7−→ σ(T ) Theorem 5.2.9]. Moreover, since Φ is a -homomorphism, it preserves the partial order ∗ on selfadjoint elements. Let 1 (σ(T )) denote the identity element (the map z 1) ∈ B 7→ of the algebra (σ(T )), and xi (i = 1, 2) the coordinate functions of the eigenvectors B ± obtained in (4.1.2), which are bounded Borel functions on C. Define 1 1 Φ(x+) Φ(x−) U := , 2 2 Φ(x+) Φ(x−) 2 which is unitary on because Φ is a -homomorphism and x±(z) form a basis for C H⊕H ∗ { } 1 1 2 2 1 2 2 2 for every z C. That is, because the z-functions x+x− +x+x− 0 and x± + x± 1 ∈ ≡ ≡ and Φ(1) = I is the identity on . And for what follows, recall Φ(Id ) = T . H σ(T ) Furthermore, because Φ(1) Φ(0 1) · D = , Φ(Id ) Φ(0 1) σ(T ) · 69 where here denotes multiplication by scalars in the algebra (σ(T )) and hence 0 1 is · B · simply the zero function, and so also Φ(2 cos θ 1) Φ(e−iθ Id ) iθ · · σ(T ) 2(Re e D) = , Φ(eiθ Id ) Φ(0 1) · σ(T ) · one obtains Φ(λ+) Φ(0 1) ∗ iθ · U 2(Re e D)U = . Φ(0 1) Φ(λ−) · When π/2 < θ < π/2 one has cos θ > 0, and therefore λ+ 0 and λ− 0. Hence − ≥ ≤ ∗ iθ ∗ iθ (U 2(Re e D)U)+ = Φ(λ+) 0 and (U 2(Re e D)U)− = 0 Φ( λ−). (4.1.3) ⊕ ⊕ − Moreover, for all z C, ∈ p 2 2 λ+(z) = cos θ + cos θ + z 2 cos θ. | | ≥ Furthermore, for the same range of θ, and for all z lying inside the closed ball B¯(0; T ) k k ⊇ σ(T ), 2 p 2 2 z λ−(z) = cos θ + z cos θ = p| | − | | − cos θ + cos2 θ + z 2 | | z 2 p| | , ≥ cos θ + cos2 θ + T 2 k k and z 2 z 2 λ−(z) = p| | | | . − cos θ + cos2 θ + z 2 ≤ 2 cos θ | | 70 From these inequalities, as Borel functions on the spectrum of T , we have the following z-function inequalities for π/2 < θ < π/2: − 2 2 λ+ 2 cos θ 1 and C1 Idσ(T ) λ− C2 Idσ(T ) , (4.1.4) ≥ · · ≤ − ≤ · 1 where C1,C2 are the positive constants given by C1 := and C2 := cos θ+√cos2 θ+kT k2 1 2 . After applying Φ to these inequalities, one has Φ(λ+) (2 cos θ)I and C1 T 2 cos θ ≥ | | ≤ 2 Φ( λ−) C2 T . Applying the trace yields − ≤ | | ∗ iθ TrH⊕H(U 2(Re e D)U)+ = TrH⊕H(Φ(λ+) 0) ⊕ (4.1.5) = TrH Φ(λ+) (2 cos θ) TrH I = , ≥ ∞ and 2 2 C1 TrH T TrH Φ( λ−) C2 TrH T . (4.1.6) | | ≤ − ≤ | | Because ∗ iθ TrH⊕H(U 2(Re e D)U)− = TrH⊕H(0 Φ( λ−)) = TrH Φ( λ−) (4.1.7) ⊕ − − and ∗ iθ iθ TrH⊕H(U 2(Re e D)U)± = TrH⊕H(2(Re e D))±, (4.1.8) inequalities (4.1.5)–(4.1.6) prove (ii) and (iii). To prove (i), simply notice that when θ = π/2, we have λ+ = λ− = Id and apply the same arguments as above in (4.1.5) − | σ(T )| and (4.1.7) along with the fact that Φ( Id ) = T | σ(T )| | | The following remark shows that idempotents can be decomposed even further than the 2 2 matrix of Lemma 4.1.1. × Remark 4.1.3 ([LW16, Remark 2.3]). With the same notation as Lemma 4.1.1, we may further decompose the underlying space as ker⊥ D = ker T ker⊥ T and ker D = ⊕ 71 ran⊥ T ran T , where ker⊥ T := ker⊥ D ker T and ran⊥ T := ker D ran T . With ⊕ respect to the ordering of subspaces = ker T ran⊥ T ker⊥ T ran T one can write H ⊕ ⊕ ⊕ I 0 0 0 0 0 0 0 D = , 0 0 I 0 0 0 T˜ 0 where T˜ B(ker⊥ T, ran T ), and the identity operators act on the appropriate spaces. In ∈ the decomposition above we have used the ordering of subspaces ker T ran⊥ T ker⊥ T ⊕ ⊕ ⊕ ran T , which makes it clear that D can be written as the direct sum of a projection and another idempotent. It is possible for this decomposition to degenerate into simpler ones if, say, ker T = 0 , in which the first row and column would disappear. Other rows and { } columns would disappear if their corresponding subspaces were zero, but none of this is problematic. ⊥ If Q3 : ker T denotes the (linear) inclusion operator and Q4 : ran T the → H H → projection operator, then T˜ = Q4TQ3. From this it is clear that T˜ is injective and has dense range. Furthermore, if T˜ = U T˜ is the polar decomposition for T˜, then | | U : ker⊥ T ran T is unitary (i.e., a surjective isometry, see [Hal82, Problem 134 and → corollaries]). Conjugating D by the unitary V := I I I U ∗ B( , 0), where ⊕ ⊕ ⊕ ∈ H H 0 = ker T ran⊥ T ker⊥ T ker⊥ T , one obtains H ⊕ ⊕ ⊕ I 0 0 0 0 0 0 0 0 ∗ D := VDV = . 0 0 I 0 0 0 T˜ 0 | | We need one more lemma before we can prove our main theorem for this section. 72 Lemma 4.1.4 ([LW16, Lemma 2.4]). Let be a two-sided ideal of B( ) and let I H ∗ ∗ B = B and A = A B( ). Then A+ if and only if (A + B)+ . Similarly, ∈ I ∈ H ∈ I ∈ I A− if and only if (A + B)− . ∈ I ∈ I Proof. Let RA+ be the range projection of the positive part A+ of A. Then since A + B (A + B)+, one has A (A + B)+ B. Therefore ≤ ≤ − A+ = RA ARA RA ((A + B)+ B)RA , + + ≤ + − + and hence A+ whenever (A+B)+ . Here we are using the fact that two-sided ideals ∈ I ∈ I of B( ) are hereditary, which is a well-known consequence of Calkin’s characterization H of ideals of B( ) in terms of their s-numbers in [Cal41]. H For the other implication, make the substitutions A A + B, B B and apply the 7→ 7→ − result just proved. More precisely, one obtains (A + B)+ P ((A + B) B)+ + B)P = P (A+ + B)P, ≤ − where P := R . Hence (A + B)+ if A+ . (A+B)+ ∈ I ∈ I To see that A− if and only if (A + B)− , note that A− = ( A)+ and apply the ∈ I ∈ I − result just proved. We are now in a position to prove our first main theorem. Theorem 4.1.5 ([LW16, Theorem 2.5]). For D B( ) an infinite rank idempotent ∈ H the following are equivalent: (i) D is not a Hilbert–Schmidt perturbation of a projection; (ii) the nilpotent part of D is not Hilbert–Schmidt; (iii) R Tr D = C; { } (iv) D is zero-diagonal; 73 (v) D has an absolutely summable diagonal; (vi) D has a summable diagonal (i.e., R Tr D = ). { } 6 ∅ Proof. The implication (i) = (ii) is clear, as are the implications (iv) = (v) = (vi). ⇒ ⇒ ⇒ The implication (iii) = (iv) is a direct consequence of Theorem 2.2.5, for if R Tr T = C, ⇒ { } then there exists a basis e with respect to which Tre T = 0, and thus by Theorem 2.2.5, T is zero-diagonal. Hence the main thrust of this theorem is proving the implications (vi) = (ii) = (iii) and (ii) = (i). ⇒ ⇒ ⇒ The remainder of the proof is structured as follows. We use Lemma 4.1.4, Remark 4.1.3, Proposition 4.1.2, and Theorem 2.2.7 to prove the implications (vi) = (ii) = (iii) which, ⇒ ⇒ with the above paragraph, establishes the equivalences (ii)–(vi). Having demonstrated these equivalences, we prove (iv) = (i) in lieu of (ii) = (i). ⇒ ⇒ (vi) = (ii). We will prove this via the contrapositive, that the nilpotent part of D ⇒ being Hilbert–Schmidt implies R Tr D = . So suppose the nilpotent part of D is { } ∅ Hilbert–Schmidt. Case 1: The nilpotent part of D has finite rank. By Lemma 4.1.1, D has the form I 0 D = . T 0 Set I 0 0 T ∗ 1 A = and B = , 0 0 2 T 0 and so Re D = A + B. By hypothesis, T has finite rank hence B has finite rank. Since 1 1 1 A = A+ / and B because B has finite rank, (Re D)+ = (A + B)+ / by ∈ L ∈ L ∈ L 1 Lemma 4.1.4. However, A− = 0 and so again Lemma 4.1.4 ensures (Re D)− = ∈ L 74 1 (A + B)− . Therefore, ∈ L 1 1 (Re D)+ / and (Re D)− . (4.1.9) ∈ L ∈ L Then Theorem 2.2.7(iv) with θ = 0 ensures R Tr D = . { } ∅ Case 2: The nilpotent part of D has infinite rank. By Remark 4.1.3, write I 0 0 0 0 0 0 0 0 D = , 0 0 I 0 0 0 T˜ 0 | | and from T˜ = Q4TQ3 we know that T˜ is Hilbert–Schmidt, and since T˜ has dense range | | in ran T which is infinite dimensional T˜, and hence also T˜ , have infinite rank. Define | | J := ker T ran⊥ T and K := ker⊥ T , then set P B(J) and D˜ B(K K) to ⊕ ∈ ∈ ⊕ I 0 I 0 ˜ P := and D := 0 0 T˜ 0 | | 0 1 Then D˜ satisfies the conditions of Proposition 4.1.2 and so (Re D )+ = P (Re D˜)+ / ⊕ ∈ L because TrH(P (Re D˜)+) = TrJ P + TrK⊕K (Re D˜)+ TrK⊕K (Re D˜)+ = . ⊕ ≥ 4.1.2(ii) ∞ 0 1 Furthermore, (Re D )− because ∈ L 0 (Re D )− = 0 (Re D˜)− ⊕ 75 1 and (Re D˜)− by Proposition 4.1.2(iii) since the nilpotent part T˜ of D˜ is Hilbert– ∈ L | | 0 1 0 1 Schmidt. Therefore (Re D )+ / and (Re D )− , and also via unitary equivalence ∈ L ∈ L 1 1 (Re D)+ / and (Re D)− . (4.1.10) ∈ L ∈ L Thus by Theorem 2.2.7(iv), one has that R Tr D = . { } ∅ (ii) = (iii). Suppose the nilpotent part of D is not Hilbert–Schmidt. Then just like ⇒ in Case 2 above use Remark 4.1.3 to decompose D0 = P D˜, with D˜ satisfying the ⊕ conditions of Proposition 4.1.2(ii). Then for π/2 < θ < π/2 − iθ 0 iθ iθ TrH(Re e D )+ = TrJ (cos θP ) + TrK⊕K (Re e D˜)+ TrK⊕K (Re e D˜)+ = . ≥ ∞ Furthermore, since the nilpotent part T of D is not Hilbert–Schmidt, and hence T˜ is | | not Hilbert–Schmidt, one has iθ 0 iθ TrH(Re e D )− = 0 + TrK⊕K (Re e D˜)− = ∞ by Proposition 4.1.2(iii). Finally, since T˜ is not Hilbert–Schmidt, neither is it trace-class. | | Therefore, by Proposition 4.1.2(i) iπ/2 0 0 TrH(Re e D )± = TrH(Im D )± = 0 + TrK⊕K (Im D˜)± = . ∞ iθ iθ 0 Thus we have proven that Tr(Re e D)± = Tr(Re e D )± = for all π/2 < θ π/2 ∞ − ≤ and hence also for all θ R, and so by Theorem 2.2.7(iv) one has R Tr D = C. ∈ { } Having established the equivalence of (ii)–(vi) and the implication (i) = (ii), it suffices ⇒ to prove (iv) = (i). We will in fact prove the contrapositive. To this end, suppose D ⇒ is a Hilbert–Schmidt perturbation of a projection. That is, D = P + K where P is a 76 projection and K 2. Because D is idempotent one has ∈ L P + K = D = D2 = P 2 + PK + KP + K2 = P + PK + KP + K2, and so K = PK + KP + K2 and PKP = 2PKP + PK2P, (4.1.11) so PKP = PK2P 1. Similarly for P ⊥ one has P ⊥KP ⊥ = P ⊥K2P ⊥ 1. − ∈ L ∈ L Therefore, with respect to the decomposition = P P ⊥ , one has H H ⊕ H K K 1 2 K = , K3 K4 1 2 where K1,K4 and K2,K3 . A technical note is that P must have infinite rank. ∈ L ∈ L Otherwise, if P were finite rank, then so also K2,K3 would be finite rank. Hence K would be trace-class, and so also would D = P + K, which contradicts the fact that D is an infinite rank idempotent because of Lemma 4.1.1. Thus relative to = P P ⊥ H H ⊕ H we may write D1 D2 z }| { z }| { IK K 0 2 1 D = + . K3 0 0 K4 Moreover, because D˜1 z }| { I 0 Re D1 = Re ∗ K2 + K3 0 ∗ 2 and K +K3 , by the proof of (vi) = (ii) (see (4.1.9) and (4.1.10) for Cases 1 and 2), 2 ∈ L ⇒ 1 1 (Re D1)+ = (Re D˜1)+ / but (Re D1)− = (Re D˜1)− . So by Theorem 2.2.7(iv), ∈ L ∈ L R Tr D1 = and hence D1 does not have an absolutely summable diagonal in any basis. { } ∅ 77 1 Because D2 , its diagonal in any basis is absolutely summable. Therefore, there is ∈ L no basis in which D = D1 + D2 has a zero diagonal, which completes the proof. The following corollary answers Question 2.2.4 due to Jasper. Corollary 4.1.6 ([LW16, Corollary 2.6]). A nonzero idempotent D is zero-diagonal if and only if it is not a Hilbert–Schmidt perturbation of a projection. Proof. If D has infinite rank, this is handled by Theorem 4.1.5. If D has finite rank, then so does the nilpotent part of D. Thus D is a finite rank (and hence Hilbert–Schmidt) perturbation of the zero projection. Furthermore, by Lemma 4.1.1 Tr D = rank D > 0 for finite rank idempotents, and so D is not zero-diagonal. In the case of infinite rank projections with infinite dimensional kernel, the next corollary is a strengthening of the result due to Fan [Fan84, Theorem 3] that an operator T is a norm limit of zero-diagonal operators if and only if 0 We(T ), the essential ∈ ⊥ numerical range. For P a projection, 0 We(P ) if and only if Tr P = , and thus ∈ ∞ Fan’s result guarantees such projections are a norm limit of zero-diagonal operators. However, we take this a step further by proving these zero-diagonal operators may be taken to be idempotent so long as Tr P = as well. ∞ Corollary 4.1.7 ([LW16, Corollary 2.7]). Every projection P with Tr P = Tr P ⊥ = ∞ is a norm limit of zero-diagonal idempotents. Proof. For P = I 0 consider idempotents I 0 whose nilpotent part has arbitrarily ⊕ T 0 small norm but is not Hilbert–Schmidt and apply Theorem 4.1.5 (ii) (iv). ⇐⇒ Constructing bases to achieve zero diagonal The proof of Theorem 4.1.5 was existential in the sense that it did not explicitly construct a basis in which a given idempotent has zero diagonal. The remainder of this section is devoted to providing an algorithm for constructing such a basis when it exists (i.e., when 78 the idempotent is not a Hilbert–Schmidt perturbation of a projection, which is included in the case when dim ker D = = dim ker⊥ D). As with the proof of Proposition 4.1.2, ∞ a careful consideration first of the 2 2 case is in order. × Remark 4.1.8 ([LW16, Remark 2.8]). Consider a 2 2 idempotent matrix, D, and the × counterclockwise rotation matrix through an angle θ, Rθ, given by the formulas 1 0 cos θ sin θ − D = and Rθ = , d 0 sin θ cos θ 2 where d 0. Conjugating D by Rθ is equivalent to changing the basis for C : ≥ cos θ sin θ 1 0 cos θ sin θ − R−θDRθ = sin θ cos θ d 0 sin θ cos θ − cos2 θ + d sin θ cos θ sin θ cos θ d sin2 θ − − = sin θ cos θ + d cos2 θ sin2 θ d sin θ cos θ − − 1+cos 2θ+d sin 2θ sin θ cos θ d sin2 θ 2 − − = . sin θ cos θ + d cos2 θ 1−cos 2θ−d sin 2θ − 2 arctan d Elementary calculus shows that the minimum diagonal entry occurs when θ = 2 and corresponds to a negative value of 2 − 1 p 2 d d := 1 1 + d = − . − 2 − 2 1 + √1 + d2 Since the trace is basis independent, the other diagonal entry is necessarily 1 + d−. Furthermore, by continuity of the diagonal entries as a function of θ, for any value x with d− x 0, there is some θ for which one of the diagonal entries is x. − ≤ ≤ 79 We require the following elementary result in linear algebra [HJ90, Page 77, Problem 3]. It’s proof by induction is straightforward and we include it here for completeness. Lemma 4.1.9 ([LW16, Lemma 2.9]). Let X Mn(C). Then Tr X = 0 if and only if ∈ there is a basis in which X has zero diagonal. Proof. One direction is clear, so suppose Tr X = 0. We proceed by induction on the size n n of the n n matrix X. The case n = 1 is clear. Given any basis ej , one has × { }j=1 n X 0 = Tr X = (Xej, ej), j=1 Pn (Xej ,ej ) and therefore also 0 = . Thus zero is in the convex hull of (Xej, ej) j=1 n { } ⊆ W (X). By the Toeplitz–Hausdorff Theorem, the numerical range W (X) is convex, so 0 W (X). So there is some unit vector f1 for which (Xf1, f1) = 0. Let P be the ∈ projection onto the orthogonal complement of f1. Then we find 0 = Tr X = (Xf1, f1) + Tr(PXP ) = Tr(PXP ). The matrix PXP can be viewed as being of size (n 1) (n 1) by expressing it in a basis − × − which contains f1 and deleting the row and column corresponding to f1 (which consist solely of zeros). By applying the inductive hypothesis to PXP we obtain orthonormal vectors f2, . . . , fn which are orthogonal to f1 and satisfy (Xfj, fj) = (XP fj, P fj) = n (P XP fj, fj) = 0 for 2 j n. Therefore fj is a basis with respect to which X ≤ ≤ { }j=1 has zero diagonal. We will use the following obvious corollary of Lemma 4.1.9 extensively in the next section. Corollary 4.1.10 ([LW16, Corollary 2.10]). Let X Mn(C). Then Tr X = nλ if and ∈ only if there is a basis in which X has constant diagonal sequence λ. More generally, m if X B( ) with basis en n∈ and nk a finite subsequence of N with restricted ∈ H { } N h ik=1 80 Pm m trace (Xen , en ) = mλ, then there is an orthonormal set fn for which k=1 k k { k }k=1 m m (Xfn , fn ) = λ for k = 1, . . . , m and span fn = span en . k k { k }k=1 { k }k=1 Proof. For X Mn(C) apply Lemma 4.1.9 to X λI and note that λI has constant ∈ − diagonal sequence λ with respect to any basis. m For the general case X B( ), let P be the projection on span en and apply ∈ H { k }k=1 the matrix result to PXP . Then simply notice that (P XP fnk , fnk ) = (XP fnk , P fnk ) = (Xfnk , fnk ). We are now ready to provide our algorithm. It requires an elementary theoretical first step with all succeeding steps algorithmic. Algorithm 4.1.11 ([LW16, Algorithm 2.11]). Suppose that D B( ) is not a Hilbert– ∈ H Schmidt perturbation of a projection. Then the following explicitly constructs (i.e., gives an algorithm for producing) a basis in which D is zero-diagonal. Construction. By Theorem 4.1.5, the nilpotent part of D is not Hilbert–Schmidt. Then by the contrapositive of Fact 1.2.3 (i), there exists a basis in which the diagonal of the nilpotent part is not square-summable. That is, there exists a basis for for which H 1 0 0 0 ··· ··· . . . . . . 1 . . .. . .. .. 0 . 0 . D = ··· ··· (4.1.12) d1 0 0 · · · ∗ ··· . . . . . . d . . .. . 2 . . .. 0 .. ∗ · · · ··· ∞ 2 with dn ` ` . Furthermore, by conjugating by a unitary U of the form U = h i ∈ \ I diag un , we may even assume without loss of generality that dn 0. Let the basis ⊕ h i ≥ 0 which gives the form of equation (4.1.12) be e := en, e n∈ . We will transform these { n} N 81 0 0 0 into a new basis f := fn, fn n∈ for which span en, en = span fn, fn for each n N. { } N { } { } ∈ 0 0 0 Specifically, fn = cos θnen + sin θne and f = sin θnen + cos θne form a rotation of n n − n 0 the pair en, en through an angle θn which we will choose momentarily. Recall Remark 4.1.8 and notice that ∞ ∞ ∞ X X d2 1 X d− = n d2 = . n p 2 p 2 n 2 1 + 1 + d ≥ 2 1 + 1 + dn ∞ n=1 n=1 n kh ik∞ n=1 Pm1 − − Let m1 be the smallest integer for which d 1 + d . Necessarily m1 2. Now n=1 n ≥ 1 ≥ arctan dn − 0 0 define θn = for 1 n < m1, hence by Remark 4.1.8, d = (Df , f ) and 2 ≤ − n n n − 1 + dn = (Dfn, fn). Our choice of m1 guarantees m −1 m m −1 X1 X1 X1 d− < 1 + d− d−, and thus d− 1 d− + d− < 0. n 1 ≤ n − m1 ≤ − − 1 n n=1 n=1 n=1 For the latter, using the continuity described in Remark 4.1.8 (last sentence) choose θm1 so that m −1 X1 (Df 0 , f 0 ) = 1 d− + d−, m1 m1 − − 1 n n=1 and therefore m1 m1−1 X 0 0 X − 0 0 − (Df , f ) = d (Df , f ) = 1 + d = (Df1, f1). − n n n − m1 m1 1 n=1 n=1 We will now inductively define the sequences mk and θn in the following interwoven h i h i fashion. Suppose that these sequences are already defined up to mk−1 and θmk−1 . Let mk be the smallest positive integer for which mk X − 0 0 d 1 (Df , f ) = (Dfk, fk). n ≥ − k k n=mk−1+1 82 arctan dn Then for mk−1 < n < mk, let θn = 2 , and as above let θmk be chosen so as to satisfy mk mk−1 X 0 0 X − 0 0 0 0 (Df , f ) = d (Df , f ) = 1 (Df , f ) = (Dfk, fk). − n n n − mk mk − k k n=mk−1+1 n=mk−1+1 0 Finally observe from this that with respect to the basis fn, f n∈ the diagonal { n} N sequence of D can be partitioned into finite subsets Ak k∈ for which the sum over { } N each subset is zero. Indeed, let Ak consist of the diagonal entries corresponding to 0 0 the basis elements fk := fk, fm +1, . . . , fm . So for each k N we may apply { k−1 k } ∈ Lemma 4.1.9 to the collection fk to obtain a new collection of orthonormal vectors gk with span fk = span gk and the diagonal of D with respect to gk is constantly zero. Thus S D has a zero diagonal with respect to the basis g := k gk. We stated in the introduction that Theorem 4.1.5(v) and (vi) are equivalent for any bounded operator, not merely idempotents, which we now prove. Proposition 4.1.12 ([LW16, Proposition 2.12]). An operator T B( ) has an abso- ∈ H lutely summable diagonal in some basis if and only if it has a summable diagonal in some basis. Proof. One direction is trivial. For the other direction, suppose that T B( ) and ∈ H e := en n∈ is a basis with respect to which the corresponding diagonal dn is summable { } N h i with sum s. Then there exists a strictly increasing sequence of positive integers nk with h i −k Pm the property that sn s 2 , where sm denotes the partial sum dj. | k − | ≤ n=1 Pn1 n1 Since dj = sn , by Corollary 4.1.10 there is an orthonormal set bj for j=1 1 { }j=1 n1 n1 s which span bj = span ej and (T bj, bj) = n1/n1. Similarly for each k N, { }j=1 { }j=1 ∈ Pnk+1 nk+1 because dj = sn sn there is an orthonormal set bj for which j=nk+1 k+1 − k { }j=nk+1 nk+1 nk+1 (s −s ) ∞ span bj = span ej and (T bj, bj) = nk+1 nk /nk+1−nk. Thus b := bj { }j=nk+1 { }j=nk+1 { }j=1 is a basis since span b = span e. For convenience of notation, set n0 = 0 = sn0 . Then 83 with respect to the basis b, the diagonal sequence is absolutely summable since ∞ ∞ nk+1 X X X (T bj, bj) = (T bj, bj) | | | | j=1 k=0 j=nk+1 ∞ nk+1 X X snk+1 snk = | − | nk+1 nk k=0 j=nk+1 − ∞ X = sn sn k+1 − k k=0 ∞ X = sn + sn s + s sn | 1 | k+1 − | − k | k=1 ∞ X −(k+1) −k sn + 2 + 2 ≤ | 1 | k=1 3 = sn + . | 1 | 2 4.2. Diagonals of the class of idempotents and applications In this section we investigate Jasper’s initial frame theory problem concerning dual frame pairs via its equivalent operator-theoretic formulation: Problem 4.2.1. Characterize the diagonals of the class of idempotent operators. In particular, we prove that every bounded sequence appears as the diagonal of some idempotent (Theorem 4.2.7). We prove this result in stages. First we consider diagonals of idempotents in M2(C) (Lemma 4.2.2). Then we give a direct sum construction of an idempotent with constant diagonal (Proposition 4.2.3). From this we show that any bounded sequence with at least one value repeated infinitely many times appears as the diagonal of some idempotent (Proposition 4.2.4). We conclude by showing that we may obtain any bounded sequence as the diagonal of an idempotent. The following technical lemma is a trivial corollary of Theorem 2.2.2 except for its norm bound which we require for the forthcoming results. 84 Lemma 4.2.2 ([LW16, Lemma 3.2]). If d C, then there is a 2 2 idempotent ∈ × D M2(C) with norm D 6 d + 4 which takes the values 3d 1, 3d + 2 on its ∈ k k ≤ | | − − diagonal. Proof. Start with the idempotent 1 0 Dz = , z 0 with z C to be chosen later. Conjugating by the (unitary) rotation matrix Rπ/4, one ∈ obtains √1 √1 1 0 √1 √1 1−z 1−z 2 − 2 2 2 2 2 Rπ/4DzR−π/4 = = √1 √1 z 0 √1 √1 1+z 1+z 2 2 − 2 2 2 2 Choosing z = 6d 3 gives the correct diagonal values. Furthermore, Dz 1+ z 6d+4. − k k ≤ | | ≤ Then D = Rπ/4DzR−π/4 gives our required idempotent. In the next proposition we exhibit an idempotent with constant diagonal d. The idea is to take an infinite direct sum of the 2 2 matrix D from Lemma 4.2.2 (whose × diagonal entries d1, d2 satisfy 2d1 + d2 = 3d), regroup the diagonal entries and apply Corollary 4.1.10 repeatedly. Proposition 4.2.3 ([LW16, Proposition 3.3]). Given d C, there is an idempotent ∈ Dd B( ) with norm Dd 6 d + 4 with constant diagonal d in some basis. ∈ H k k ≤ | | Proof. Let D0 be the 2 2 idempotent matrix obtained from Lemma 4.2.2 and set × L∞ 0 D = D . Then the diagonal of D consists of the values d1 = 3d 1 and d2 = 3d+2, i=1 − − each repeated infinitely many times. With respect to the basis e := ej j∈ , the diagonal { } N 85 entries are d1 if j is odd, (Dej, ej) = d2 if j is even, and these diagonal entries satisfy 2d1 + d2 = 3d. Let π be any permutation of N which sends 2N onto 3N (i.e., maps the positive even integers to positive multiples of three). Create a new basis f := fj j∈ by fj := e −1 . Then we have { } N π (j) d1 if j N 3N, ∈ \ (Dfj, fj) = (Deπ−1(j), eπ−1(j)) = d2 if j 3N. ∈ For each j 3N, the sum of the diagonal entries corresponding to fj−2, fj−1, fj is ∈ 2d1 + d2 = 3d. Thus for each j 3N we may apply Corollary 4.1.10 to obtain new ∈ orthonormal vectors gj−2, gj−1, gj with span fj−2, fj−1, fj = span gj−2, gj−1, gj (hence { } { } g := gk k∈ is a basis) and (Dgk, gk) = d for any k N. Taking Dd := D with respect { } N ∈ to the basis g is the required idempotent. Using Proposition 4.2.3 we will now prove that any bounded sequence with at least one value repeated infinitely many times appears as the diagonal of some idempotent. ∞ Proposition 4.2.4 ([LW16, Proposition 3.4]). Suppose d := dj ` and for some m h i ∈ one has dm = dk for infinitely many k N. Then there exists an idempotent D B( ) ∈ ∈ H with diagonal d for which D 18 d + 4. k k ≤ k k∞ Proof. Observe that the direct sum of idempotents from Proposition 4.2.3: ∞ M D := (Dd D−d +2d ), j ⊕ j m j=1 86 is a bounded operator whose norm satisfies D = sup Ddj , D−dj +2dm k k j {k k k k} sup 6 dj + 4, 6 dj + 2dm + 4 ≤ j { | | |− | } 18 d ∞ + 4. ≤ k k The idempotent D comes with an associated basis e := ei,j,k i = 1, 2; j, k N with { | ∈ } respect to which the diagonal is dj if i = 1, (Dei,j,k, ei,j,k) = dj + 2dm if i = 2. − Create a new basis by the following procedure. Set fj := e1,j,1, so that (Dfj, fj) = dj. Then for each j, k N, apply Corollary 4.1.10 to the pair e1,j,k+1, e2,j,k to obtain ∈ orthonormal vectors g1,j,k, g2,j,k with the same span and corresponding diagonal entries 1 dm = (dj + ( dj + 2dm)). Then g := fj j∈ gi,j,k i = 1, 2; j, k N is a basis with 2 − { } N ∪ { | ∈ } diagonal entries d = dj (from the fj) along with dm with infinite multiplicity (from h i the gi,j,k). Since the value dm is repeated infinitely many times in the sequence d, after a suitable relabeling (permutation of the basis), the diagonal is precisely the sequence d. Before we prove our main result for this section we need Fan’s quantitative version of the Toeplitz–Hausdorff Theorem on the convexity of the numerical range. As a matter of notation, throughout the remainder of this paper we will use [a, b] to denote the complex line segment joining a, b C. Then each d [a, b] has a convexity coefficient λ defined by ∈ ∈ d = λa + (1 λ)b for 0 λ 1, with the convention that λ = 0 when a = b. Equivalently, − ≤ ≤ λ = b−d if b = a and λ = 0 if b = a. b−a 6 87 Lemma 4.2.5 ([Fan84, Lemma 3]). Let d1 A = ∗ M2(C) ∈ d2 ∗ be a matrix with respect to the basis e1, e2 and let d [d1, d2] with convexity coefficient { } ∈ λ. Then there exists a basis b, f for which (Af, f) = d, (Ab, b) = d1 + d2 d and { } − 2 (e1, f) λ. | | ≤ We bootstrap this lemma to modify diagonals in an interesting useful way in Lemma 4.2.6. A main tool is to use the following lemma in the case all convexity coefficients λn 1/2 ≡ to prove both Theorem 4.2.7 and Theorem 4.3.3. ∞ Lemma 4.2.6 ([LW16, Lemma 3.6]). Suppose that T is an operator and e = en { }n=0 ∞ an orthonormal set. Let rn := (T en, en) and suppose dn is a sequence such that h in=1 for n 1, dn [dn−1, rn] with convexity coefficient λn and where d0 := r0. Then there ≥ ∈ ∞ is an orthonormal set b = bn for which (T bn, bn) = rn + dn−1 dn. Moreover, if { }n=1 − Q∞ λi = 0 for all n N, then span b = span e. i=n ∈ Proof. Set f0 := e0. Since d1 [d0, r1] = [r0, r1], by Lemma 4.2.5 with diagonal ∈ entries r0, r1, there exist orthonormal f1, b1 for which span f1, b1 = span f0, e1 and { } { } 2 (T f1, f1) = d1 and (T b1, b1) = r1 + r0 d1 = r1 + d0 d1 and (f1, f0) λ1. − − | | ≤ ∞ Iterating this procedure produces an orthonormal set b = bn and a sequence of { }n=1 ∞ unit vectors fn n=0 satisfying, for each n N, { } ∈ (i)n span fn, bn = span fn−1, en ; { } { } (ii)n (T fn, fn) = dn and (T bn, bn) = rn + dn−1 dn; − 2 (iii)n (fn, fn−1) λn; | | ≤ (iv)n b1, . . . , bn, fn is an orthonormal set; { } (v)n span b1, . . . , bn, fn = span e0, . . . , en . { } { } 88 We prove this via induction. The case n = 1 is handled in the first paragraph. Suppose that (i)n–(v)n hold for some fixed n N. Then by hypothesis and (ii)n ∈ one has dn+1 [dn, rn+1] = [(T fn, fn), (T en+1, en+1)], so we may apply Lemma 4.2.5 ∈ to obtain orthonormal fn+1, bn+1 for which (i)n+1–(iii)n+1 hold. By (iv)n we know that fn is orthogonal to span b1, . . . , bn , and by (v)n we know en+1 is orthogonal to { } span b1, . . . , bn . Thus we obtain span b1, . . . , bn is orthogonal to span fn, en+1 = { } { } { } span bn+1, fn+1 by (i)n+1, thereby establishing (iv)n+1. Finally, by (i)n+1 and (v)n we { } find span b1, . . . , bn+1, fn+1 = span b1, . . . , bn, fn, en+1 = span e0, . . . , en+1 , { } { } { } proving (v)n+1. Hence by induction we have shown (i)–(v) for all n N. ∈ Q∞ Suppose now that λi = 0 for each n N. Let Pn be the projection on b1, . . . , bn i=n ∈ { } and let P be the projection onto span e. Observe span b span e by item (v), and so to ⊆ prove span b = span e it suffices to show that (P Pn+k)en 0 in norm as k for − → → ∞ each n Z+. ∈ Since fj span fj−1, ej for all j N by (i), one has ∈ { } ∈ (en, fn+k) = en, (fn+k, fn+k−1)fn+k−1 + (fn+k, en+k)en+k (4.2.1) = (en, fn+k−1) (fn+k, fn+k−1), · and from (iv)–(v), P Pn+k is the projection onto span fn+k, en+k+1, en+k+2,... . This, − { } along with (iii) and repeated use of (4.2.1) proves 2 2 2 (P Pn+k)en = (en, fn+k−1) (fn+k, fn+k−1) k − k | | · | | k n+k 2 Y 2 2 Y = (en, fn) (fn+i, fn+i−1) (en, fn) λi. | | · | | ≤ | | · i=1 i=n+1 As k the latter product converges to zero by hypothesis. → ∞ 89 Our main result for this section characterizes the diagonals of the class of idempotents to be `∞. This, according to Jasper, also characterizes all inner products of dual frame pairs. ∞ Theorem 4.2.7 ([LW16, Theorem 3.7]). Every dn ` admits an idempotent D h i ∈ ∈ B( ) whose diagonal is dn with respect to some orthonormal basis. H h i Proof. F Let N = j∈N Nj be any partition of N such that each Nj is infinite. Let ϕj : N Nj be any bijection. Then for each j define dj,n := dϕ (n); in this way we → j partition the desired sequence into infinitely many infinite sequences. By Proposition 4.2.4 F there is an idempotent D B( ) and a basis e = ej where ej := ej,n n∈ for which ∈ H j { } Z+ dj,0 := 0 = (Dej,0, ej,0) and 2dj,n dj,n−1 = (Dej,n, ej,n) for n N. − ∈ In the above we have assigned dj,0 = 0, and since there are infinitely many zeros, we can apply Proposition 4.2.4. Note however that dj,0 bears no relation to the sequence dn , h i unlike dj,n when n > 0. The remainder of the argument is independent of j. For each j we will employ a judicious use of Lemma 4.2.6. Our initial orthonormal set will be ej with diagonal entries rj,n = (Dej,n, ej,n) = 2dj,n dj,n−1. We then note that dj,n [dj,n−1, 2dj,n dj,n−1] with − ∈ − 1 convexity coefficient λj,n = 1/2 since dj,n = (dj,n−1 + (2dj,n dj,n−1)). Thus for any 2 − n N. ∈ ∞ ∞ Y Y 1 λj,i = = 0. 2 i=n+1 i=n+1 ∞ By Lemma 4.2.6 there exists an orthonormal set bj = bj,n for which { }n=1 (Dbj,n, bj,n) = rj,n + dj,n−1 dj,n = (2dj,n dj,n−1) + dj,n−1 dj,n = dj,n, − − − 90 S S and span bj = span ej. Thus b := j bj is a basis because e = j ej is a basis. With respect to the basis b the idempotent D has diagonal dj,n which is precisely dn after h i h i a suitable relabeling. 4.3. Diagonals of the class of finite rank idempotents Recall that Lemma 4.1.1 is valid for both finite and infinite dimensional . As a result, H for D Mn(C) with 0 = D = I, Tr D = rank D 1, . . . , n 1 . Theorem 2.2.2 shows ∈ 6 6 ∈ { − } that this trace condition is the only restriction for a given sequence to be the diagonal of a nonzero non-identity idempotent matrix. Because not all idempotent operators D B( )( infinite dimensional) are trace-class, it is unnatural to expect there to be ∈ H H any sort of trace restriction on the diagonals of idempotent operators in B( ). In this H light, Theorem 4.2.7 is naturally expected: if the only restriction in the n n matrix × case was the trace, there should be no restrictions in B( ). H However, there is another perfectly reasonable class to consider: the trace-class idem- potents. Again, Lemma 4.1.1 ensures that trace-class idempotents are actually finite rank idempotents. The restriction that Tr D = rank D N is still applicable for finite rank ∈ idempotents D B( ). In this section we prove that, as for Mn(C), this trace condition ∈ H is the only restriction for an `1 (absolutely summable) sequence to be the diagonal of a finite rank idempotent, which is Theorem 4.3.3 below. A corollary of the next lemma verifies Theorem 4.3.3 when restricted to rank-one idempotents. That is, the diagonals of the class of rank-one idempotents are precisely those absolutely summable sequences which sum to one. Lemma 4.3.1 ([LW16, Lemma 4.1]). If T B( ) is a rank-one operator then T 2 = ∈ H Tr(T )T , hence T is idempotent if and only if Tr T = 1. Proof. We may write any rank-one operator as an infinite matrix with entries aibj where 2 P∞ ai , bj ` . Since the trace is independent of the choice of basis, Tr T = akbk. h i h i ∈ k=1 91 Finally, ∞ ! ∞ ! ! 2 X X T = (aibk)(akbj) = ai akbk bj = Tr(T )(aibj) = Tr(T )T. k=1 k=1 Another proof which is less, but not entirely, coordinate dependent: since T is rank-one, there are x, y for which T z = (z, x)y. By expanding T in a basis for which ∈ H H contains y/kyk, it is clear that Tr T = (y, x). Thus 2 T z = T (z, x)y = (z, x)(y, x)y = (y, x)T z = Tr(T )T z. 1 Corollary 4.3.2 ([LW16, Corollary 4.2]). An absolutely summable sequence dj ` is h i ∈ P the diagonal of some rank-one idempotent D if and only if j dj = 1. P Proof. One direction is trivial since j dj = Tr D = rank D = 1 by Lemma 4.3.1. 1 For the other direction, let dj ` be any absolutely summable sequence which h i ∈ iθ 2 sums to one. Write dj = rje j with rj 0 and j R. Then define √d ` as ≥ ∈ ∈ iθj/2 (√d)j := √rje . Then define D = (√d)i(√d)j = √d √d. By Lemma 4.3.1, D is ⊗ idempotent since its diagonal is dn which sums to one. h i We now prove Theorem 4.3.3 by two distinct methods. The first uses Theorem 2.2.2, Corollary 4.3.2, and Lemma 4.2.6. The second proof is an inductive argument analogous to the proof of Theorem 2.2.2 by Giol, Kovalev, Larson, Nguyen and Tener in [Gio+11]. It uses Corollary 4.3.2 as the base case and exploits the fact that the class of finite rank idempotents is similarity invariant. Theorem 4.3.3 ([LW16, Theorem 4.3]). The diagonals of the class of nonzero finite rank idempotents consist precisely of those absolutely summable sequences whose sum is a positive integer. 92 Proof using Lemma 4.2.6. Lemma 4.1.1 makes this sum condition necessary, so suffi- 1 ciency is all that is needed. Let d := dn ` be an absolutely summable sequence h i ∈ P whose sum dn = m is a positive integer. If m = 1, then dn is the diagonal of a n h i rank-one idempotent by Corollary 4.3.2. So suppose m > 1, in which case m 1 N. − ∈ 0 Pm−1 Set d := (m 1) dn. By Theorem 2.2.2, there is an idempotent matrix m − − n=1 (1) 0 D1 Mm(C) with diagonal d := d1, . . . , dm−1, dm . Now consider the sequence ∈ h i (2) 0 (2) 1 1 d := 2dm d , 2dm+1 dm, 2dm+2 dm+1,... . It is clear that d ` since d ` . h − m − − i ∈ ∈ Furthermore, ∞ ∞ ∞ ∞ X (2) 0 X 0 X X d = 2dm d + (2dn+1 dn) = d + dn = dn (m 1) = 1. n − m − − m − − n=1 n=m n=m n=1 Therefore, by Corollary 4.3.2, there is a rank-one idempotent D2 with diagonal sequence (2) d . Defining D = D1 D2, we find that D has a basis e := en n∈ in which its diagonal ⊕ { } N is 0 0 d1, . . . , dm−1, d , 2dm d , 2dm+1 dm, 2dm+2 dm+1,... . h m − m − − i 0 0 That is, (Den, en) = dn for 1 n < m;(Dem, em) = d ;(Dem+1, em+1) = 2dm d ; ≤ m − m and (Den, en) = 2dn−1 dn−2 for n > m + 1. − We will now apply Lemma 4.2.6 to the orthonormal set em, em+1,... . So { } 0 dm if n = m 0 rn := (Den, en) = 2dm d if n = m + 1 − m 2dn−1 dn−2 if n > m + 1 − Since dm [rm, rm+1] and dn [dn−1, rn+1] for n > m (with convexity coefficients all ∈ ∈ λn 1/2), after a suitable relabeling of the sequences involved (rn rn−m; dn dn−m+1) ≡ 7→ 7→ 93 we may apply Lemma 4.2.6 to obtain an orthonormal set bm, bm+1,... satisfying { } ) rm+1 + rm dm if n = m (Dbn, bn) = − = dn. rn+1 + dn−1 dn if n > m − Moreover, by Lemma 4.2.6, since the convexity coefficients are λn = 1/2, we have ∞ ∞ ∞ span bn = span en . Setting bn := en for n < m, we find that b = bn is a { }n=m { }n=m { }n=1 basis with respect to which D has diagonal dn . h i Proof by induction using techniques from [Gio+11]. We proceed by induction on the P∞ 1 sum m := dn where dn ` is an absolutely summable sequence whose sum is a n=1 h i ∈ positive integer. The base case m = 1 is handled by Corollary 4.3.2. Now suppose m > 1 and for any absolutely summable sequence whose sum is m 1, − there is a finite rank idempotent with that sequence on its diagonal. By possibly permuting the sequence dn, we may assume without loss of generality that d1 + d2 = 2. Since 6 P∞ P∞ dn = m, then (d1 + d2 1) + dn = m 1. So by the induction hypothesis n=1 − n=3 − there exists a finite rank (in fact, rank-(m 1)) idempotent D˜ with diagonal sequence − d1 + d2 1, d3, d4,... . Then consider the rank-m operator h − i 1 0 0 1×∞ D = , 0∞×1 D˜ ∞ 0 which is obviously idempotent. With respect to the basis e = ej , D has diagonal { }j=1 1, d1 + d2 1, d3, d4,... . Then consider the invertible S which is the identity on h − i ∞ span ej and whose compression to span e1, e2 has the matrix representation { }j=3 { } λ λ 1 − , 1 1 94 0 0 −1 where λ := (d2−1)/(d1+d2−2). Conjugating D by S produces an idempotent D := SD S whose diagonal with respect to e is precisely the sequence d. 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Sh¯oda, Einige S¨atze¨uber Matrizen, Japanese Journal of Mathematics 13.3 (1937), pp. 361–365. [Ste13] E. Steinitz, Bedingt konvergente Reihen und konvexe Systeme. (Teil I.)., Journal f¨urdie Reine und Angewandte Mathematik 143 (1913), pp. 128–175, doi: 10.1515/crll.1913.143.128. [Toe18] O. Toeplitz, Das algebraische Analogon zu einem Satze von Fej´er, Mathema- tische Zeitschrift 2.1-2 (1918), pp. 187–197, doi: 10.1007/BF01212904. 101 Appendix A Prerequisite material This appendix is meant to provide the reader with a brief overview of the basic facts necessary for a thorough understanding of this dissertation. Proofs are omitted for brevity and this is intended only to be a reference for the major topics. For a detailed treatment of these topics please consult such references as Kadison and Ringrose [KR97a; KR97b]. A.1. Hilbert spaces A Hilbert space is a Banach space (i.e., a complete normed complex vector space) H with a sesquilinear map (i.e., linear in the first coordinate, conjugate-linear in the second) 2 ( , ) : C called an inner product such that (x, x) = x for all x . In this · · H × H → k k ∈ H situation we say that the norm of is induced by the inner product. It is possible to H consider Hilbert spaces over R instead of over C, and in this case the inner product is actually a bilinear map. However, we concern ourselves (almost) entirely with Hilbert spaces over C. The key property of Hilbert spaces is that the inner product allows for the definition of the angle between two vectors. In particular, two vectors are orthogonal if and only if their inner product is zero, and in general the angle θ between vectors x, y is given ∈ H 102 |(x,y)| by arccos kxk·kyk . A collection of orthogonal unit vectors whose linear span is dense in is said to be an orthonormal basis, or simply basis when we are too lazy to write H “orthonormal”. Take note that an orthonormal basis is in general very different from an algebraic (Hamel) basis which one encounters in linear algebra. In fact an orthonormal basis is an algebraic basis if and only if it contains finitely many vectors. It is a standard exercise to verify that all orthonormal bases for a Hilbert space have the same cardinality. This H common cardinality is called the dimension of and written dim . Moreover, Hilbert H H spaces are uniquely determined (up to isomorphism) by their dimension. That is, if dim = dim 0, then there is a bijective isometric linear map U : 0. Such a map H H H → H U preserves all the structure of , including its inner product, and is an isomorphism in H the category of Hilbert spaces. n The simplest example of a Hilbert space is C with the Euclidean norm and the dot product as the inner product. This Hilbert space is n-dimensional since the standard n basis e1 = 1, 0,..., 0 , . . . , en = 0,..., 0, 1 is an orthonormal basis for C . Since h i h i Hilbert spaces are uniquely determined by their dimension, these constitute all the finite dimensional Hilbert spaces (up to isomorphism). For the most generic example, consider an arbitrary index set J and the C-vector space 2 J X 2 ` (J) := xj C xj < , h ij∈J ∈ | | ∞ j∈J J where C denotes the space of functions from J C which we think of as sequences → even though J may not be N (or even countable or ordered, for that matter). An inner product on `2(J) is given by analogy with the Euclidean dot product: X xj , yj := xjyj. h ij∈J h ij∈J j∈J 103 Up to isomorphism, `2(J) is the unique Hilbert space with dim = J since an orthonor- H | | 2 mal basis for ` (J) is given by ej j∈J where ej is the sequence with a 1 in the j-th { } position and 0 elsewhere. A separable Hilbert space is one which has a countable orthonormal basis; this is equivalent to the topological separability of . Thus the unique (up to isomorphism) H 2 2 separable infinite dimensional Hilbert space is ` (N), which we abbreviate as ` . A.2. Bounded operators on Hilbert spaces An operator on a Hilbert space is a linear transformation T : . The following H H → H are equivalent for an operator T : (i) T is continuous. (ii) T is uniformly continuous. (iii) T is bounded, i.e., T := sup kT xk < . k k x6=0 kxk ∞ The set of all bounded operators on is denoted B( ) and forms an algebra over C H H with multiplication given by composition. It is complete with respect to the norm given above and satisfies TS T S for any S, T B( ). k k ≤ k k · k k ∈ H Each bounded operator T has a corresponding bounded operator T ∗, called its adjoint, which is the unique linear operator satisfying (T x, y) = (x, T ∗y) for all x, y . Thus ∈ H T ∗∗ := (T ∗)∗ = T . The adjoint is an involution on B( ), which turns B( ) into a H H -algebra. The adjoint interacts with the algebraic structure by ∗ (i) (T + S)∗ = T ∗ + S∗; (ii) (ST )∗ = T ∗S∗; and with the analytic structure by (iii) T ∗T = T 2. k k k k 104 This interaction with the norm is called the C∗-property, and B( ) is a C∗-algebra. H In general, an abstract C∗-algebra is just a complete normed -algebra satisfying the ∗ C∗-property. Because of the Gelfand–Naimark–Segal (GNS) construction, any abstract C∗-algebra can be realized (via a -isomorphism) as a concrete C∗-algebra (i.e., a norm- ∗ closed -subalgebra of B( ) for some Hilbert space ). ∗ H H The spectrum σ(T ) of an operator T B( ) is defined as the set of λ C such that ∈ H ∈ T λI is not multiplicatively invertible, where I denotes the multiplicative identity of − B( ). It is a standard exercise to prove that σ(T ) is a nonempty compact subset of C. H A special class of operators is the collection of selfadjoint operators. It is easy to prove ∗ that if T = T then σ(T ) R. Every operator can be written as a linear combination ⊂ T +T ∗ T −T ∗ of selfadjoint operators. Indeed, we may write T = 2 + i 2i . The selfadjoint T +T ∗ T −T ∗ operators Re T := 2 and Im T := 2i are called the real and imaginary parts of T , respectively. Those selfadjoint operators with σ(T ) R+ are said to be positive; although ⊂ the term nonnegative is perhaps more accurate, there is no use fighting convention. A selfadjoint operator T may always be written (uniquely) as the difference of two positive operators T+,T− with the property that T+T− = T−T+ = 0. This is easily proven by an appeal to the continuous functional calculus which states that for a selfadjoint operator T (or more generally, for a normal operator, i.e., TT ∗ = T ∗T ) there is a -isomorphism ∗ Φ from C0(σ(T ) 0 ), the complex-valued continuous functions on the spectrum of T \{ } which vanish at zero (if it is in the spectrum), onto C∗(T ), the C∗-algebra generated by ∗ T (i.e., the smallest C -algebra containing T ). Then we may define T± := Φ(f±) where f±(x) := max 0, x . { ± } It is similarly easy to show that any operator is a linear combination of four unitary operators (i.e., U B( ) for which U ∗U = UU ∗ = I) using the continuous functional ∈ H calculus. In particular, it suffices to prove that every selfadjoint operator T = T ∗ of norm T 1 is the sum of two unitaries. In particular, define functions on [ 1, 1] by k k ≤ − 105 2 u±(x) := x i√1 x , which are unitary because they take values in the unit circle. ± − Setting U± := Φ(u±) we find T = U+ U−. − We denote that an operator T is positive by T 0. While we defined positive operators ≥ (on page 6) as those selfadjoint elements whose spectrum is contained in the nonnegative real numbers, there are actually several useful equivalent conditions. In particular, the following are equivalent: (i) T 0. ≥ (ii) T = S2 for some S = S∗. (iii) T = X∗X for some X B( ). ∈ H (iv) (T x, x) 0 for all x . ≥ ∈ H The positive operators form a cone in B( ). That is, if S, T 0 and c R+ then H ≥ ∈ S + T 0 and cS 0. The fact that the positive operators form a cone leads to a ≥ ≥ natural partial order on selfadjoint elements. Specifically, if S = S∗ and T = T ∗ then we write S T if and only if T S 0. ≤ − ≥ Another very special class of operators in B( ) is the collection of so-called compact H K operators. These are operators which are approximately finite in the sense that they are the norm closure of the finite rank operators (i.e., those operators whose range is finite dimensional). Compact operators adhere to several other properties which are similar to their finite dimensional counterparts. For example, if T then unlike many operators ∈ K in B( ): H (i) σ(T ) 0 is discrete; \{ } (ii) σ(T ) 0 consists of eigenvalues of finite multiplicity; \{ } (iii) if T = T ∗, then T is diagonalizable. There is other useful information concerning , such as the fact that it is the only K norm-closed two-sided ideal of B( ) and it contains all other two-sided ideals. H 106 A.3. Trace-class and Hilbert–Schmidt operators On B( ) there is a trace Tr (defined on positive operators) which is the sum of the H diagonal entries: ∞ X Tr T := (T ej, ej), (A.3.1) j=1 ∞ where ej is an orthonormal basis and T 0. By an application of Tonelli’s { }j=1 ≥ theorem to interchange the summations, the argument on page 5 ensures that this value is independent of the choice of orthonormal basis and is therefore well-defined. The trace can then be extended by linearity to the trace-class operators 1 := span T 0 Tr T < . L { ≥ | ∞} When T 1, we can define Tr T by formula (A.3.1) because of Fubini’s theorem. Less ∈ L obvious is the converse, which guarantees that T 1 if and only if the value of these ∈ L sums is independent of the choice of basis. An equivalent definition of the trace-class operators is that they are those compact operators A whose absolute values A have | | summable eigenvalue sequences when repeated according to multiplicity. Moreover, 1 is L an ideal in B( ). H The set of Hilbert–Schmidt operators is the ideal 2 := T B( ) Tr T ∗T < . L { ∈ H | ∞} This is related to the trace-class ideal by ( 2)2 = 1. The Hilbert–Schmidt operators L L also have an alternative description in terms of their matrix representations: ∞ X 2 T 2 := (T ei, ej) < . k kL | | ∞ i,j=1 It is straightforward to check that this value is independent of the choice of orthonormal basis. Of course, just like the trace-class operators, the Hilbert–Schmidt operators are those compact operators A whose absolute values A have square-summable eigenvalue | | sequences when repeated according to multiplicity, which justifies the notation. 107 Appendix B A primer on zero-diagonal operators The purpose of this chapter is to provide corrected proofs of [Fan84, Theorem 1] and [FFH87, Theorem 4], herein quoted as Theorem 2.2.5 (page 27) and Theorem 2.2.7 (page 27). The reason for the inclusion of these modifications is two-fold. Firstly, while Theorem 2.2.5 is true, the proof in [Fan84] has a flaw, and we correct this issue here. Secondly, the proof of Theorem 2.2.7 in [FFH87] fails to address one case[1]. Thirdly, the proof of each of these theorems provided in [Fan84; FFH87] is lacking sufficient detail in the opinion of this author, so we fill in the remaining details here. The flaw in the proof of Theorem 2.2.5 in [Fan84] mentioned above is easily described. Consider a set S C with the property that, for every line ` through the origin, the ⊂ intersection of S with each of the two open half-planes determined by ` is infinite. Fan claims [Fan84, Proof of Theorem 1, page 243] “it is easy to see that we can find three [points of S] such that the origin lies in the interior of the triangle formed by these three points.” At first this sounds somewhat reasonable, but in fact this claim is false. In particular if S is included in the union of two distinct lines passing through the origin, then zero is in the triangle determined by three points of S, but it lies on the boundary [1]In the construction, a polygon is produced which contains a line segment in its interior and Fan–Fong– Herrero do not consider the case when the line determined by this segment intersects the polygon at a vertex. 108 of any such triangle, and being in the interior is vital to his proof. However, it is true that zero lies in the interior of the convex hull of finitely many points of S, and by a triangulation argument the number of points necessary may be reduced to four (see Lemma B.1.4). Moreover, an examination of the proof in [Fan84] shows that this is all that is necessary to complete the proof, which we establish below. B.1. Zero-diagonal operators are “trace-zero” operators Theorem 2.2.5 may be regarded, and was indeed regarded in [Fan84], as an extension of the finite dimensional result mentioned earlier in the introduction: a matrix X Mn(C) ∈ is zero-diagonal if and only if Tr X = 0. We prove Theorem 2.2.5 via several useful lemmas. ∞ Lemma B.1.1 ([Fan84, Lemma 1]). Let bj be an orthonormal set, let Pn be the { }j=1 n ∞ projection onto span bj . If Pn I strongly, then bj forms a basis. { }j=1 → { }j=1 Lemma B.1.2 ([Fan84, Lemma 2]). Let aj be a sequence of positive numbers such h i ∞ ∞ ∞ P 0 P P an 0 that aj < and let s = aj. Then /sn = . j=1 ∞ n j=n n=1 ∞ Proof. Choose nk N inductively such that n1 = 1 and ∈ nk+1−1 0 X snk aj . ≥ 2 j=nk Then we have ∞ ∞ nk+1−1 ∞ nk+1−1 ∞ X an X X aj X 1 X X 1 0 = 0 0 aj = . sn sj ≥ sn ≥ 2 ∞ n=1 k=1 j=nk k=1 k j=nk k=1 P∞ Lemma B.1.3. Given a sequence of positive numbers an , the series an = s < h i n=1 ∞ Q∞ if and only if the product (1 + an) < . n=1 ∞ 109 Proof. Suppose the series is finite. Then since 1 + x ex, we find ≤ k k Pn Y Y an ( an) s (1 + an) e = e n=1 e . ≤ ≤ n=1 n=1 The partial products are therefore a bounded monotone sequence and so converge to a finite value. Now suppose the series diverges. Without loss of generality, we may assume all the terms are less than or equal to one. For if there are infinitely many terms greater than one, then the corresponding product obviously diverges. Because log(1 + x) x/2 on 0 x 1, we find ≥ ≤ ≤ k ! k k Y X 1 X log (1 + an) = log(1 + an) an. ≥ 2 n=1 n=1 n=1 Since the right-hand side approaches infinity, so does the left-hand side, and therefore so also do the partial products. ± Lemma B.1.4. Let H` denote the two open half-planes determined by a line ` in C. If + − D C with the property that for every line ` through zero D H = = D H , then ⊆ ∩ ` 6 ∅ 6 ∩ ` ◦ there exist finitely many points d1, . . . , dn D such that zero lies in (co d1, . . . , dn ) . ∈ { } Proof. We first claim that 0 co D. To see this, note that co D is a closed convex ∈ set, and every closed convex set is an intersection of (possibly infinitely many) closed half-planes (see e.g., [Hof75, p. 90]). If zero were not in co D, then there would be some 0 − + 0 line ` with co D H 0 and 0 H 0 . Let ` be the line parallel to ` passing through zero ⊆ ` ∈ ` + + + so H H 0 . Then D H = , which contradicts the properties of D. Hence 0 co D. ` ⊂ ` ∩ ` ∅ ∈ Now either 0 (co D)◦ or 0 ∂(co D). However, for every x ∂(co D), there is ∈ ∈ ∈ 00 + some line ` passing through x with (co D) H 00 = by the supporting hyperplane ∩ ` ∅ theorem (see e.g., [BV04, p. 51]). Thus, by the properties of D, 0 / ∂(co D) and therefore ∈ 0 (co D)◦. So, there exist x, y, z co(D) such that 0 (co x, y, z )◦. And thus ∈ ∈ ∈ { } 110 there also exist x1, . . . , xj, y1, . . . , yk and z1, . . . , zl in D such that x co x1, . . . , xi and ∈ { } similarly for y, z. Finally, this gives ◦ 0 (co x1, . . . , xj, y1, . . . , yk, z1, . . . , zl ) . (B.1.1) ∈ { } Although not necessary, we can even reduce the set x1, . . . , xj, y1, . . . , yk, z1, . . . , zl to { } four elements by [Gus47]. For an explicit proof, let d1, . . . , dn D be the vertices of the ∈ convex polygon in (B.1.1). Let K denote this polygon co d1, . . . , dn . Since the boundary { } of a convex polygon is a simple closed curve, we may assume the vertices are ordered, Sn−1 say, counterclockwise, and hence K = co d1, di, di+1 (see Figure B.1). Indeed, this i=2 { } union clearly contains ∂K since the boundary is the union of the line segments [d1, dn] ◦ and [di, di+1] for 1 i < n. Furthermore, if x K , let y ∂K be the (unique) point ≤ ∈ ∈ not equal to d1 on the boundary of K and on the line determined by d1, x. Then there is some 1 < i < n such that y [di, di+1]. Therefore x co d1, di, di+1 . Thus, either x ∈ ∈ { } lies in the interior of some triangle co d1, di, di+1 , or x lies on the line segment [d1, di] { } for some 2 < i < n. Hence x lies in the interior of co d1, di−1, di, di+1 and we may take { } these to be our four points. Theorem 2.2.5 ([Fan84, Theorem 1]). If T B( ) and there exists some basis ∈ H ∞ ej for for which the partial sums { }j=1 H n X sn := (T ej, ej) j=1 have a subsequence converging to zero, then T is zero-diagonal. Proof. Let dj = (T ej, ej) for j N. According to the way the entries (counting multiplic- ∈ ∞ ities) of the diagonal D = dj lie in the plane, the proof will be divided into three { }j=1 cases. 111 Figure B.1.: Convex polygon K. d1 d2 d3 x d7 y d4 d6 d5 Case 1: For every line ` passing through the origin, there are infinitely many diagonal ± entries (counting multiplicities) in each of H` . By Lemma B.1.4 there exist diagonal entries d1,1, d1,2, . . . , d1,k1 such that zero lies ◦ in (co d1,1, d1,2, . . . , d1,k ) . Since a subsequence of the sn converges to zero, we may { 1 } assume without loss of generality that sn1 is also in the convex hull of these diagonal entries (and all these diagonal entries occur prior to dn1 ). Then because the numerical range W (T ) is convex, sn W (T ) and so there exists some unit vector f1 which is 1 ∈ a linear combination of e1,1, . . . , e1,k such that (T f1, f1) = sn . Let b1, . . . , bn −1 be 1 1 { 1 } a basis for the orthogonal complement of f1 with respect to span e1, . . . , en . Then { 1 } f1, b1, . . . , bn is a basis for span e1, . . . , en and so { 1 } { 1 } n −1 n X1 X1 (T f1, f1) + (T bj, bj) = dj = sn1 j=1 j=1 112 because of preservation of the trace under a change of orthonormal basis in the finite Pn1−1 dimensional setting. This implies that j=1 (T bj, bj) = 0, and so by Lemma 4.1.9, we can actually assume the stronger condition that (T bj, bj) = 0 for 1 j n1 1 (by ≤ ≤ − choosing a possibly new set of bj with the same span). { } ∞ Now then, the set of diagonal entries sn dj still satisfies the conditions of { 1 } ∪ { }j=n1+1 ◦ Lemma B.1.4, and so there exist some d2,1, d2,2, . . . , d2,k with 0 (co d2,1, d2,2, . . . , d2,k ) , 2 ∈ { 2 } and some n2 such that sn2 also lies in the interior of the convex hull of these diagonal entries, and also that these diagonal entries occur between dn1+1 and dn2 . Just like before, because of the convexity of the numerical range, there exists some f2 of norm one with (T f2, f2) = sn . Then let bn , . . . , bn −1 be a basis for the orthogonal complement 2 { 1 2 } of f2 with respect to span f1, en +1, en +2, . . . , en . Again by preservation of the trace { 1 1 2 } for finite dimensional Hilbert spaces, we obtain n −1 n X2 X2 (T f2, f2) + (T bj, bj) = (T f1, f1) + dj = sn2 , j=n1 j=n1+1 which means that we can assume (T bj, bj) = 0 for each n1 j < n2 again by Lemma 4.1.9. ≤ ∞ Continuing inductively in this manner we obtain an orthonormal sequence bj and { }j=1 ∞ a sequence fk where fk is orthogonal to each ej for nk < j nk+1, and hence also { }k=1 ≤ is orthogonal to fk+1. That is, fk span bn , . . . , bn −1 . Thus we have ∈ { k k+1 } span b1, . . . , bn −1 span b1, . . . , bn −1, fk = span e1, . . . , en . { k+1 } ⊇ { k } { k } ∞ ∞ ∞ Therefore span bj = span ej and so the orthonormal set bj forms a basis { }j=1 { }j=1 { }j=1 in which T has zero diagonal. Case 2: For every line ` passing through the origin, H+ H− has infinitely many diagonal ` ∪ ` ± entries and for at least one such line, one of H` has finitely many diagonal entries. First notice that these conditions on the diagonal entries force the set of diagonal entries to satisfy the conditions on Lemma B.1.4. Indeed, if ` is any line through the origin, 113 then at least one of D H− or D H+ is nonempty; then, in order for a subsequence of ∩ ` ∩ ` the partial sums sn to converge to zero, D cannot be contained entirely in the closed − + half-plane H` (or H` ) unless it were contained entirely on the line ` (which it isn’t). Therefore, both D H− = = D H+. ∩ ` 6 ∅ 6 ∩ ` Next we reduce to the case where the special line ` is the real axis. To do this consider the operator e−iωT in place of T , where ω is the angle between the real axis and ` measured counterclockwise. Then simply note that T has a zero diagonal if and only if e−iωT has a zero diagonal. Finally we reduce to the case where only one diagonal entry, say, d1 lies in the open upper half-plane and the rest lie in the (closed) lower half-plane. Suppose, after a finite permutation which necessarily ensures a subsequence of partial sums still converges to zero, d1, . . . , dk are the finitely many diagonal entries lying in the open upper half-plane. Then there are infinitely many entries lying in the open lower half-plane. Since a subsequence of the partial sums converges to zero, the imaginary parts of the diagonal entries are absolutely summable (because there are only finitely many terms with nonnegative imaginary part) and so must therefore sum to zero. Therefore, all the partial sums sn lie in the upper half-plane because the only terms with nonnegative imaginary part occur at the beginning of the series. By Lemma B.1.4, there is some n1 > k for ◦ which zero lies in (co d1, . . . , dn ) . Moreover, since a subsequence of the partial sums { 1 } converges to zero, by increasing n1 we can also guarantee that sn1 lies in this open ◦ convex set (co d1, . . . , dn ) . By the convexity of the numerical range, there is some { 1 } unit vector f such that (T f, f) = sn1 (remember (Im sn1 ) > 0). As in case 1, we can find b1, . . . , bn1−1 a basis for the orthogonal complement of f with respect to e1, . . . , en1 such that (T bj, bj) = 0 (using again Lemma 4.1.9). Then with respect to the basis f, b1, . . . , bn −1, en +1, en +2,... , among the diagonal entries of T there is only one in { 1 1 1 } the upper half-plane, namely, (T f, f) = sn1 . 114 This completes the reduction that we may assume that there is only one diagonal entry, say d1, in the open upper half-plane. Again, the set of diagonal entries D satisfy the conditions of Lemma B.1.4 (since in the previous paragraph we only modified finitely many of the diagonal entries of T and a subsequence of the partial sums still converges to zero). P∞ Note that Im d1 = Im dj, and hence all the diagonal entries lie in the closed − j=2 strip z C Im d1 Im z Im d1 . By Lemma B.1.4, there exist finitely many { ∈ | − ≤ ≤ } ◦ diagonal entries with 0 (co d1,1, . . . , d1,k ) = P (d1 is necessarily among d1,1, . . . , d1,k ∈ { 1 } 1 since it is the only entry in the upper half-plane, and we may assume d1 = d1,1). Let t denote the intersection point of the boundary ∂P of the convex set P with the line through d1 and the origin. Since all the diagonal entries lie in the horizontal strip (which is convex) P also lies in this horizontal strip. Thus it is easy to see that t 0 1/2 t d1 | − | ≤ | − | because it is closer to zero than the reflection of d1 through the origin. Now, by a limiting argument we can see that we can choose n1 large enough so that d1,1, . . . , d1,k1 are among d1, . . . , dn1 , and if t1 is the intersection of ∂P with the line through sn1 and d1 then t1 sn 2/3 t1 d1 . | − 1 | ≤ | − | Let e1 = f0. Note that we can write t1 = (T g, g) for some g a (possibly degenerate) linear combination of say e1,r and e1,s, neither of which is e1. Thus g is orthogonal to e1 and so we may apply Lemma 4.2.5 to obtain an f1 such that (T f1, f1) = sn1 and 2 2 (t −s ) 2 (f0, f1) = (e1, f1) 1 n1 /(t1−d1) /3. We then use Lemma 4.1.9 to perform the | | | | ≤ ≤ now standard procedure of choosing a basis b1, . . . , bn −1 for the orthogonal complement { 1 } of f1 with respect to span e1, . . . , en such that (T bj, bj) = 0 for 1 j n1 1. When { 1 } ≤ ≤ − we look at T with respect to this new basis b1, . . . , bn −1, f1, en +1,... , we find that its { 1 1 } diagonal has all the same properties that T started with, but now sn1 replaces d1, and we can ignore the first finitely many terms corresponding to b1, . . . , bn −1 . Thus we { 1 } ∞ can repeat the above procedure ad infinitum to obtain an orthonormal sequence bj { }j=1 115 iR z C Imz = Imd1 d1 = d1,1 { ∈ | } K1 sn1 R d1,2 t1 d1,4 t d1,3 z C Imz = Imd1 { ∈ | − } Figure B.2.: The inclusion of zero and sn1 in the interior of the polygon P . ∞ and a sequence of unit vectors fj such that for all j N we have (T bj, bj) = 0 and { }j=1 ∈ fj span fj−1, en +1, . . . , en and (fj, fj−1) 2/3. ∈ { j−1 j } | | ≤ ∞ We must now show bj forms a basis. Let Pn be the projection onto span b1, . . . , bn . { }j=1 { } Note that each fj is a linear combination of fj−1 and enj−1+1, . . . , enj . So we compute nj+k X ej, fj+k = ej, (fj+k−1, fj+k)fj+k−1 + (fj+k, em)em | | m=nj+k−1+1 = (ej, fj+k−1) (fj+k−1, fj+k) | | · | | = (ej, fj) (fj, fj+1) (fj+k−1, fj+k) | | · | | · · · | | 2k (ej, fj) , ≤ | | 3 2 and thus as k , ej, fj+k 0. Note that I Pn −1 is the projection onto → ∞ | | → − j+k span fj+k, en +1,... . Then we find that { j+k } 2 2 (I Pn −1)ej = (fj+k, ej) 0. − j+k | | → 116 ∞ Therefore Pn I and so bj form a basis and with respect to this basis T has a zero ↑ { }j=1 diagonal. Case 3: There exists some line ` passing through the origin with only finitely many diagonal entries not lying on the line `. By arguments which are virtually identical to those presented at the beginning of case 2 (first four paragraphs), without loss of generality we may assume that this line ` is the real axis and all the diagonal entries lie on the real axis. Now, we have two subcases. Either there are infinitely many positive and negative diagonal entries or not. If so, simply use an argument analogous to case 1. If not, by again applying arguments similar to those at the beginning of case 2, we may assume without loss that there is only one negative entry, say d1 < 0, and the rest positive (since we may ignore those that are already zero if we wish). 0 P∞ 0 P∞ Let sn := dj, so that sn + sn+1 = dj = 0 for each n N. Note that sn < 0 j=n j=1 ∈ 0 0 for all n N, and sn > 0 for all n > 1, and moreover, sn = sn+1. It is clear that ∈ − sn+1 [sn, dn+1] for all n N with convexity coefficient ∈ ∈ 0 dn+1 sn+1 sn sn+1 1 λn = − = − = 0 = 0 −1 . dn+1 sn dn+1 sn dn+1 + s 1 + dn+1(s ) − − n+1 n+1 ∞ P dj+1 0 Furthermore, j=n /sj+1 diverges by Lemma B.1.2, and hence by Lemma B.1.3 we have ∞ ∞ ! Y Y dj+1 λ−1 = 1 + = . j s0 ∞ j=n j=n j+1 Q∞ ∞ Therefore λj = 0. Thus by Lemma 4.2.6, there is a basis bn in which the j=n { }n=1 diagonal entries are given by (T bn, bn) = dn+1 + sn sn+1 = 0. − 117 B.2. The shape of the set of traces In this section we prove Theorem 2.2.7 due to [FFH87] and attempt to fill in many of the details left to the reader or completely omitted in their proof. We first collect several results which are vital to the proof. Theorem B.2.1 ([FSW72, Theorem 5.1]). The following statements are equivalent for T in B(H): T (i) 0 We(T ) = W (T + K) K + (where W (A) denotes the numerical range ∈ { | ∈ K } of the operator A); (ii) 0 T W (T + F ) F is a finite rank operator (or more generally, F runs over ∈ { | } some ideal of compact operators); ∞ (iii) There exists a sequence xn of unit vectors such that xn 0 weakly, and { }n=1 → (T xn, xn) 0 as n ; → → ∞ ∞ (iv) There exists an infinite orthonormal system en such that (T en, en) 0 as { }n=1 → n ; → ∞ (v) There exists an infinite-rank projection P such that PTP +. ∈ K Lemma B.2.2. 0 We(T ) if and only if for every ω R and every > 0 the spectral ∈ ∈ iω projection χ(−,∞)(Re e T ) is infinite rank. Proof. Suppose there is some ω R and some > 0 such that the spectral projection ∈ iω χ(−,∞)(Re e T ) is finite rank. Then let F be the finite rank operator Z ∞ iω iω F = e λ dχλ(Re e T ). − − Then we find Z ∞ −iω iω iω T + F = e e T λ dχλ(Re e T ) − − 118 Z ∞ −iω iω iω iω = e (Re e T ) + i(Im e T ) λ dχλ(Re e T ) − − Z − −iω iω iω = e λ dχλ(Re e T ) + i(Im e T ) . −∞ Therefore, the numerical range of T + F is Z − −iω iω iω W (T + F ) = e W λ dχλ(Re e T ) + i(Im e T ) −∞ Z − −iω iω iω e W λ dχλ(Re e T ) + iW ((Im e T )) ⊆ −∞ −iω e (( , ] + iR), ⊆ −∞ − −iω and 0 / e (( , ] + iR), so 0 / W (T + F ). Therefore by Theorem B.2.1, 0 / We(T ). ∈ −∞ − ∈ ∈ iω Now suppose that for every ω R and every > 0 the spectral projection χ(−,∞)(Re e T ) ∈ is infinite rank. Notice that this statement remains true if we replace T by T + F for any finite rank operator F . Claim: If an operator S has the property that for every ω R and every > 0, the ∈ spectral projection χ (Re eiωS) is nonzero, then 0 W (S). (−,∞) ∈ To prove the claim, suppose that 0 / W (S). Then there is some separating line ` with ∈ W (S) H− and 0 H+. Let ω be such that the angle between ` and the positive real ⊆ ` ∈ ` axis is ω + π and let = dist(0, `) > 0. Since χ (Re eiωS) is nonzero, we can find − 2 (−,∞) some unit vector f such that ((Re eiωS)f, f) = α for some α > . Then notice that − (Sf, f) = e−iω[((Re eiωS)f, f) + i((Im eiωSf, f)] = eiω(a + ir) H+, ∈ ` − for some r R. Thus W (S) H , which is a contradiction. Therefore 0 W (S). ∈ 6⊆ ` ∈ Since T + F satisfies the conditions of the claim for every finite rank F , we know that 0 W (T + F ), and therefore by Theorem B.2.1, 0 We(T ). ∈ ∈ The following definition will prove useful. 119 Definition B.2.3. For T B(H), and a finite dimensional subspace, define ∈ M m X m ΓM(T ) = (T ej, ej) ej finite orthonormal sequence . { }j=1 ⊥ M j=1 For ease of notation, let Γ(T ) := Γ∅(T ). Remark B.2.4. We claim that Γ(T ) is completely included in one of the two half-planes iω determined by a certain line `ω with slope equal to cot ω if and only if (Re e T )+ ∈ 1 . To see this, note that for a positive operator A, Γ(A) [0, A 1]. Furthermore, L ⊆ k k Γ(X + Y ) Γ(X) + Γ(Y ) and Γ(eiωT ) = eiωΓ(T ). Then we have ⊆ iω iω iω iω Γ(e T ) = Γ((Re e T )+ (Re e T )− + iIm e T ) − iω iω iω Γ((Re e T )+) + Γ( (Re e T )−) + iΓ(Im e T ) ⊆ − iω [0, (Re e T )+ 1] + R− + iR ⊆ k k iω = z C Re z (Re e T )+ 1 . { ∈ | ≤ k k } iω iω Therefore Γ(T ) z C (Re e z) (Re e T )+ 1 . The above computations ⊆ { ∈ | ≤ k k } iω 1 iω 1 prove two things. Firstly, (Re e T )+ and (Re e T )− / for some ω imply that ∈ L ∈ L iθ 1 R Tr T = . Secondly, if R Tr T = C then (Re e T )+ / for all θ. { } ∅ { } ∈ L Finally, we remark that these statements hold also for ΓM(T ) when is a finite M dimensional subspace. To see this, let V be some isometry from onto ⊥. Then clearly H M ∗ ∗ ⊥ ⊥ ΓM(T ) = Γ(V TV ). Furthermore, V B( )V = P B( )P via the natural map. Thus H ∼ M H M iω ∗ 1 iω ⊥ ⊥ 1 ⊥ (Re e V TV )+ if and only if (Re e P TP )+ . Note that F := P TPM + ∈ L M M ∈ L M ⊥ iω iω iω ⊥ ⊥ PMTPM + PMTPM is finite rank and Re e T = Re e F + Re e PMTPM Therefore, iω ⊥ ⊥ 1 iω 1 by Lemma 4.1.4 we obtain (Re e P TP )+ if and only if (Re e T )+ . To M M ∈ L ∈ L summarize, + ∗ + ΓM(T ) H Γ(V TV ) H ⊆ `ω ⇐⇒ ⊆ `ω 120 ∆ α β0 γ0 λ γ β Figure B.3.: [λ, α] ∆◦ ∆ (∆0)◦ ⊂ ⊂ ⊂ iω ∗ 1 (Re e V TV )+ ⇐⇒ ∈ L iω ⊥ ⊥ 1 (Re e P TP )+ ⇐⇒ M M ∈ L iω 1 (Re e T )+ ⇐⇒ ∈ L Γ(T ) H+ . ⇐⇒ ⊆ `ω iθ 1 Lemma B.2.5 ([FFH87, Lemma 5]). If (Re e T )+ / for 0 θ < 2π, then Γ(T ) = ∈ L ≤ C. Proof. From Remark B.2.4 and Lemma B.1.4, we conclude co Γ(T ) = C. Moreover, for every finite dimensional subspace , co ΓM(T ) = C. Let λ C be arbitrary and let M ∈ Pm m α = (T fj, fj) for some orthonormal set fj . If α = λ, there is nothing more to j=1 { }j=1 m prove. Otherwise, let = span fj . Then we can find a convex polygon ∆ including M { }j=1 the linear segment [λ, α] in its interior and whose vertices lie in ΓM(T ). There are then two cases. Case 1: The ray from α through λ intersects ∆ strictly between two vertices β, β0. 121 Pn 0 Pn0 0 0 0 Here we take β = j=1(T gj, gj) and β = j=1(T gj, gj). The line through β, β determines two open half-planes. In one of these lie λ, α, and by Remark B.2.4 the other m n 0 n0 contains some element γ ΓN (T ) where = span fj , gj , g . We write ∈ N {{ }j=1 { }j=1 { j}j=1} Pp γ = j=1(T hj, hj) iω 1 By hypothesis, for every ω R we have (Re e T )+ / and by Remark B.2.4 also ∈ ∈ L iω ∗ 1 ∗ ⊥ (Re e V TV )+ / , where V is some isometry with VV = P . Thus the spectral ∈ L N iω ∗ projection χ(−,∞)(Re e V TV ) is infinite-dimensional. By Lemma B.2.2, we have that ∗ 0 We(V TV ). Then by Theorem B.2.1 (iv), there exists an orthonormal sequence ∈ e0 ∞ with e0 such that (T (V e0 ), V e0 ) 0. Since V is an isometry with final space { k}k=1 k k k → ⊥ 0 , setting ek := V e provides us with an orthonormal collection orthogonal to and N k N satisfying (T ek, ek) 0. → ∞ 0 Using some of the terms from ek , we can get new points α, β, β , γ arbitrarily { }k=1 close to the originals, so that λ still lies in the interior of their convex hull, with the additional property that the new systems have the same number of vectors. That is, we can simply assume m = n = n0 = p. By construction, the line through β, β0 intersects the line segment [α, γ] at some point γ0. Since λ co α, β, β0 , at least one of β, β0 lies in the same half-plane (determined ∈ { } by α, γ as λ. Choose the one in this half-plane that is further away from γ0, which we assume is β without loss of generality. Then certainly λ co α, β, γ0 and hence also ∈ { } λ co α, β, γ . ∈ { } Then λ = c1α + c2β + c3γ, where c1, c2, c3 0 and c1 + c2 + c3 = 1. Let Rj = ≥ span fj, gj, hj for j = 1, 2, . . . , m. By the convexity of the numerical range (the { } Toeplitz–Hausdorff Theorem), we can find unit vectors dj Rj such that ∈ (T dj, dj) = c1(T fj, fj) + c2(T gj, gj) + c3(T hj, hj), 122 m for each j = 1, 2, . . . , m. Since Rj Rk if j = k, then dj is an orthonormal sequence ⊥ 6 { }j=1 and so m m m m X X X X (T dj, dj) = c1 (T fj, fj) + c2 (T gj, gj) + c3 (T hj, hj) j=1 j=1 j=1 j=1 = c1α + c2β + c3γ = λ. Therefore, we conclude that Γ(T ) = C. Case 2: The ray from α through λ intersects ∆ at a vertex β. ∞ As in the first case, we can find some infinite orthonormal set ek orthogonal to { }k=1 m n fj gj with (T ek, ek) 0. We can use finitely many of these terms to modify { }j ∪ { }j=1 → α, β so that they are arbitrarily close to the originals and m = n. However, it may no longer be the case that λ lies on the segment [α, β]. If not, we are back in case 1. If so, then λ = c1α + c2β for c1 + c2 = 1, and the argument is the same as the end of case 1. Note that using Remark B.2.4 the proof of Lemma B.2.5 actually shows the stronger assertion that ΓM(T ) = C for any finite dimensional subspace . Fan–Fong–Herrero M use the above to prove the next corollary and their main result. iθ 1 Corollary B.2.6 ([FFH87, Corollary 6]). If (Re e T )+ / for all 0 θ 2π, then ∈ L ≤ ≤ ∞ P∞ there exists an orthonormal basis ej such that the series (T ej, ej) is convergent { }j=1 j=1 and m4k+s+1 X is (T ej, ej) = (s = 0, 1, 2, 3; k = 0, 1, 2,...), k j=1 ∞ for a certain sequence mk of natural numbers. { }k=1 Theorem 2.2.7 ([FFH87, Theorem 4]). Suppose T B( ). Then there are four ∈ H possible shapes that R Tr T can acquire. More specifically, R Tr T is: { } { } iθ 1 (i) the plane C if and only if for all θ R, (Re e T )+ / (the trace-class); ∈ ∈ L iθ 1 iθ 1 (ii) a line if and only if for some θ R, both (Re e T )± / but (Im e T ) ; ∈ ∈ L ∈ L 123 (iii) a point if and only if T 1; ∈ L iθ 1 iθ 1 (iv) the empty set if and only if for some θ R, (Re e T )+ / but (Re e T )− . ∅ ∈ ∈ L ∈ L 124 Index ampliation, 48 of finite rank idempotents, 10 analysis operator, see frame transform of finite spectrum normal operator, arithmetic-mean closed, 7 23 of idempotent matrices, 9, 24 Birkhoff’s Theorem, 61 of idempotent operators, 10, 29, 83 Borel functional calculus, 68 of nilpotent operators, 8 Carpenter Theorem, 23 of orthogonal matrices, 8 commutator, 5 of positive compact operator, 53 conditional expectation, 2 of quasinilpotent operators, 8 convexity, 61 of rotation matrices, 8 of diagonal sequences, 10 of unitary matrices, 8 of expectation of partial isometry or- summable, 82 bit, 62 difference of kernel dimension, 38 of expectation of unitary orbit, 64 eigenvalue sequence, 8 decreasing rearrangement, 11 faithful, 2 diagonal invariance, 7 frame, 29 diagonal sequence, 1–3 canonical dual, 29 absolutely summable, 82 125 dual, 24, 29, 83 matrix representation, 1 idempotent characterization, 30 monotonization, 11 frame operator, 29 nilpotent part, 28, 67, 73 frame transform, 29 normal map, 2 Grammian, 29 numerical range, 6, 26 Hilbert space, 1 operator Hilbert–Schmidt, 10, 68 compact, 6 perturbation of a projection, 73 diagonal, 2 normal, 6, 67 ideal, 72 trace-class, 6 idempotent decomposition of, 66, 71 partial isometry orbit, 19, 39 Jasper’s questions, 24, 66, 77, 83 range projection, 12, 72 Riemann Rearrangement Theorem, 27 L´evy–SteinitzTheorem, 27 Schur–Horn Theorem majorization, 61 approximate, 13 approximate p , 9, 54 − positive compact, 20 hierarchy of, 22 + positive compact nonzero finite di- in c0 , 14 mensional kernel, 21, 60 in `1, 15 n positive trace-class, 17 in R , 7 selfadjoint trace-class, 16 p , 9, 37, 38, 45, 47, 53, 54 − classical, 7, 13, 61 Riemann and Lebesgue, 14 exact strong, 14, 47, 53 positive compact infinite dimen- uniform Hardy–Littlewood, 14 sional kernel, 21, 58 weak, 15 positive compact kernel zero, 21 masa, 2 126 positive finite rank, 17 in type II factors, 14 on real Hilbert spaces, 35 orthostochastic, 35 set of traces, 27, 73 singular value orbit, 19 positive trace-class, 17 singular value sequence, 11 stochastic matrices, 33 doubly stochastic, 33 orthostochastic, 33, 45, 46, 58, 61 synthesis operator, 29 Toeplitz–Hausdorff Theorem, 6, 26, 79 quantitative, 87 trace, 4 with respect to a basis, 27 trace-class, 68 unitary orbit, 19 positive trace-class, 17 zero-diagonal, 5 idempotent, 28 with respect to a nest, 25 idempotents, 10, 73, 77 intrinsic characterization, 28 trace zero, 26, 79 127 128