Approximation of Hilbert Space Operators (Research Notes in Mathematics Series)

Domingo A Herrero Arizona State University

Approximation of Hilbert space. operators VOLUME I

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© Domingo A Herrero 1982 First published 1982 AMS Subject Classifications: Primary 47ASS, 41A6S, 47A60; Secondary 47A15, 47AS3, 81C12 British Library Cataloguing in Publication Data Herrero, Domingo A. Approximation of Hilbert space operators. Vol. 1-(Researcb notes in mathematics; 72) 1. Hilbert space 2. Operator theory I. Title II. Series 515. 7'33 QA329 ISBN 0-273-08579-4 Library of Congress Cataloging in Publication Data Herrero, Domingo A. Approximation of Hilbert space operators. (Research notes in mathematics; 72- ) Bibliography: v. 1, p. Includes index. 1. Operator theory. 2. Hilbert space. I. Title. II. Series: Research notes in mathematics; 72, etc. QA329.H48 1982 515.7'24 82-10163 ISBN 0-273-08579-4 (v. 1) All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechaniall, photocopying, recording and/or otherwise without the prior written permission of the publishers. This book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, without the prior consent of the publishers. This book is sold subject to the Standard Conditions of Sale of Net Books and may not be resold in the UK below the net price. Reproduced and printed by photolithography in Great Britain by Biddies Ltd, Guildford

To Buenos Aires, on her four-hundred first birthday

"A mi se me hace cuento que empez6 Buenos Aires, la juzgo tan eterna como el agua o el aire" (Jorge Luis Borges)

Contents

1. Stability and approximation

1

1.1 Lower estimated derived from the Riesz-Dunford functional calculus 1.2 Lower estimates for the distance to Nk(H) 1.3 Lower semicontinuity of the rank 1.4 Stability properties of semi-Fredholm operators

2 6 8 9

1.5 On invariance and closure of subsets of L(H)

10

1.6 Notes and remarks

11

2. An aperitif:

approximation problems in finite dimensional

spaces

12

2.1 Closures of similarity orbits in finite dimensional spaces

13

2.1.1 The nilpotent case

15

2.1.2 Proof of Theorem 2.1 2.1.3 The lattice (N(Ek)/i,<)

16 17

2.1.4 Closures of similarity orbits of finite rank operators 2.2 The distance from the set of all non-zero orthogonal projections to N(H)

19

20

2.2.1 The limit case

20

2.2.2 On the exact values of 6k and nk 2.2.3 A companion problem: the distance from the set of

23

all non-zero ide!_11potents to N{H) 2.3 On the distance to Nk{H) 2.3.1 A general upper bound 2.3.2 Two illustrative examples

25 27 27 30

2.3.3 An example on approximation of normal operators by nilpotents

33 vii

2.3.4 On the distance to a similarity orbit 2.4 On the distance from a compact operator to N(H) 2.5 Notes and remarks 3. The main tools of approximation

3.1 The Rosenblum operator:

X

+

AX -XB

3.2 Generalized Rota's universal model 3.3 Apostol triangular representation 3.4 Correction by compact perturbations of the singular behavior of operators 3.5 Apostol-Foia~-Voiculescu's theorem on normal restrictions of compact perturbations of operators 3.5.1 Schatten p-classes 3.5.2 Normal restrictions 3.5.3 Density of sets of operators with bad properties 3.6 Notes and remarks 4. Two results borrowed from the theory of C*-algebras 4.1 Essentially normal operators

4.2 4.3 4.4 4.5

Brown-Douglas-Fillmore theorem Berger-Shaw trace inequality Examples of essentially normal operators An application to approximation problems

Matrix models for operators Spectra of compact perturbations of operators Voiculescu's theorem Closures of unitary orbits 4.5.1 Operator-valued spectrum and unitary orbits 4.5.2 Concrete examples of closures of unitary orbits 4.5.3 On normal and quasinilpotent restrictions

viii

37 38 41

3.1.1 Linear operator equations 3.1.2 Approximate point spectrum of a sum of commuting operators 3.1.3 Local one-side resolvents in L(H) 3.1.4 The left and the right spectra of TAB 3.1.5 Rosenblum-Davis-Rosenthal corollary 3.1.6 The maximal numerical range of an operator 3.1.7 The norm of TAB

4.1.1 4.1.2 4.1.3 4.1.4

35

41 41 42 44 50 53 54 56 58 62 69 74 75 76 78 79 84 84 84 85 89 90 92 94 96 99 99 100 101

4.6 Irreducible operators

104

4.7 Notes and remarks

106

5. Limits of nilpotent and algebraic operators 5.1 Limits of nilpotent operators

108 108

5.1.1 Normal limits of nilpotents

108

5.1.2 Spectral characterization of N(H)

111

5.2 Closures of

~imilarity orbits of normal operators with perfect spectra 5.3 Limits of algebraic operators 5.4 Normal operators in closures of similarity orbits 5.5 Sums of two nilpo~ents 5.6 The Apostol-Salinas approach: an estimate for the distance to Nk(H) 5.7 Salinas' pseudonilpotents 5.8 Limits of nilpotent and algebraic elements of the Calkin algebra 5.9 On the spectra of infinite direct sums os operators 5.10 Notes and remarks

6. Quasitriangularity

113 114 115 117 120 124 128 130 132 135

6.1 Apostol-Morrel simple models

135

6.2 Quasitriangular operators

140

6.2.1 Equivalence between the formal and the relaxed

definitions of quasitriangularity 6.2.2 Two lower estimates for the distance to (QT)

141 142

6.2.3 Spectral characterization of quasitriangularity

145

6.3 Biquasitriangular operators

146

6.3.1 Block-diagonal and quasidiagonal operators

146

6.3.2 Characterizations of biquasitriangularity

147

6.4 On the relative size of the sets (QT), (QT)*, (BQT),

[N(H)+K(H)] and N(H)-

153

6.5 A Riesz decomposition theorem for operators with

disconnected essential spectrum 6.6 Notes and remarks 7. The structure of a polynomially compact operator

154 157 162

7.1 Reduction to the (essentially) nilpotent case

162

7.2 The structure of a polynomially compact operator

164

ix

7.3 7.4 7.5 7.6 7.7

Restrictions of nilpotent operators 167 Operators similar to Jordan operators 171 A similarity invariant for polynomially compact operators 173 Nice Jordan operators 177 Notes and remarks 188

8. Closures of similarity orbits of nilpotent operators 8.1 Universal operators 8.1.1 Universal quasinilpotent operators 8.1.2 Universal compact quasinilpotent operators 8.2 Compact perturbations of not nice operators 8.3 Quasinilpotents in the Calkin algebra

189 189 194 194 198

8.3.1 General quasinilpotents 8.3.2 Nice elements of the Calkin algebra

198 204

8.4 Compact perturbations of nice Jordan operators

205

8.4.1 8.4.2 8.4.3 8.4.4 8.4.5

Nice Jordan nilpotents Nilpotents of order 2 Quasinilpotent perturbations Universal operators in N~,h(H) A general criterion for universality

8.5 Separation of isolated points of the essential spectrum affiliated with nilpotents 8.6 Notes and remarks REFERENCES INDEX SYMBOLS AND NOTATION

X

189

206 211 212 215 222 230 236 239 249 253

Preface

The last decade has been fruithful in results on approximation of Hilbert space operators, due to a large extent to the impulse given by Paul R. Halmos in his famous survey article "Ten problems in Hilbert space". The purpose of this monograph (and a second one, by C. Apostol, L. A. Fialkow, D. A. Herrero and D. Voiculescu that will follow and complete the results contained here) is to provide a set of general arguments to deal with approximation problems (in the norm-topology) related to those subsets of the algebra L!HI of all operators acting on a complex separable infinite dimensional Hilbert space that are invariant under similarities. Many interesting subsets of L(HI have this property: nilpotent operators: algebraic operators (satisfying a fixed polynomial): polynomially compact operators: triangular, quasitriangular and biquasitriangular operators: cyclic and multicyclic operators: semi-Fredholm operators (with fixed given indices): operators whose spectrumis equal to a fixed compact subset of the complex plane ~. or whose spectra are contained in a fixed nonempty subset of ~: any bilateral ideal of compact operators, etc, etc. The following list illustrates the kindsofproblems to be considered here: a) Given a subset R of L(HI invariant under similarities, defined in algebraic, geometric or analytic terms (e.g., the set of all algebraic operators, the set of all operators T such that T3 is compact, the set of all cyclic operators), characterize its norm-closure in "simple terms". Since the spectrum and its different parts are the most obvious similarity invariants of an operator, these "simple term~' will usually be expressed in terms of properties of the different subsets of the spectra of the operators in the closure of R. b) More generally, obtain a formula for the distance from a given operator to R or, al least, upper and/or lower estimates for this distance. c) In a surprisingly large number of interesting cases, either R is invariant under compact perturbations, or its closure is contained xi

in (or equal to) the set of all compact perturbations of R. Compact perturbations will be used as a useful tool for approximation and will be also analyzed with respect to the above mentioned peculiar properties. d) Analogues to the problems in a) and b) for subsets of the quotient Calkin algebra. In Chapter I, we shall obtain all the necessary conditions for approximation that can be easily derived from the Riesz-Dunford functional calculus and the well-known stability properties of semi-Fredholm operators. This chapter is followed by "an ap~ritif": the solution of several approximation problems in finite dimensional spaces, which only depend on the results of Chapter I and some "handwork" with matr~ces.

It is interesting to observe that, for a large number of approximation problems (but, unfortunately, not for all of them) the "obvious" necessary conditions derived from the results of Chapter I are actually sufficient, but the proofs of their sufficiency are very hard.These proofs are constructive to a large extent and the "tools" for these constructions are developed in Chapters III and IV: Rosenblum's coro~ lary, Rota's universal model and extensions, Apostol's triangular representation, results on compact perturbations and results "borrowed" from the theory ofC*-algebras (the Brown-Douglas-Fillmore theorem, Voiculescu's theorem, results on closures of unitary orbits). Except for those results related to the theory of C*-algebras (Chapter IV) , the monograph is essentially self-contained. Results on approximation of Hilbert space operators really begin in Chapter V with the characterization (due to c. Apostol, C. Foia~ and D. Voiculescu) of the closure of the set of all nilpotent operators. The closure of the set of all algebraic operators is then obtained as a corollary and this result is used to characterize biquasitriangularity, the closure of the similarity orbit of a normal operator with perfect spectrum and to give several results about the closure of the similarity orbit of an arbitrary operator. Combining the above results with Apostol-Morrel "simple models", we shall obtain the Apostol-Foia~-Voiculescu theorem on the spectral cha; acterization of quasitriangularity. Since algebraic op~rators and operators that "look like backward shifts" are the simple3t examples of triangular operators, this approach to quasitriangularity is, perhaps, more natural than the original one. Finally, the last two chapters are devoted to a deep analysis of the structure of a polynomially compact operator and to the clos~e of xii

its similarity orbit, respectively. A large part of the material contained in the last chapter consists of unpublished results of c. Apostol-D. Voiculescu, J. Barr!a-D. A. Herrero and D. A. Herrero on closures of similarity orbits of essentially nilpotent operators. The author is deeply indebted to Professors Constantin Apostol, I. David Berg, Charles A. Berger, Ronald G. Douglas, Alain Etcheberry, Lawrence A. Fialkow, Ciprian Foia,, Carl M. Pearcy, Allen L. Shields and Dan Voiculescu, and to his professors and ex-fellow students from the University of Buenos Aires, Alejandro de Acosta, Mischa Cotlar, Beatriz Margolis, Lazaro Recht, Norberta Salinas and, very especially, to his wife Marta B. Pecuch de Herrero for infinitely many informal discussions and suggestions and, most important, for their friendly support during all these hard years. The contents of this monograph have been developed in a "Seminar on Approximation of Hilbert Space Operators• at the University of Georgia (Athens, Georgia, USA), during the academic year 1980-1981. The author wishes to thank the authorities of the University of Georgia for their support and to his colleagues Edward Azoff, Richard Bouldin, Kevin Clancey, Douglas N. Clark and Derming Wang for their assistance during the preparation of the manuscript. Many of their valuable observations have been included here. Domingo A. Herrero

Tempe, Arizona June, 1981

xi~i

1 Stability and approximation

As explained in the Preface, in "most" approximation problems related to similarity-invariant sets of operators, the "obvious" necessary conditions for approximation that can be derived from the RieszDunford functional calculus and the stability properties of semi-Fredholm operators turn out to be also sufficient. In a certain sense,this short first chapter contains all the necessary conditions and the remaining of the monograph (and the second one that will follow [16]) is devoted to explain why these necessary conditions are also sufficient. Throughout this monograph, the word operator will always denote a bounded linear transformation mapping a complex Banach space into another. If X and Y are complex Banach spaces, the Banach space of all operators mapping X into Y will be denoted by L(X,Y). We shall write L(H) for the Banach algebra L(H,H), where His a complex separable Hilbert space. Unless otherwise stated, H (H 0 , H1 , H2 , ••• , etc) will always denote an infinite dimensional Hilbert space. The algebra L(H) contains the open subset G(H) ={WE l(H):

W is invertible}

(the linear group of L(H)), which will play a very important role here. A subset R of L(H) is called invariant under simiLarities (or simiLarity-invariant) if it is invariant under conjugation by elements of the group G(H), i.e., T

€

R => S (T)

c

R,

where -1 .S (T) = { WTW :

W

E

G(H)}

is the simiLarity orbip of T. If K(H) denotes the ideal of all compact operators acting on Hand ~:l(H) + A(H) = L(H)/K(H) is the canonical projection of L(H) onto the (quotient) Calkin algebra, then the image ~(T) = T+K[H) ofT € l(H) in A(H) will also be denoted by T. The reader is referred to [77], [119] for the general theory of Hilbert space operators. 1

1.1 Lower estimates derived from the Riesz-Dunford functional calculus A nonempty bounded open subset n of the complex plane a: is a Cauchy domain if the following conditions are satisfied: (i) n has finitely many components, the closures of any two of which are disjoint, and (ii) the boundary an of n is composed of a finite positive number of closed rectifiable Jordan curves, no two of which intersect. In this case, r = an will be assumed to be positively oriented with respect to n in the sense of complex variable theory, i.e., so that 1 2'11'i

J

d>..

r >..-~

{1, if =

o,

~ e n 1 n- = nur

if ~

(the upper bar will always denote closure with respect to the metric topology of the underlying space). Clearly, r is uniquely determined by n (and conversely). We shall say that r is a rectifiabZe contour. If all the curves of r are regular analytic Jordan curves, we shall say that r is an analytic contour (or n is an anaZytic Cauchy domain). If A is a Banach algebra with identity 1 and a e A, the spectrum of a will be denoted by a(a). The complement p(a) = 0:\o(a) of a(a) in the complex plane is the resoZvent set of a and the function >.. ~ (X-a)-l (from p(a) into A) is the resolvent of a. It is well-known that (X-a)-l is an analytic function of >.. in that domain that it satisfies the first resoZvent equation: (>..-a) -l_ (p-a) -l = (ll- X) (>..-a) -l (p-a) -l (X rll e p (a) ). Furthermore, if a, be A and>.. e p(a)np(b), then (>..-a)- 1 - (X-b)-l = (X-a)- 1 (a- b) (X-b)-l (second resolvent equation). If a is a nonempty clopen subset of a(a), then there exists an analytic Cauchy domain n such that a c nand [a(a)\a]nn-= -; in this case 1 Jr (>..-a) -1 dX E(o;a) = 2 '1fi

is an idempotent of A commuting with every b in A such that ab = ba. (E(a;a) is the Riesa idempotent corresponding to a.) The following theorem is just a quantitative form of the classical result on upper semi-continuity of separate parts of the spectrum. The reader is referred to [76], [153], [172], or [173,Chapter XIV] for the basic properties of the Riesz-Dunford functional calculus. THEOREM 1.1. Let a and b be two elements of the Banach algebra A 2

~ith

identity 1. Assume that the spect~um o(a) of a is the disjoint u~ ion of two compact subsets a 0 and o 1 such that o 1 is nonempty (o 1 t< ~)

a~d Zet n be a Cauchy domain such that a 1

lla-brJ< minfmc;~.-a)-~J- 1 : (where JI.Jidenotes the norm of A)

II = II a- at II

;~.

If

an},

€

= min{ 11!>.-a) -~1- 1 :

A "- an} and

= t6 < li -~~- 1

for all A E an, so that o(at)nan = fJ for all t E Thus, the idempotent

....!..-f 2w1 an

et

nn- = -·

then o(b) nO 'I ffand o(b) nan =fJ.

PROOF. Assume that II a-b II = 6 < m let at= (1-t)a+tb, 0 ~ t ~ 1: then 11!>--at> - <><-a>

0 and a 0

c

(>.

-at

~0,11.

) -1 d.>.

is a well defined element of A for all t E [0,1]. Furthermore, if 0 ~ t < s ~ 1, the second resolvent equation implies that

whence it readily follows that t + et is a continuous mapping from [0,1] into A. Since o 1 = o(a)nn 'I fJ, it follows that 1

eo = 2w1

f

(;i.-a)

-1

d).

F o,

an so that II e 0 II :2; 1 (Recall that et is idempotent, 0 ity, lle 1 11 :2; 1 and therefore el

= 2!i f

(>.-b)-1 d). an This is clearly impossible, unless o(b)nn

~

~

t

1). By continu

F 0. F fJ.

0

Recall that if (X,d) is a metric space and B(Xl (Bc(X]) is the family of all nonempty bounded (closed bounded, respectively) subsets of X, then ~(A,B)

= inf{e

>

0:

B

c

AE, A

c

BE},

where AE = {x EX: dist[x,AJ s e},defines a pseudometric in B(Xl (a metric in Bc(X], resp.);dH(A,B) is the Hausdo~ff distance between A and B, A, B € Bc(X]. The qualitative form of Theorem 1.1 is the following COROLLARY 1.2. (i} Let a be an element of a Banach algebra A with identity. Assume that o(a) is the disjoint union of two compact subsets a0 and a 1 such that o 1 F fJ. and that a 1 is contained in abounded openset 0. Then there ezists a constant C = C(a,o 1 ,n) > 0 such that 3

o(b)nO t: ~foro aZZ bin A satisfying lla-bll 0, there e~ists 6 > 0 suah that a(b) a(a)E, provided lla-bll < 4, i.e., the mapping a+ a(a) from A into Be(~) (Hausdorff distanae) is upper semiaontinuous.

c

PROOF. (i) follows immediately from Theorem 1.1: Given a 1 and 0 (as above) , there exists a Cauchy domain o1 such that a 1 c n1 c n 1 - c S2 and a 0 no 1 - =~-Take c = min{lj(A-a) -1II -1 : Ac: an 1 }. (ii) Apply (i) with a0 ~and 0 =interior a(a)E. 0 COROLLARY 1. 3 . (i) If S2 is an open subset of ~. then {a c: A: c S'l} is an open subset of A. (ii) If E is a G6 subset of ~. then {a € A: a(a) c E} is a G6 subset of A. In par-tiauZar, the set {a e; A: a (a) = {0}} of aZZ quasiniZpotent eZements of A is a G6 in A. a(a)

PROOF. (i) follows from Corollary 1.2(ii) and (ii) is an immediate consequence of (i) • 0 The limit case of Theorem 1.1 yields the following COROLLARY 1.4. Let a, o 1 (o 1 7: ~) and assume that b c: A satisfies the inequality

n be

II a- b II s min{ II< A-a) -11 I -1 :

as in Theorem 1.1 and >.

c:

anl;

PROOF. Define at as in the proof of Theorem 1.1; then Theorem 1.1 implies that a(atlnan = g and o(at)nS'l t: ~for 0 s t < 1. Since lim(t _,. 1) lib- at II= 0, i t follows from Corollary 1.2(i) that o(b)nO- cannot be empty. 0 It is convenient to observe that the result of Corollary 1.4(and, a fortiori, the result of Theorem 1.1 too) is sharp. In fact, we have EXAMPLE 1.5. Let P c: L(H) be a non-zero orthogonal projection and let Q = {A: I A-ll < ~}; then II< A-P) -lll-l = ~ (A € an ) • Thus' by Corol lary 1.4, a(B)nS'l- 7: ~ for all B in L(H) such that liP- Bll s ~On the other hand, if A = ~. then II P- A II = ~ and o (A) = {~} c an. If T c: L(H), o is a clopen subset of o(T) and E(o;T) is the corresponding Riesz idempotent, then the range ran E(o;T) and the kernel ker E(o;T) of E(a;T) aresubspacesofH invariant under every B-in L(H) commutinq with T (i.e., hyperinvariant forT), and H can be written as 4

the algebraic (not necessarily orthogonal!) direct sum H = ran E(a:T)+ ker E(a:T). (Here and in what follows, subspace will always denote a closed linear manifold of a Banach space.) Furthermore, the spectrum of the restriction T)ran E(a:T) ofT to ran E(a:T) coincides with a and the spectrum of the restriction T!ker E(a:T) coincides with a(T)\a [l73,Chapter XIV]. In what follows, ran E(a:T) will be denoted by H(a:Tl. If a = {X} is a singleton, we shall simply write H(X:T) (E(X:T)) instead of H({X}:T) (E({X}:T), resp). If a = {X} and dim H(X:T) is finite, then X is called a normaL eigenvaLue ofT: in this case, H(X:T) coincides with ker(X-T)n for some n ~ 1. The set of all normal eigenvalues ofT will be denoted by a 0 (T). Clearly, a 0 (T) is contained in the point spectrum ap(T) ofT (i.e., the set of all eigenvalues ofT). The esse~ tiaL spectrum ofT, i.e., the spectrum ofT in A(H) will be denoted by ae (T).

COROLLARY 1.6. Let A, Be L(H); then (i)

Assume that a is a nonempty

~Lopen)subset

of a(A) and Let Q

~Cauchy domain) be a neighborhood of a such that

[a(A)\a]nO- =-.If X dO}. then a 1 =a (B) nO -,; ~; (ii) furthermore, dim H(a:A) =dim H(a 1 :B) (0 s dim H(a:A) s ~>. (iii) If a is a nonempty cLopen subset of a (A) and the Cauchy e domain n is a neighborhood of a such that [a e (A)\a]nO- = ~. then a e (B) nO-,;- for aZZ Bin L(H) suah that IIA-BII < min{I!CX-A)-~1-l: A e an}.

II A- B II

< min{

II< X-A) -~~-l:

PROOF. (i) and (iii) follow immediately from Theorem 1.1, applied to A= L(H) and to A= A(H), respectively. (ii) This follows from the proof of Theorem 1.1. Observe that, if At= (1-t)A+tB, then the continuity of the mapping t + E(a(At)nO:At) (0 s t s 1) implies that the idempotents E (a :A) = E (a (A 0 ) nO :A0 J and E (a 1 :B) E (a (A1 ) nO:A1 ) necessarily ha-ve the same (finite or infinite) rank.O Until now, we have only applied the arguments of functional calc~ lus to a very particular class of functions analytic in a neighborhood of the spectrum a(a) of an element of the Banach algebra A: namely, the characteristic function of a suitable neighborhood of a clopen su~ set of a(a). Analogous results hold in a much more general setting: namely, PROPOSITION 1.7. Let a be an eLement of the Banach algebra A with identity 1 and let f be an anaLytic function defined in a neighborhood n of a(a). Given £ > 0, there ezists ~ > 0 such that f(b) is weLL-de5

fined for aZZ b in A satisfying

II a- bll

II f (a) - f

< ti and. mol'eovel'.

(b)

II

< e.

PROOF. Let n 1 be a Cauchy domain such that a(a) c n 1 c Ql- c Q. By Corollary 1.2(i), there exists o1 > 0 such that a(b) c n 1 for all b inA satisfying lla-bll < ti 1 • Clearly, f(b) is well-defined for all these b, by means of the integral f(b)

= 2!i

Ianl f(~) <~-b)-ld~.

The second resolvent equation implies that max{ 11<~-b) -~1:

~ " an 1 }

< l+max{

11<~-a) -11:

~ " an 1 l,

and llf(a)-f(b) II s (1/2'11") .length(an 1 > .max{ If(~) I·II-11 x


-1

II>:

A.

E

an 1 l ·II a- blj,

provided II a - b II < o2 (for some o2 , 0 < o2 s ti 1 ) • It follows that, i f !Ia -bll < o for a suitably chosen ti, 0 < then f (b) is well-defined and II f {a) - f (b) II < e. fJ

o < oz

The following particular case is especially important for our purposes. COROLLARY 1.8. Let

a~:

A.

let~

be an isolated point of a(a) and

let k

={(A.-~} • in some neighborhood of~

f(A.) for some k

2:

0

• in some

neighborhoodofa(a)\{~}.

1.

Let {an\,,:1 be a sequence of elements in A such that !Ia- ani!-+ 0 (n-+oo). Then f(an) is IJJell-defined for all n large enough and

f(a) =0 if and only if lim(n-+ oo) l!f(anlll

=

0.

1.2 Lower estimates for the distance to Nk(H)

Let Nk (A l = {a E A: a"· = 0} denote the set of all nilpotent elements of order at most k {k = 1,2, ••• ) of the Banach algebra A, and let N(A) = uk:l Nk(A) be the set of all nilpotents ·of A. In order to simplify the notation, the set of all nilpotents (of order at most k) in L(H) will be denoted by N(H) (Nk(H), resp), or simply by N if His understood. It is a trivial consequence of Proposition 1.7 that, given a non6

constant polynomial p,the set {a E particular, i f we choose pk(A) COROLLARY 1.9. (i)

=

A:

p(a) = 0} is closed in A. In

Ak (k = 1,2, ••• ), then we obtain

Nk(A) is alosed in A for aZZ k = 1,2, •••

N(A) is an F0 subset of A.

(ii)

The following result provides a partial answer to the problem of estimating the distance from a given operator to Nk(H). PROPOSITION 1.10. (i) Bkx = O,

for some

x

II A- B II

Let A, BE L(H). If IIAkxll = llxll = 1, but in H and some k l! 1, then

~

max{
(ii) If IIAkxll = llxll = 1, but Bkx = 0 for some x in H and some k

~

1, and max{IIAII,IIBIIl = M, then IIA-BII ;:: 1/(kM).

(iii) for some k

Let a, b e: A (a Banaah algebra). If llakll = 1, but bk = O, ~

1, then

II a- b II ;:: max{ l/k-11 all, l/k-11 bIlL

(iv) If II akll =1, but bk = 0, for some k M,

~

1, and max{ll aJI,IIblll

=

then IJa-bll ~ 1/(kM). PROOF.

(i)

Let IIAII = M and IIA-BII = cS. I t is completely apparent

that II B II ~ M+cS, and therefore 1 = II Akx II = II
~I~:~

~

II Ak-Bk II

11Air-j-11A-BII-11BIIj

~ I~:~

~ cSI~:~

II Ak-jBj-Ak-j-lBj+lll

Mk-j-l(M+cS)j =

cSMk-li~:~j

cSMk-l[ ( l+.Q) k-1] (cS/M) - l = (M+o) k_t;c, M so that (M+cS)k

~ Mk+l. Hence, cS ;:: (Mk+l)l/k_M.

I f we assume that IIBII = M and IIA-BII =

o,

then IIAII ~ M+cS and we

arrive at the same inequality, whence the result follows. The remaining statements follow by the same argument. REMARKS. 1.11. (i)

0

Since, in Proposition l.lO(ii) and (iv), M

cannot be smaller than 1 and (by the mean-value theorem) 1/ ( 2kt.t)

< (~+1) l/k_M s 1/ (kM) ,

for all k;:: 1, the estimates of (i) and (ii)

((iii) and (iv)) are of

the same order. (ii)

The argument of the proof of Proposition l.lO(i) also ap-

plies to the case when IIP(A)xll = llxll = 1, but p(B)x = 0, for some pol2_ m k· nomial p, p(A) = rrj=l(A-Aj) J. Since ll[p(A)-p(B)Jxll = 1, IIA.-BII cannot

7

be "too small", where the words "too small" have a concrete numerical expression in terms of p and A . The same applies, of course, to (ii), (iii) and (iv).

1.3 Lower semicontinuity of the rank PROPOSITION 1.12. Let A e L(H) and Zet {An}n:l be a sequence of opePatoPs such that II A- A II -+- 0 (n + oo); then n (i) rank As lim inf(n + oo) rank An. If IIAnxll ~ £11xll foP some £ > 0 and aZZ x in a subspace Hn with dim Hn ~ d, then II Axil ~ £11 xll fop aZZ x in a subspace H0 with dim H0 ~ d. (iii) If An e KIHI foP aZZ n ~ n 0 , then A e K(HI. (ii)

PROOF. (i) If lim inf(n + oo) rank An=oo, then there is nothing to prove. Assume that lim inf (n + oo) rank An= d < oo. Passing, if necessary, to a subsequence, we can directly assume that rank An= d for all n = 1. 2, ••.• If rank A~ d+l, then there exist d+l linearly independent vectors y 1 , y 2 , ••• , yd+l e H such that {Ayj}~!i is a line~rly indepen~ ent set. Clearly, Y =linear span{y 1 ,y 2 , •.• ,yd+l} has dimension d+l and therefore Ynker An~ {O} for each n = 1,2, ••.• It is easily seen that {An}n:l is a bounded sequence and there exists y e Y and a subsequence {An.}J.:l such that IIAYII = 1 but lim(j + oo)IIA Yll = 0, so that "" J nj {An.lj=l cannot converge to A, even in the strong operator topology, a J

contradiction. Therefore, rank A s d. (ii) Observe that our hypothesis implies that IIA*-A~II + 0 and i IIA*A-A~Anll + 0 (n + oo). Now the result follows from an elementary ana~ysis of the spectral decompositions of the hermitian operators A*A and A*A (see, e.g., [117]). n n The third statement is trivial. 0 It is convenient to recall that A e LIHI is compact if and only if ran A does not contain an infinite dimensional subspace [71],[96]. we shall use the following non-standard notation: If A e KIHI is not a finite rank operator, then rank A will be defined as oo-. Thus, rank T = oo will be an equivalent way to say that T is not compact. The different possible ranks will be linearly ordered by 0 ~ 1 ~ 2 ~ ••• ~ n ~ n+l ~ ••• ~ 8

oo- ~ ""•

1.4 Stability properties of semi-Fredholm operators Recall that T € L(H) is a semi-FredhoZm operator if ranT is clos ed and either nul T = dim ker T or nul T* = dim ker T* = dim H/ran T

is finite (where T* denotes the adjoint ofT in L(H)). In this case, the index of T is defined by ind T = nul T - nul T*. The following theorem resumes the main properties of the semiFredholm operators. The reader is referred to [106] and [153,Chapter IV] for details. THEOREM 1.13. Let T E L(HI be a semi-FredhoZm operator; then (i) T* is aZso a semi-FredhoZm operator and ind T* = -ind T; (ii) There exists a constant 6 > 0 such that IITxll ~ 6llxll for aZZ x 1 ker T and II T*y II ~ 611 y II for aZZ y .L ker T*; moreover, 6 can be chosen as min{). E o([T*T]~)\{0}}; (iii) There exists 6 = 6(T) > 0 such that if A E: L(H), tiAlls land j).j < 6, then T+).A is semi-Fredho Zm ar~d ind (T+AA) = ind '1', nul (T+AA) s nul T and nul (T+AA) * s nul T*;moreover,nul(T+I.A) and nul (T+).A) *are constant for 0 <-~).I < &. (iv) In particuZar, if {An}n:l is a sequence of operators such that IIT-Anll _,. 0 (n->- co), then An is semi-FredhoZm for all. n <-: n 0 and nul T "' lim sup(n _,. co) nul An' nul T* "' lim sup(n _,.co) nul A~; (v) If K € K(H), then T+K is semi-FredhoZm and ind(T+K) = ind T. (vi) If B is another semi-FredhoZm operator and ind T+ind B is ~ezr-defined (i.e., {ind T,ind B} ~ {oo,-oo} or {-oo,oo}J, then TB is a semi-FredhoZm operator and ind TB = ind T+ind B. In particular, ~ is semi-Fredholm and ind ~ = k (ind T), for aZZ k "' l. A semi-Fredholm operator T is a FredhoZm operator if -co < ind T < The well-known Atkinson's theorem asserts that Tis Fredholm if and only if Tis invertible in A(H)[ll9], [153]. Hence,

~.

pF(T)

=

11:\ae(T)

=

{).€11::

).-Tis Fredholm}

(the FredhoZm domain of T) is an open subset of a:. The Zeft ~ight) spectrum of an element a of a Banach algebra A will be denoted by a 1 (a) (ar(a), resp) and its complement p1 (a) =II:\ a (a) (p (a) = 11:\0 (a)) is the Zeft (right, resp) resoZvent set of a. R. r r Thus, a e (T) = a",e (T) ua re (T) , where a".. e (T) = a".. (T) ( l.eft essential. spectrum) and a (T) = a (~) (right essential. spectrum). It is well-known re r that the intersection aR.re(T) = a 1e(T)nare(T) (some authors call oR.re(T) the WoZf spectrum ofT) contains the boundary aae(J) of aJTl 9

and therefore, it is a nonempty (compact) subset of ~. Its complement ~\aire(T) = pie(T)upr~(T), where pie(T) = ~\~ie(T) and pre(T) =~\are (T), coincides with ps-F(T) ={A e ~: A-T is semi-Fredholm}, the semiFredholm domain of T.

ll

The following results are an immediate consequence of Theorem 1. and its proof (see [153,Chapter IV]).

COROLLARY 1.14. Let T E L(H); then (i) Ps-F(T) is the disjoint union of the (possibly empty) open n ® -oo sets {ps-F(T)}_oo
Ps-F(T) = {A €, ~: A-T is semi-Fredholm with ind(A-T) = h}, -oo~h~oo. (ii) For eaah h, -oo ~ h ~ oo, ph s- F(T) is a lower semiaontinuous funation of T in the sense that, if II T- Tnll + 0 (n + oo) and e: > 0, then h h p 6 _F(T) c [p 6 _F(Tn)]e: for all n ~ n0 (e:). (iii) pOs- F(T) aontains the resolvent set p(T) = ~\a(T) ofT, and a0 (T).

(iv) (v)

h

If h 'F O, then p 6 _F(T) is a bounded set. If the minimal index of A-T, A " ps-F(T), is defined by

min.ind(!.-T) = min{nul(A-T),nul(A-T)*}, then the funation A + min.ind(A-T) is aonstant on every component of Ps-F(T}, except for an at most denumerable subset p:_F(T) without limit points in ps-F(T) . Furthermore, if~ E p;_F(T) and A is a pointof p F(T} in the same component as ~ but A i p F(T), then

s-

s-

min.ind(~-T)

(vi)

A

i

If A

> min.ind(A-T).

ps-F(T) and nul(A-T} < oo (nul(A-T)* < oo, resp), then oo n ps-F(T) <=> ker(A-T} c nn=l ran(A-T) oo n (ker(A-T)* c nn=l ran(A-T)* , resp.). E

s

p:_F(T) is the set of singular points of the semi-Fredholm domain p s- F(T) ofT~ pr s- F(T) = p s- F(T)\pss- F(T) is the set of regular points. It is completely apparent that pr F(T) is open and contains p(T). On sthe other hand, it is easily seen that p:_F(T) contains a 0 (T). InCha£ ter III we shall return to the analysis of these sets.

1.5 On invariance and closures of subsets of L(H) A subset R of L(H) is called invariant under unitary equivalence 10

if

T

E

R

=> U(T) c

R,

where U(H) = {U € L(H): U is unitary} and U(T) = {UTU*: U € U(H)} is the unitary orbit of T. It is completely apparent that U(T) c S(T). We shall establish without proofs some very elementary facts that will be frequently used in the future. PROPOSITION 1.15. If R is a subset of L(H) invariant either under simiZarities, or under unitary equivaZenae, or under aompaat perturbations (i.e., R+K(H) = RJ, then R- has the same property.

1.6 Notes and remarks Theorem 1.1 is just the quantitative version of [153,Theorem 3.16, p.212J (see also [177], or [132,Theorem lJ). Corollaries 1.3(ii) and 1.9(ii) are two elementary observations due to S. Grabiner [108] and D. A. Herrero [132], respectively. Proposition 1.10 is a mild improvement of a result due to D. A. Herrero rlSO,Lemma 4.3] (see also [44, Lemma 4.1]). The notion of "rank T ;= oo-" for a compact operator T, not \ of finite rank, was introduced by J~ Barr!a and D. A. Herrero in [44] in connection with the analysis of '~e similarity orbit of a nilpotent operator (See also Chapter VIII). The notion of "minimal index" is due to C. Apostol [10]. The fact that the singular points of Ps-F(T) are isolated points of this set was discovered by I. c. Gohberg and M. G. Krein [107]. In the above mentioned article, C. Apostol proved that p:_F(T) is, precisely, the set of points of discontinuity of the function that maps A€~-F(T) into the orthogonal projection of H onto ker(A-T) (see also t2S,Lemma 1.6 and Corollary 1.7]). This result will be analyzed in Section 3.3.

11

2 An aperitif: approximation problems in finite dimensional spaces

In this chapter we shall analyze several intrinsically finite dimensional problems, as well as infinite dimensional ones which can be solved through an essentially finite dimensional approach or by an argument in Which the (finite or infinite) dimension of the underlying Hilbert space plays absolutely no role. It will be convenient to introduce some notation: H will always denote a complex separable Hilbert space of dimension d, 0 s d s ~. If 0 s d < ® 1 then we shall also write ~d (with its canonical inner product) instead of H. If A, B E L(H), A- B (A~ B) will mean that A and B are similar (unitarily equivalent, resp.). A 7 B will be used as an alternative . . _ s1m _1 way to 1nd1cate that B € S(A) I i.e., that IIB-W AN II~ 0 (n ~ oo)for ~ n n in G(H). If A ~ Band B ~ A (equivaa suitable sequence {W} n n= 1 s1m s1m lently, S(A)- = S(B)-), then we shall say that A and Bare asymptotically similar. (In symbols: A# B.) It is completely apparent that s!m is a reflexive and transitive relation and that # is, indeed, an equi~ alence relation in L(H) .(Use Proposition 1.15. It is well-known and trivial that - and ~ are also equivalence relations.) If A E L(H 1 ) and B E L(H 2 ), where H1 and H2 are isomorphic Hilbert spaces (in symbols: H1 ~ H2 ) , then A# B will be understood as "up to a unitary mapping U from H2 onto H1 ", i.e., S (A)-= S (UBU*)-. The same observation applies to the other relations. The relation sim induces a partial order < in the quotient set L(H)/#, defined by: Let [A]= {T E L(H): S(B)

T#A}; [B] <[A] i f A sim B.(Equivalently,

c S(A)-.)

Given a (finite or denumerable) uniformly bounded family {AV }V€ r of operators such that A" E L(H") for all v in r, we shall denote by 49 v E r Av the direct sum of· the operators A" acting in the usual fashion on the orthogonal direct sum H = •vEr Hv of the spaces Hv, i. e., if x = 8lV€ r x V is a vector of H, then (8lV£f Av)x = •vEr Avxv. Clearly, IIIBVEf A)l = sup(v 12

E

r> IIA)I <

oo

If r = {1,2, ••. ,n}, we shall also write A1eA 2e •... eAn. If AV ~A£ L(H) for all v in r and card r =a (0 ~a~~>, then A(a)will de H(a) (orthogonal directnote the operator eV£ r Av acting on •v, r Hv sum of a copies of H) . If M is a subspace of H, then ML = H9M is the ortho9onal complement of M in H. Given f, g 10 H, f8g" L(H) is the rank one operator defined by (f9g)x = <x,g>f, where<.,.> denotes the inner product of H. FIHl = {Lj~l fj6gj:

fj, gj

€

H, j = 1,2, ••• ,n; n = 1,2, •.• }

is the ideal of all finite rank operators acting on H. Let {e 1 ,e 2 , ••• ,ek} be the canonical orthonormal basis (ONB) of ~k and let qk" L(~k) be the operator defined by (2 .1)

(k

= 0,1,2, .•• ;

q 0 is the 0 operator acting on the trivial space {0},

q 1 is the 0 operator acting on the one-dimensional Hilbert space ~.

~l ~

and qk admits the matrix representation 0 0 0

1 0

0 1

0

0

0 0 0

0 0 0

(k

qk

0

0

0

0

1

0

0

0

0

0

X

k)

with respect to the canonical ONB, for k =2,3, •.• ). These operators will play a very important role throughout this monograph. A £cr(a)} will denote the spectral raFinally, sp(a) =max{ IXI: dius of a £ A (a Banach algebra) •

2.1 Closures of similarity orbits in finite dimensional spaces

As remarked in the introduction, for many approximation problems the "obvious" necessary conditions derived from the results of Chapter I turn out to be sufficient too. Here is a concrete example of this situation: THEOREM 2.1. Let T

£

L(~

d

) and let p(X)

m

k"·

ITj=l (X-Xj) Jrxi ~ Aj• 13

for i F j) be its minimal (monic) polynomial; then the closure of the similarity orbit of T is equal to S(T)

= {A£ L(~d):

rank q(A) s rank q(T) for all qlpl,

~here

q!p denotes a monic polynomial q dividing p. Furthermore, if L £ L(~d), then L # T if and only if rank q(L) rank q{T) for all q!p if and only if L ~ T.

COROLLARY 2.2. Let T e L(~d); then the follo~ing are equivalent: (i) S(T) is maximal ~ith respect to inclusion (equivalently, [T] is a maximal element of (L(~d)/#,<)J; (ii)

T ~ lB.~~\ ]-

(), ].+qk j )

0 ..] and k.] as in Theorem 2.1, j

= 1,2, ••. ,m),

(iii) Lj~l kj = d (~here the kj's have the same meaning as in Theorem 2.1); (iv) T is a cyclic operator; (v) S(T)- ={A£ L(~d): o(A) o(T) and dim H(A;A) dim H(A:T) for all A in o(T)}. COROLLARY 2.3. LetT e L(~d); then the follo~ing are equivalent: (i) S(T) is minimal (equivalently, [T] is a minimal element); (ii) kj (defined as in Theorem 2.1) is equal to 1 for all j = 1, 2, ••• ,m; (iii) T is similar to a normal operator; (iv) S(T) is closed in L(~d); (v) T £ S(L) for all Lin L(~d) such that o(L) o(T) and dim H(A;L) =dim H(A;T) (A e o(T)J.

d -1 LetT, A£ L(~ I and assume that IIA-WnTWn II+ 0 (n + aa) for a suitable sequence {Wn}n:l of operators in G(~d). Clearly, q(A) and q(WnTWn -1 ) = Wnq(T)Wn -1 are well-defined (for all n = 1,2, .•• ) for all q!p and (by Proposition 1.7) l!q(A) -Wnq(T)Wn-lll + 0 (n + -1

aa).

Since rank Wnq(T)Wn =rank q(T) (for all n = 1,2, ... ), it follows from Proposition 1.12(i) that rank q(A) s rank q(T). Hence the conditions of Theorem 2.1 are necessary. The sufficiency of these conditions will be proved in several steps. The second statement of the ~eorem is a trivial consequence of the first one: It is obvious that S(L) = S(T) if and only if o(L) = o(T) and rank q(L) = rank q(T) for all qlp (Use Proposition 1.15). On the other hand, a simple analysis of the Jordan forms of L and T shows that rank q(L) = rank q(T) for all qlp if and only if L and T 14

are similar. (In particular, this implies that they have the same spectrum.)

2.1.1 The nilpotent case LEMMA 2.4. If 1 s m s k-1, then ~eqk sim ~+leqk-l"

computations show that W is invertible, e: .•• ,m), W -1 f 1 = (l/e:)f 1 and W -1 f.= f.(j = 2,3, .•. ,k) and e: e: ] ] -1 Qe: = We:(qmeqk)We: = Q+e:f 1ef 2 , where Qf 1 = 0, Qe 1= f 1 , Qei= ei_ 1 (i = 2,3, ••• ,m), Qf 2= 0 and Qfj= fj-l (j = 3,4, •.• ,k). It is immediate that Q = ~+ 1 eqk-l and that Q = lim(e: + 0) Qe:(unless otherwise stated, lim must always be understood as a limit in the noPm-topoZogy). D LEMMA 2.5. Let T S(T)-= {A

E

E

L(~d):

L(~d) be a nilpotent of oPdeP m; then rank Aj s rank Tj foP j = 1,2, ••• ,m}.

PROOF. We have already observed that the condition "rank Aj s; rank Tj for j = 1, 2, ..• ,m" is necessary. (Observe that the minimal po!, ynomial of T is p(A) = Am.) Assume that rank Aj s; rank Tj for j = 1,2, ••• ,m. Clearly, we can directly assume that T and A are Jordan forms, i.e., (t 1 l (t 2 l (tk) (cx 1 l (cx 2 l (cxh) T q $q e •.• eq and A = ~ eq $ ••• e'Im nl n2 nk 1 m2 h (tj' cxi

> 0).

We shall proceed by induction on m(T,A) = d+Lj:l (rank Tj- rank Aj).

The case m(T,A) = 1 is trivial. Assume that m(T 1 ,A1 ) s; n implies that T1 7 A1 , whenever A1 and T1 are nilpotent operators acting on a s~m . · finite dimensional space and satisfy rank A1 J s; rank T1 J for all j =. 1,2, .•• , and let A, T E L(~d) be nilpotent operators such that rankAJ s rank Tj for all j = 1,2, ••• , and m(T,A) = n+l. If Tan~ A have a.common Jo~dan block.qr' then T = qreT 1 , A= qr $A1 , rank A1 J-rank T1 J =rank AJ- rank TJ s; 0 for all j = 1,2, ••• , 15

m(T1 ,A1 ) s (n+l)-r s nand, by induction, T1 sim A1 . A fortiori, T=qr $Tl sim qr$A1= A. If T and A have no common Jordan blocks and if r ~ 1 is the minimum index such that ~ < nr, then T has the form T = qn

$qn $T' r-1 r (where n 0 = 0 and q 0 acts on a {0}-space, if r = 1) and we have rank As < rank T8 for nr-l + 1 s s s mh- 1. Indeed, if rank As= rank Ts and if as (ts, resp.) denotes the number of Jordan blocks of A (T, resp.) with order of nilpotency grea! er than or equal to s, then it is obvious that as > ts and this yields the contradiction rank As-l = as+rank As> ts+rank Ts =rank Ts-l. Setting T1 = qn

+1 $qn _ 1$T' , we can check that rank T1 j = rank r-1 r . Tj for j = 1,2, •.• ,nr-l and rank T1 J= rank Tj-1 for nr_ 1 +1 s j s mh -1, so that s rank T1 j for j = 1,2, ••.• m -1 On the other hand, m(T 1 ,A) s m(T,A)- (rank Tmh-l- rank T1 h ) m(T,A) - 1 = n, and consequently, T1 sim A. If r > 1, then T sim Tl by n1-1 n1 !.emma 2.4._If r=l, then qn 1 "' }:j=l ej6ej+l E: L(a: ) is similar to e:e 1 ee 2 +-!~! 2 1 ej6ej+.l (« ~ 0) and, lett-Ulg e: + 0, we conclude that T s!m 0 T1 • In either case, Ts!m T1 sim A, and therefore A £ S (T)-.

2.1.2 Proof of Theorem 2.1 L (Q: d )

(0 S d < oo) 1 the minimal polynomial of k· J (A.-I' A., i f i -F j) and rank q(A) s rank T is p, p(A) = nj=l (;\-A.) J ~ J q(T) for all qiP· We want to show that T simA. Clearly, we can directly assume (without loss of generality) that T and A are unitarily equivalent to their Jordan forms: let T = $j:l (Aj+Qj), where a:d = $j:l Hj and Qj is a Jordan nilpotent acting on the subspace H., 0
Assume that T, A m

16

E:

A~ e r= ml (A r +Rr ), where Rr is a Jordan nilpotent acting on Hr and satisfying the conditions

rank (Rr)s s rank (Qr)s, s = 1,2, •.• ,kr,r=l,2, ••• ,m. By Lemma 2.5, Or s!m Rr and, a fortiori, Ar+Or sim Ar+Rr for all r = 1,2, ••• ,m, whence it readily follows that m

m

T = ej=l (Aj+Qj) sim ej=l (Aj+Rj) ~ A. The proof of Theorem 2.1 is complete now.

0

REl-1ARK 2. 6. Since nul q (T) = d-rank q (T) , the conditions "rank q(A) s rank q(T) for all qlp" can be replaced by the conditions "nul q(A) ~ nul q (T) for all ql p". PROOF OF COROLLARIES 2.2 AND 2.3. It is easily seen that T has the form of Corollary 2.2(ii) if and only if S(T) is maximal; (v) => (i) is a trivial implication and (ii) => (v) follows from Theorem 2.1. Now Corollary 2.2 fol~ows from the well-known algebraic fact that (ii), (iii) and (iv) are equivalent. Similarly, S(T) is minimal if and only if k.= 1 for all j = 1,2, J d ..• ,m, if and only if T is similar to a normal operator acting on ~ ; (v) => (i) is a trivial implication and (ii) => (v) follows from Theorem 2.1. On the other hand, it easily follows from Theorem 2.1 and its proof that if k.= 1 for all j = 1,2, •.• ,m, then the minimal polynomial J of A € S(T) is necessarily equal top and that A-T, i.e., S(T) S(T) is a closed subset of L(~d). Conversely, if S(T) is closed and T s!m A, then A € S(T) = S(T) and therefore A - T. A fortiori, A ~ T, whence it readily follows s1m d that [AJ = [TJ = S(T) is a minimal element of (L(~ )/#,<). The proof of Corollary 2.3 is complete now. 0

2.1.3 The lattice (N(~d)/i,<) The partially ordered set (poset) (L(H)/i,<) is never a lattice (unless dim H = 0). However, it contains many interesting lattices. Co~ sider the case when H = ~d is a finite dimensional space and let n (T.) d T ej=l qj J € L(~ ) ; then rank Tj = 'j+l+2'j+ 2+ .•• +(n-j)Tn and this implies that . 1 . "+1 T. =rank TJ- - 2rank TJ +rank TJ :<: 0, for j = 1,2, ••• ,n. (2.2) J

Let Ed be the set of all (d+l)-tuples (m 0 ,m1 ,m 2 , ••• ,md) of non17

negative integers such that {

d = m0 > m1 ~ m2 mj-l- 2mj + mj+l

0, and (2. 3) ~

0 for all j = 1,2, ••. ,d-1.

Observe that these two conditions imply that mj > mj+l' unless mj = 0, and mj-l- mj
0

~ v ~·

It is easily seen that (I:d,s) is a poset, d = max{m 0 ,m

0} >

max{m 1 ,mil

2

•••

~

max{md,md}

0

and 2 max{mj,mj} s max{mj_ 1+mj+l'mj_ 1 +mj+ 1 } s max{mj_ 1 ,mj_ 1 }+max{mj+l'mj+l}, so that p v ~· e I:d. It is completely apparent that~ v ~· is the least upper bound (l.u.b.) of~ and u' with respect to the partial orders. Since I:d is finite, every subset of I:d has a l.u.b .. In particular, ~A~·

= l.u.b.{v e I:d:

v s

~and

v s

~·}

is the (unique) greatest lower bound of~ and~·· It readily follows that (I:d,s) is a finite lattiae with supremum (d,d-l,d-2, ... ,2,1,0) and infimum (d,O,o, ... ,o,o,o). Given~= (m0 ,m 1 , .•• ,md) e I:d' define T e N(~d) by d (Tj) ~ T~ = IBj=l qj where Tj = mj_ 1 -2mj+mj+l for j = 1,2, ... ,d-1 and Td =md_ 1 : (2.2) and (2.3) guarantee that the mapping ~ ~ [T~] (2.4) is a bijection from I:d onto N(~d)/# and, moreover, that rank T j =mj for j = 0,1,2, ••• ,d. Combining these observations with Theorem 2.1, we obtain THEOREM 2.7. The mapping (2.4) defines an order-preserving bijeation from (I:d,s) onto (N(~d.)/#,s). In partiaular, (N(~d)/#,s) is a finite lattiae with supremum [qd] and infimum [0] (0 = ql (d)) .

18

2.1.4 Closures of similarity orbits of finite rank operators Let H be an infinite dimensional Hilbert space and let T, A E F(H); then T and A are algebraic operators with nul T =nul A= oo and there exists a finite dimensional subspace H(T,A) reducing both, A and T, such that TIH(T,A)L = AIH(T,A)L = 0 (H(T,A) can always be defined so that dim H (T ,A) s 2 rank T + 2 rank A) • Assume that A E S(T)-; then we can prove exactly as in the finite dimensional case that rank q(A) :S rank q(T) and, by using Proposition 1.12(ii), that nul q(A) ~ nul q(T) for all qJp, where p is the minimal polynomial of T. Conversely, if A satisfies those conditions, then it is not difficult to check that AIH(T,A) satisfies the same conditions with respect to TIH(T,A) and therefore, by Theorem 2.1, TIH(T,A) sim AIH(T,A). A

fortiori, T = TIH(T,A)$0 simA= AIH(T,A)$0; hence, we have

COROLLARY 2.8. Let T E L(H) be a (necessarily algebraic) finite Pank operator with minimal polynomial p, then S (T)

= {A E L (H) : rank q (A) s rank q (T) and nul q (A) for all qlp}.

Let L

E

L(H);

then L#T if and onl-y if L

~

:?:

nul q (T)

T.

REMARK 2.9. Since H is infinite dimensional, the conditions "rank q(A) s rank q(T) and nul q(A) ~ nul q(T) for all qlp" of Corollary 2.8 still imply that cr(A) = cr(T). However, the following example shows that those two conditions cannot be replaced by "cr(A) = cr(T) and rank q(A) s rank q(T) for all qJp": Let T be an orthogonal projection of rank 2 and let A be an orthogonal projection of rank 1; then cr(A) = cr(T) = {0,1}, and rank q(A) :S rank q(T) for all qlp (p(;\) = J.(J.-1) ),but A cannot belong to s(T)~, because 1 =dim H(l;A) ~dim H(l;T) = 2. Reversing the roles of A and T, we see that conditions of Corollary 2.8 cannot be replaced by "cr(A) = 0 (T) and nul q(A) ;:, nul q(T) forall qlp: Let NF(H) = N(HinF(H) and NFk(H) Since H is infinite dimensional, NF 1 (H)

c

NF 2 (H)

c

...

c

NFk (H)

c

{T

E

NF(H):

NFk+l (11)

c

rank T

:S

k-1}.

...

is an infinite chain (all the inclusions are proper: no two sets in this chain coincide), and this chain naturally induces a chain of lattices

19

(NF 1 (H)/*,<) c (NF 2 (H)/lt,<) c ... c (NFk(H)/It,<) c (NFk+l(H)/#,<)c •. , where (as can be easily checked by using Theorem 2. 7) ( NFk (H)/#,<) is order-isomorphic with (Ek,s); moreover, the above chain is in correspondence with a chain of natural order-preserving inclusions between the lattices {(Ek,s)}~=l ( (E 1 ,s) c (E 2 ,s) c ••• c (Ek,s) c (Ek+l's) c ••• ) . Let (E,S) be the set union of the Ek's with the induced partial order, i.e., if~. ~· £ E, then~. ~· E Ek for all k ~ k 0 (~,~·) and we define~ s ~· in (E,s) if~ s ~· holds in (Ek ,s). 0

COROLLARY 2.10. (NF(H)/#,<) is a lattice with infimum [0] and no maximal element. This lattice is oPdeP-isomoPphic with (E,s). REMARK 2.11. However (as we shall see in Chapter VIII),

(NF(H)/#,

<) does have a supremum in (L(H)/#,<).

2.2 The distance from the set of all non-zero orthogonal projections to N(H)

2.2.1 The limit case

Let P(H) be the set of all non-zero orthogonal projections in L(H). What is the exact value of the distance n.. = inf{ll P - Q II: from P(H) to N(H)

(1-1

P

E

P (H) , Q

(2.5)

N (H) }

E

an infinite dimensional space)?

According to Example 1.5, liP-Oil> l:i for all Q in L(H) such that o(Q)n{)..: p. -11 : ; l:i} = [8. Since o(Q) = {0} for every nilpotent Q, the above inequality holds, in particular, for all Q in N(/-1). Hence, noo ~ l:i. It will be shown that n.. = l:i; more precisely, if nk

inf{IIP-OII:

P

E

P(/-1), Q

1\

= inf{IIP- Oil:

P

E

P(«:k), 0

Nk(H)}

E

and E

N(U:k) },

then it is completely apparent that l:i ::; noo ::; nk ::; ok is non-increasing with k. We have the following THEOREM 2.12. l:i < ok $ l:i +sin (m:l) I whel'e m integral pal't of (k-1)/2), for all k ~ 3. In particular , lim(k + oo) ok = lim(k + oo) nk 20

$

1 and nk (ok)

((k-1)/2]

nco = Js.

I= the

We shall need some auxiliary results. LEMMA 2.13. Let 1 s m < n < ~ and let m < r < n, s be such that r+s = m+n. If {ei}i~l ({fj}j! 1 J is an ONB of ~r (~s. :r>esp. ), r-1 s-1 T =
and

r-1 s-1 .,m-1 L = e exists a unitrs rs n111 ary ope:r>ator V € L(~ ~~ ) such that IIVTV* - L II = II T- V*LV II = s (m), where s (m) =2sin (m:l) , Vei = ei> for m+1 s i s r, and Vfj = fj, for m+l s j s s. PROOF. Define

gt

r+l-t'

m+l-r $ t s 0, t7T t7T (cos 2 (m+l)) em+1-t + (sin 2 (m+l)) ft' 1 s t s m, m+1 s t s s,

ft'

t1T t7T ) kt = (-sin 2 (m+ 1 )) em+ 1-t + (cos 2 (m+l) ft, 1 s t s m,

and s-1 m-1 R = is also an ONB of ~r~s, the pair {gt,kt} generates the same two-dimensional subspace as the pair {em+l-t'ft} (for each t such that 1 s t s m), T = qr$qs and L = qn~~ = R. Now a straightforward computation shows that L = VRV*, where V is the unitary operator defined by Vgt = {

em+ 1-t' ft'

m+l-r s t s 0 1 s t

s s

and em+1-t'

1

s

t

s m.

On the other hand, (T-R)gt

0,

(T-R) g 0

(1-cos 2 (~+1)) em- (sin 2 (~+l)) f 1 ,

(T-R)gt (T-R)gm

(cos 2 (~+1) - l) gt+l- (sin 2(~+1))kt+1' 1 s t s m-1, m7T (sin 2 (m+l) - l) fm+l

(T-R) gt

0,

(T-R)kt

(sin 2(~+l))gt+1+(cos 2 (~+1) - 1 l kt+1 I

(T-R) km

mTI (cos 2 (m+l)) fm+l •

and

m+1-r s t s -1,

m+1

$

t s s,

1 s t

$

m-1,

21

Since V{gt,kt} = V{em+l-t'ft} (1 s t s m; V { ••• } denotes the oZosed Zinear span of{ •.• }), these two-dimensional subspaces are pa: wise orthogonal and (T-R)gt L (T-R)kt for 1 s t s m-1, we conclude that . 2 'If 11 2 ~ II T-R II -- max{[ sl.n 2 (m+l) + (cos 2 (m+l) - 1) J ,

.

I}

Ia (s1.n 2 (m+l) - 1) +bees 2 (m+l) max lal2+lbl2=1 ( 11 1) 2 ~ 2 . 'If ( ) . 2 'If = [ Sl.n 2 (m+l) + cos 2 (m+l) J = Sl.n (m+l) = s m • mw

m1r

Since IIVTV*-LII = IIT-V*LVII = IIT-RII, we are done.

D

The above proof admits a very simple geometric description: Th1 action of q r (e r + e r- 1 + e r- 2 + ••• + e 2 + e 1 + 0) can be described by an arrow of length r, and the action of T = qreqs by two arrows o1 lengths r and s, as follows m

r s

Similarly, R can be indicated as m

n

T maps em+l to em' the pair {em+l-j'fj} onto the pair {em-j'fj+: of "parallel" vectors (j = 1,2, ••• ,m-l) and the pair {e 1 ,fm} onto {0, fm+l}. On the other hand, R maps em+l to (cosn/2(m+l))em+(sinn/2(m+l: £ 1 , which is equal to the result of a "slight twist" of the vector er in the two-dimensional space V{em,f 1 }, the pair {em+l-j'fj} onto a "slight twist" of the "parallel pair" {e .,f.+l} (j = 1,2, ••. ,m-l)aJ m-J J a "slight twist" (in the reverse sense) of fm to fm+l" The accumulation ofall these "slight twists" after m+l steps wi: map em to fm+l and f 1 to 0. This fact will be described by the follo' ing scheme m (T + R)

r

:;:::::><:::::::

-~

s

Conversely, if we take ({gt}t~+l-ru{kt}t~l) as the original ONl of Er&E8 ~ En~ and modify R to obtain T, then the corresponding scheme will be (R + T)

:::::::><::

n

m

In this case, the twist will move g 0 to 22

~

to 0 and k1 to -gm+l

In what follows, we shall freely use these schemes instead of ana lytic descriptions in the proofs. COROLLARY 2.14. If k ~ 3, 1 s m s ({k-1)/2], p = k-m and U = ,p-1 p * p-1 p . L j=l ej6ej+l + ep6el E L (a: ) • then up "' qp + ~qp ) "' vp. lJhePe vp t.S the unitary operator defined by V e.= (w )le. (lJhere -wp is a primip ] p ] k tive p-th root of lJ and there e:r:ists a unitary operator V E L (a: ) suah that !!VqkV*-qmEDUpll = s(m).

PROOF. It is (Observe that the determinant(X-Vp) Represent qk

straightforward to check that Up ::: qp+(qp*)p-l == Vp three operators are unitary and determinant(X-Up) = = xP- (-l)P.) by a "curled arrow" and modify the action of the op-

erator in the subspace

V{e 1 ,e 2 , ••• ,em,em+l'ek-m'ek-m+l'"""'ek} as in

Lenuna 2 • 13

(qk

-+

m

qmEDUP)

We conclude as in the above lemma that there exists an operator Lm,p"' qmEDUP such that l!qk-Lm,pll = s(m).

0

PROOF OF THEOREM 2.12. With the notation of Corollary 2.14: Let L(O:k) be the orthogonal projection of a:k onto the one-dimensional kernel of ((-1)-~EDUP), where m = [(k-1)/2] and p = k-m: then P e

liP- (-l:z)VqkV*I! sliP+ (l:z)qmEDUpil+ (l:zli!VqkV*- ~EDUPI! s l:z + l:z s (m) = l:z + sin (m:l) • 7T

Hence, cSk s l:z +sin (m+l) • It readily follows that cSk=inf{I!P-QI!:

P c P(O:k), Q

E

N(O:k) and !lUlls 3/2+sin (m:l)}.

(2.6) Since P(O:k) x {Q E N(O:k): I!QII s 3/2+sin (m:l)} is a compact sue_ set of L(O:k) ~ L(O:k), it is easily seen that the infimum in (2.6) is actually attained for some pair (Pmin'Gmin) in this set. Thus, by our previous remarks (Example 1.5), 0

2.2.2 on the exact values of cSk and nk

PROPOSITION 2.15. If 2 s h = dim H s oo,

th~n

23

inf{IIP- Oil:

p

P(H),

€

a

E

N2(H)} = /2/2.

Furthermore, the above infimum is aatually attained by the operators Pmin =

[~ ~)eoh_ 2

and Qmin =

c P(H)

(~ =~Jeoh_ 2

E

N2 (H),

where the 2 x 2 matriaes aat an a two-dimensional subspaae H2 of H and Oh_ 2 denotes the 0 operator aating on (H 2 )l. PROOF. It is straightforward to check that Pmin

E

P(H), Qmin

E

N2(H) and JIPmin-Qminll = /2/2. On the other hand, if Q e N2 (H) and P E P(H), then either 0 = 0 and II P - o II = II P II = 1, or o ~ o • Assume that JJP- Oil< /2/2. If x e ker Q and y l ker Q, then IIPxJJ IJ
II P -

Q II

?:

It readily follows that 62=n2= ue of 6k or nk (k ?: 2). Ifk 3,

[11 ']

p3 = 1/3 1 1 1 1 1 1

and

03 =

0

/2/2.

/2/2. This is the only known val

2/3[: 11] 0 1

2 = 2/3[q3+(q3)

]

,

0 0

then P 3 e P (a: 3 ) (P is a rank one projection), 0 3

N (a: 3 ) and II P 3

-o

311 is equal to 2/3. Thus, ~ ~ n 3 ~ 15 3 ~ 2/3. This result is certainly better than the poor estimate 15 3 ~ ~ + sin ~~ = 3/2 given by Theorem 2.12 and the same is true for the k-dimensional analogues pk = 1/k ~i~l~ '~1 ei&eJ. and ok = (2/k) [qk+(qk) 2 + 3 k-1 J . (qk) + ••• +(qk) J of P 3 and Q3 , respect1ve1y, for smaZZ values of k. E

However, there is some numerical evidence that the estimate ~+sin(m~ 1 ) is much better than II Pk -Qk II for all k

and

24

?:

20.

2.2.3 A companion problem:

The distance from the set of all non-zero

idempotents to N(H) Clearly, P(H) is not invariant under similarities. The smallest set invariant under similarities containing P(H) is the set of all non zero idempotents acting on H, E(H) = {WPW- 1 : P E P(H), WE G(HJ} = {E E LIHI: E 2 =E~O}. We shall see later that if R1 and R2 are two subsets of L(H) , where H is an infinite dimensional space, then R~nR; is "very large", in general; namely, when R: contains an operator R. ~ J ] such that Rj is not an algebraic element of A(H), for j = 1,2.(See Chapter of the second monograph.) Very little is known about dist[Rl'R 2 J = inf{

IIR 1-R2 11:

Rj

E

Rj' j = 1,2}

for the case when one of the sets R~ or R; is a subset of the algebra~ ic operators (or, more generally, operators whose images are algebraic elements of the Calkin algebra) . In particular, what is the value of n~ = dist[E(H) ,N(H)J for the case when H is infinite dimensional? What is the answer if H = ~k or NIH) is replaced by Nk(H) (and dim H = ~> for some k ~ 2? Clearly, if and

then 0

~

nk

s

nk = distr.EIH),Nk(H) J ok' for all k ~ 2.

(H infinite dimensional),

PROPOSITION 2.17. (i) ok = 1/k, but E(ah and all Q in N(~k) (k "' 2). (iiJ 0 ~ nk s 1/k and n~ = 0.

I!E-Qi!

>

1/k for a'Ll E in

k

PROOF. (i) Let {e.} .~ 1 be the canonical ONB of ~ and let Pk k ] ]- k k-j (1/k)}:. . 1 e.t9e. and Wk - L· 1 n e.t9e. (n = 1,2, ••• ). Straight~, J= ~ J ,n J= J J forward computations show that.Pk is a rank one projection, wk ,n is in vertible, and ~ (nj-i k -l) 0 k,n = l.ld<jsk ei@ej

E

N (~k).

Since li Ek,n- Q k,n II = II~Llsjsisk (nj-ik -l) ei e ej II <1/klll

s ( 1 /k)

k-1 k-i -r I Li=OLr=1 n er+ieer I

k {II Lr=1

~k-i ei+rt9er Ill< ereerII + (l/n) ~k-111 l.i=1 l.r=1 '' · 1/k+1/n,

25

we see that 15k_ s; 1/k. On the other hand, if F o (F) c { O,l} and

€

k trace F = Lj=l = rank F ?? 1,

so that ~ 1/k for (at least) one value of j, 1 ::; j ::; k. ] ] k k Given F as above and Q € N(~), we can find an ONB {fj}j=l so that Q is strictly upper triangular with respect to that basis, i.e., = 0 for 1 s j s i s k. In particular, we have IIF-OII

~max

I<(F-Q)f.,f.>l =max II?? 1/k, lsjsk J J lsjsk so that !IF-Oil~ 1/k, i.e., &k_ ~ 1/k. Furthermore, since = 0 for 1 s j s i s k, it easily fol~ J lows that, eitheriiF-QII > 1/k or = 0 for 1 s j = 1/k for all j = 1,2, •.• ,k: i.e., F admits an upper triangular matrix with 1/k in all the diagonal entries with respect to the ONB {fj}j! 1 • But the second possibility implies that o(F) = {1/k}, a contradiction (recall that k ~ 2). Hence, IIF- Qll > ~Sk_ = 1/k for all F in E(~k) and all Q in N(~k). Since the second statement is a trivial consequence of the first one, we are done. D Clearly, the trace argument cannot be applied in the case when H is infinite dimensional, but the above result suggests the following CONJECTURE 2.18. nk = E(H) and all Q in Nk(H) (k

cSk = 1/k, ~

but liE-Oil> 1/k for all E in

2).

The last result of this section says that the above conjecture is true at least for k = 2. PROPOSITION 2.19. If 2 s h

= dim

n2 = 152 = inf{IIE- Oil:

E

H s €

ro,

E(H),

then Q

€

N2 !Hl}

but this infimum cannot be attained foP any paiP (E,Q), E

~. €

E(H),Q

€

N2 (H).

PROOF. If E = 1, then i t is clear that liE- oil = 1 for every quasinilpotent Q. Let E =

Ill- oil ~ sp(l-Q)

(~ :):::nEE)L

be the matrix of E € E(H)\{1} with respect to the decomposition H ran ES(ran E)L and let

26

lc oJ

0 =

be the matrix of Q

€

L(H)

(with respect to the same decomposition). ~ 2 (H)

if and only if

= AB+BD

= CA+DC = 0.

It is immediate that Q e

= o 2 +CB II E- Oil :S ~:

A2 +BC Assume that

then Ill- All

~ and

:S

liD II

:S

~ and therefore

o(A) c {A: Jl-AI s ~}and o(D) c {A: !AI s ~}.Thus, by the spectral mapping theorem, o (A 2 ) o(-BC) c {A: :le A ~ ~}

and 0

(D 2 )

=

o(-CB)

c

{A:

Since o(-BC)\{0} = o(-CB)\{0}

I AI

~}.

:S

(see, e.g., [ll9], [153], [172])and

A is invertible (recall that Ill-All s ~ < 1 ) , it readily follows that o(-BC) = o(A 2 ) = {~} c o(-CB) = o(D 2 ) c {0,~} and o(A) = {~} c o(D) c {0,!1}. Hence, liE- oil ~

Ill-All ~ sp(l-A)

= ~-

Assume thati!E-QJJ =~.Since o(A) = {~} = oR.(A), there exists a sequence {xn}n:l of unit vectors in ran E such that JJ(A-~)xnll-+- 0 (n +

oo) • On the other hand, A2 = -BC and A invertible imply that

Ellxll

II Cx II ~

for some e: > 0 and for all x in ran E, so that JJE- Oil

:?.:

lim sup(n

-+- co)

IJ(E- Olxnll

~

lim sup(n

-+- co) {

i!(l-A)xn11 2 +11-Cxn11 2 l~

~ (~+£2)~

>

~.

a contradiction. Hence

II E - Q II

> ~

for all 0 in ~ 2 ( H) •

On the other hand, it readily follows from Proposition 2.17 that inf {II E - 0 II:

2.3 On the distance to

E

€

E (H) , Q

€

~ 2 (H) }

s

~.

0

~k(H)

2.3.1 A general upper bound LEMMA 2. 20. Suppose that T

€

II T II s 1 and II Tk II s E: for some (T*T)~ = JLO,l]A dE (speatral deaomp£ L (H),

k ~ 2 and some e:. 0 < E s 1. Let sition) and let P = E([O,Ie:J); then

11((1-P)T(l-P)]k-41 s (k-1)/e:. PROOF. Clearly,

II TP II

s le:, and 27

e:11xlr

~

11Tkxll2 =

II~k- 1 xl1 2

=

IIPIT*T)~Tk-lx+ll-P) IT*T)~Tk-lxll2

~II (1-P) (T*T)~Tk-lxll 2

=

II
~ e:II11-P)~-lxll 2 , for all x in H. Hence, 1111-P)Tk- 1 11 s le:. On the other hand,

(1-P)~- 1 = [ ll-P)T]k-l+ll-P)TPT[Il-P)T]k-J+Il-P)T 2 PT[ 11-P)T]k- 4 + ••• +11-P)T

k-2

PT,

so that 11[11-P)Til-P)Jk-11 s 11[(1-P)T]k- 1 11 s 11(1-P)Tk-11+ { 1111-P) TPT[ 11-P) T]k-~~ + 1111-P) T 2 PT[ 11-P) T]k-~~ + ••• II11-P)Tk- 2 PTIIl s

ll(l-P)~-1I+!k-2)IITPII s lk-1)/e:. 0 k

THEOREM 2.21. Tf T E L(Hl. IITII s 1 and liT II s e: foro some k and some e:, 0 < e: s 1, then

2

;>

I 2. 7) wheroe ~kle:) is a continuous, positive, non-decroeasing function defined on 10,1] such that lim(e: + 0) $k(e:) = 0. Moreovero, ~kle:) can be inductively defined by $ 2 le:) = l2e:)~ and $k(e:) = {e:+$k_ 1 1(k-l)/e:) 2 }~. fork= 3,4, •••• PROOF. Let P be defined as in Lemma 2.20 and let T = [E 1 T1]ran P E 2 T 2 ker P be the matrix of T with respect to the decomposition If k = 2, define T' = PT(1-P) =

H

ran P$ker P.

(~ ~l).

Since

(~ ~,.2 ) Iran P and ker P

1

=

0

ran P, it follows from Lemma 2.20 and its proof that

II T- T' II

=

1 (:~ ~J 1 {II(:~ ~JII \II(~ ~JI 2 r s

s (e:+£)

~ = ( ~.

~ 3 and (2.7) holds for j s k-1, with ~ 2 (e:) and 4>.(£) = {e:+~.-l((j-2)/£) 2 }\ for j = 3,4, •• ,k-l. Assume that k

J J Clearly, IIT 2 11

28

=

11(1-P)Til-P) II s 1 and, by Lemma 2.20, IIT 2

2 £)

= (2e:)~ k-L

11 s

(k-1)/E. Thus, by our inductive construction, we can find T2 (ran P)

f

Nk-l

such that II T 2 -T211 s lj)k-l ( (k-1) /E). Define

T' It

is easily seen that T'

€

Nk(H) and a formal repetition of our

previous argument (for the case when k = 2) shows that IIT-T'Il s lj)k(E)

{E+Ij)k-l((k-l)IE) 2 }~

(def)

4

D

k.

J ().i~ Aj if i ~ jJ be a potynomiat and tet Mp(H) = {A € L(H): p(A) = 0}. Given E > 0, there e:J:ists 6 > 0 suah that, if IITII s 1 and IIP(T) II s 6, then COROLLARY 2.22.

=

Let p().)

rrj:l (A-Aj)

dist[T,Mp(H) J < E. PROOF. I f m = 1, then the result follows from Theorem 2.21. As~

2. There exists E1 > 0 such that the m open disks D1 , o2 , ••• ,Dm of radius El centered at ). 1 , A2 , ••• , Am' respectively, are pairwise disjoint. Let E0 > 0 be such that i f IP(A) I < E0 , then A is sume that m

contained in one of these disks. Let 1 I = 2ni aD. J

PJ. We shall show that

().-T)-1 d)..

is a bounded function of IIPII, IITII, e 0

Pk

for small IIP(Tlll· Observe that p(A)- p(T) = a1 q

in

(A-T)q(A,T)

,

p(J.)

for a polynom!_

the variables A, T. Thus 1-p(T)p(A)-l = ().-T)q(A,T)p(A)-l.

Now, for A such that I A-Aj I = El we have that IIP(A) small i f IIP(T)II

-1

p(Tlll

is

is. Let E=p(A)-lp(T); then

().-T) - l = p(A) -lq().,T) (1-E) - l = where II J II s II p (A) -lq (A ,T>II ·II E

IV (1-JI E II>

p(A) -lq(A,T)+J,

is small since II p (A) -lq (A ,T)II is

bounded by a funtion of II T II and p (A) • Thus the P. 's are bounded ( j

J

2, ••• ,m) • On the other hand, k. 1 II
1 = ll2n1 Since each (T-).j)

k·

1

I aD.

P(A)- ().-).j)

I aD.

(A-Aj)

J

+

k.

().-).j)

k. JJ d). II

k.+l

J Jd).ll s IIJII·E 0

J

J

Jpj is small (provided IIP
use Theorem 2.21 to perturb TP. to aT~

J

J

is a nilpotent of order at most k .•

•

k. lq().,T)

= l,

J

in

L(ran P.) J

such that (TJ~-).J.)

•

I f H =ran P.+ker P., define T. =T~+O with respect to this decomJ J J J

29

position (j

= 1,2, ••• ,m)

= Lj~l

and T'

II T- T' II ~ max{ll P j 11·11 Tj-TI ran P jll: provided II p (T) II < c5 for some c5

>

Tj. Then p(T') = 0 and 1 ~ j ~ m} < e:,

0 small enough.

0

Let T, k and e: be as in Theorem 2.21 and let n. 0 < n ~ 1, be the k-th root of e: (i.e., 11~11 ~ nk) and define ljlk(n) = cpk(e:), k = 2,3, ••• ~ then Theorem 2.21 implies that, if k = 2, then dist[T,N 2 (H)J s l2n!kif k = 3, then dist[T,N 3 (H) J ~ (n3+4n3 / 2 )~, •• , and dist[T,Nk(H) 1 = O(e: 2 = 0
po~

CONJECTURE 2.23. There exists a continuous function ljl(n), defined on [0,1], such that ljl(n) > 0 on (0,1], lji(O) = 0 and dist[T,Nk(H)J s 11J(n) for all Tin L(H) such that IITII ~ 1 and IITkll ~ nk (i.e., the functions ljlk(n) can be replaced by a single one).

2.3.2 Two illustrative-examples The rough argument of the proof of Proposition 1.10 might suggest that those estimates are very poor. However, Lemma 2.13 shows that the "very poor" lower estimates given by Proposition 1.10 are actually the best possible except, perhaps, for a constant factor independent of k. Indeed, if T and L have the form of that lemma, then nul ~+l = 2 (m+l) >nul Lm+l=2m+l. Thus, if W (U) is an invertible (unitary, resp.) operator, we can always find a unit vector x = x(W) (= x(U), resp.) such that IILm+lxll = 1, but (WTW-l)m+lx = 0 ( (UTU*)m+lx = 0, resp.), whence it readily follows from Proposition l.lO(i) ((ii), resp.) that dist[L, S(T)] :?: 2l/(m+l)- 1 (dist[L,U(T) J :?: 1/(m+l), resp.). An even more surprising example can be constructed on the same lines. We shall need the following auxiliary result (With the notation of Lemma 2.13): COROLLARY 2.24. (i) Let {gn}-m
Let ·H+ = y{gn}n~O(H_ = V{gn}n
(ii) If k ~ 3 and 1 5 m 5 [ (k-1V2J, there exists a unitary W : (~m) (oo)EBH ~ (~k) (oo) such that k,m llq (oo) -W (q (oo)$U)N* II= s(m). k k,m m k,m PROOF.

(i)

For suitably defined A'

=A

and B'

= B,

oper~

we have

m

o~=m-1

'

(-oo,oo)

----------~--------==~==~~--.~~m

whence we obtain (-oo,m)

A'EilB':

( 0 ,oo) II!~EilU)- (A'$B'lll

= s(m) and the unitary operator Vm can be chosen so

that Vm(A'EilB')v; = AEilB, Vmgo=k 1 and Vmg_ 1 =-km. (ii) Since r = k-2m ~ 1, there exists U' = U such that (

qk

~ q

( 00)

m

( 00) $U I )

k

__......,:;:._::::--><~=

. ·><

m (~

fact that r

~

.

m

)

k

...X =:;;;:. .:::::J:T-kk:- -

~

- - -

m

1 guarantees that we can consistently apply the ar-

gument of Lemma 2.13 to each step.), with llqk(oo) -qm(oo)$U'II = s(m), whence the result follows.

0

EXAMPLE 2.25. Let s be a unilateral shift of multiplicity one and letT

E

L(H)

be unitarily equivalent to s(oo)$S*(oo); then IITkll = 1 for

all k ~ 1, cr(T) = crR.re(T) ={A:

IA.I

51} and

2l/k- 1 5 dist[T,Nk(H) J s 4 sin TT/([ (k-1)/2]+1) < BTT/k, for all k

~

3.

PROOF. It is not difficult to see that IITkll = 1 for all k

~ 1 and

that nul(>.-T) =nul(>.-T)*=oo for all A. in the open unit disk D ={A: 1>-1 < 1}, whence it follows that cr(T) =crR.re(T) =D-.

Let 0 c: Nk(H). Since Qk=O, it follows from Proposition l.lO(iii) that liT-Oil'??: 2l/k_l (k =·1,2,3, ••• ). On the other hand, i f k ~ 3, m=

31

[ (k-1)/2] and the operators are defined with respect to a suitable ONB of H, it is not difficult to infer from Corollary 2.24(i) and (ii) that ( (qk (oo}) (oo) "' qk (oo)): ll
II qk (co)

-

(~am•) (oo)ll +ll(~eu•) (oo) -Til

(oo) -

~ (oo) eu'll +ll~eu• - ses* II = 2s (m)

= 4{sin 'IT/([ (k-1)/2]+1)} < B'IT/k. Our next example shows that the infinite dimensional ampliations of finite dimensional examples can produce certain surprises. (An ator T in L(H) is called an ampliation if T = A81 = A("") for some ator A € L(Ho) I 1 s dim Ho s oo.) EXAMPLE 2.26. Let k > h ~ 3: then the operators A qk (h) (A, B € L(~kh)) satisfy dist[B,S(A)]

1:

qh (k) and B = (2.8)

however, 21/k-1 s dist[B(ool,S(A(oo))] s 4{sin 'IT/([(k-1)/2]+1) (2.9) +sin 'IT/([(h-1)/2]+1)}

Bn(l/k+l/h).

<

PROOF. Observe that nul A= k >nul B =h. Thus, dist[B,S(A)] ~ 1. (Use Proposition l.lO(i) as in our previous observations at the begi~ ning of this section.) On the other hand, A~ EA for all E > 0, so that 0 € S(A)-, whence we obtain (2.8). The lower estimate of (2.9) follows from Proposition l.lO(iii) ([A(oo)]h=O, II[B(oo)Jhll = IIB(oo) II= 1.) The upper estimate follows from the proof of Example 2.25: For suitably chosen·T = s<"">es*(oo), A' =A and B' = B, we have IIB'-A'II s IIB'-Tit+JIT-A'II s 4{sin 'IT/([(k-1)/2]+1)

+sin 'IT/([(h-l)/2J+l)} < B'IT(l/k+l/h).

D

Example 2.26 suggests that, if H is infinite dimensional, then 1 N (H) = {Q € N(H): !lOll s 1} "looks like" IU+iQI (where !1J denotes the set of all rational numbers), in the following sense: Observe that Gl+iGI = uk:l (Qik +i!IJk) , where !Ilk = {m/n: "large" nowhere dense subset of !IJ+i
A

m € 2Z , 1 s n s k}, !Ilk +iQlk is a

€

Gl+ill} < 1/(k/21+ 0 (k + oo)

and (GI+iGI) (in the complex plane) coincides with the much larger set 1 1 E, and N (H) = uk=l [N (H)nNk(H)J. 00

32

CONJECTURE 2.27. max{dist[Q,Nl(H)nNk(H)J: some constant C > 0 independent of k.

0

E

Nl(H)} ~ C/k for

An affirmative answer to this conjecture would provide some heuristic explanation to the wild structure of N1 (H) - (see Chapter V).

2.3.3 An example on approximation of normal operators by nilpotents > SO, p = [,/k/2hrJ, n

aj =

p

-1

and r = [kn/2] and let Qk E k-1 L(~k) be the operator defined by Qk = Lj=l aj ej+ll8ej with respect to the ONB {ej}j~l of ~k, where Let k

l

nn, for r (n-1) < j s rn, n = 1, 2, ••• ,p, nn, for r(2p-n) < j s r(2p-n+l), n = 1,2, .•• ,p, 0, for 2rp < j ~ k-1.

(Roughly speaking: Qk is a truncated weighted shift; the weights aj grow from n to 1 through p steps of length r and then go down from 1 to 0 through p steps of length r, so that the upper step has length 2r with weights equal to 1, i.e., a. = 1 for r(p-1) < j ~ r(p+l) .) . . . . J . .. . In the f~rst mod~f~cat~on, we shall "~gnore" the coord~nates 1,2, ... ,r(p-1) and r(p+l)+l,r(p+l)+2, ••• ,k and apply Corollary 2.14 to the subspace V{er(p-l)+l'er(p-l)+ 2 , ••• ,er(p+l) }. It is easily seen that we can modify Qk in order to obtain an operator Ri = Tieur+l' where U +l ~ e 10e +l+L.: 1 e.+ 10e. is a unitary operator acting on a subspace r r JJ J 1 1 1 of dimension r+l and there exists an orthonormal system {f 1 ,f 2 , ••• ,f2t r (p-1) 1 2r k . R , _ such that {ej}j=l u{fj}j=lu{ej}j=r(p+l)+l ~san ONB of a:, T1ej-Qk ej for j i 1

1

(r(p-l),r(p+l)J, Tier(p-l) =fi, _

1_

Tif~=f~+l for h = 1,2, ••• , h 1

r-2, T1 fr_ 1 - (1-n)er(p+l)+l and Ur+lfh- (wr+l) fh for h = r,r+l, ••• ,2r (-wr+l is a primitive (r+l)-th root of 1); furthermore, IIOk-Rill = s(r-1). Let T1 be the operator obtained from Ti by replacing each weight equal to 1 by 1-n and R1 = T1eur+l; then II Qk- R111 ~ s (r-1) +n. Now we can apply the same argument to R1 in order to obtain an OE erator R2 = T2e (l-n)u 2 reur+l, where u 2 r is a unitary operator acting on a subspace of dimension 2r, whose eigenvalues are equal to minusthe 2r 2r-th roots of 1, T 2ej = ~ej for all j I. (r(p-2) ,r(p+2) ], T2er(p- 2 ) = 2

2

2

2

(l-2n)f 1 , T2 fh = (l-2n)fh+l for h = 1,2, •••,r-2, T2 fr-l = (l-2n)er(p+ 2 )+l'

,£;_

{fi,f~, ••• 1 } is an orthonormal system that spans a subspace orthor u{ej}j=r(p+ k gonal to the span of the vector ({ej}j=l 2 )+1), II R1 - R2 II ~ (1-n)[s(r-l)+n]and this second modification only affects the vectors 33

· th e su b space spanned by (e.}. { r (p-1) r (p+2) 1n J J=r ( p- 2 ) +1 u{e.}. J J=r ( p+ l) +1 ), so that IIOk- R2 JJ = max{JJok- R1 JJ,JJR 1 - R2 11} s s(r-l)+n, etc. An easy inductive argument shows that after p-1 steps we finally obtain an operator L = u mre:£?-1 (l-· )U 1...,.. (k+l-(2p-l)r) k r+l J=l Jn 2r ~~1 such that JJok-~11 s s(r-ll+n < 2n/r+[lk/2lnJ < 5(n/k)~ for all k >SO. On the other hand, if 1 s k s 50, then 5(1T/k)~ > 1. Thus, we have the following PROPOSITION 2.28. (i)

FoP eaah k

~

1 thePe exists a noPmaZ

opeP~

toP~ E L((tk) suah that JJLkJI = 1 and dist[~,N(a:k)] < 5(1T/k)~. (ii) If H is infinite dimensional, thePe exists a noPmaZ opePatoP M suah that cr(M) = D-, !Uhepe D = {;\: !AI< 1} and distrM,Nk(H)J < 5(1T/k)~ fop all k = 1,2, •••. In partiauZar, ME N(H)-.

PROOF. (i) If k > 50, define ~ as above. If 1 s k s SO, take Lk = 1, Qk=O. (ii) Let {Am}m:l be an enumeration of all those points A in Osuch that both lAml and (arg Am)/1T are rational numbers (arg 0 is defined equal to 0) and let M be a diagonal normal operator with eigenvalues A1 ,A 2 , ••• ,A , ••• such that nul(A -M) =cofor all m = 1,2, ••• , i. . m (co) m e., M = (d1ag{A 1 ,A 2 , ••• ,Am, •.. }) Given k, it is easytoseethatMcanbe written as M" (EDm:l Am Lk) (co), whence it readily follows that dist[M, Nk (H) J A

fortiori, ME N(H)-.

0

The result of Proposition 2.28(i) is, in a certain sense, the best possible. Observe that if Nk E L(a:k) is normal and there exist k Qk € N(a: ) and e:k > 0 such that JJNk- QkJJ < e:k' then (by Corollary 1.6 (i)) cr(Nk) is a connected set containing the origin. If the points e:k of cr(Nk) are more or less evenly distributed in a connected neighborhood n of the origin with smooth boundary (namely, n = D), then 2 cr(Nk) will include n and therefore k1re:k ~ m2 (n), where m2 denotes e:k the planar Lebesgue measure. Hence, e:k ~ [m 2 (0)/(1Tk)J~ = O(k-~). cannot be connect (On the other hand, if e:k is too small, then cr(Nk) e:k ed, a contradiction.) CONJECTURE 2.29. There exists a constant C > 0 (independent of k) 34

such that dist[N,N(~k)J ~ Ck-~ for every normal operator N such that II Nil= 1 (k = 1,2, ••. ) .

E

L(~k)

The following result provides some extra support to this conjecture. Observe that if A E L(~k) is hermitian and 0 ~ A ~ 1, then the points of a(A) are not evenly distributed in any set of positive measure. (More precisely, m2 (a(A)£) ~ 2c+~c 2 independently of k, and 2c+ 2 .,. 0, as £ ..,. 0.) nE k

PROPOSITION 2.30. If A € l(~ ), 0 dist[A,N(~k)] > (1/2/k), k = 1,2, ••••

~A~

1, and 1

E

a(A), then

PROOF. Assume that IIA-QII ~£for some Q E N(~k), Q = H+iJ (Cartesian decomposition); then IIA-HII = IIJR.e(A-Q) II~ IIA-QII ~£and trace (H) = trace (llle Q) = Ie trace (Q) llle 0 = 0. On the other hand, it is easily seen that a(A) ~ ~O,l]{Use Carol £ lary 1.6(i)), so that trace {A) ~ l+{l-2c)+{l-4c)+ ••• +{l-2nc), where n = [l/2c](= integral part of {l/2c)). It is clear that l/2c ~ n > l/2c-l. Hence, {n+l) > l/2c and n-1 trace {A) ~ {n+l) - 2c}:j=O j = (n+l)- {n+l) nc = {n+l) {1-nc) > l/4c. Let A f A dE and H = J A dF (spectral decompositions) • If a E [0,1], c'> c and rank F{{a-c',aa)) l > a-(a-c) =c, a contradiction. Hence, rank F{[a-c,aa)) ~rank E{[a,aa)) and, by symmetry, rank F([a,ao)) ~rank E{[a-c,ao))' for all a E [0,1]. It folows that 0 = trace (H) > trace {A) - kc > 1/ 4c - kc. Hence, c > 1/2/k. By a compactness argUMent (exactly as in the proof of Theorem 2.12), we conclude that dist[A,N{a:k)J = min{IIA-OII=

Qk =0,

11011 ~

2} > 1/2/k.o

2.3.4 On the distance to a similarity orbit LetT E L{a:d) be a ayalia operator with minimal polynomial p, p(A) = n.m1 {A-A·)kj {A. -1- A·, i f i -1- j); then }:.m1 k. =d and Tis sim J=

J

l.

J

J=

J

ilar to the Jordan form ej:1 {Aj+qk·). Let A E L{a:d) be an operator wfth spectrum a(A) = {a 1 ,a 2 , ••• ,an} 35

and dim H(a.;A) =h .• (Clearly, }:.n 1 h. =d.) Define JJ 1 =JJ 2 = •.• =JJk =A 1, J

l.

J=

1

l.

JJk +l=JJk +2= ... =JJk +k =:\2' ••• 'JJd-k +l=JJd-k +2= ... =JJd=A and 1 1 12 m m m 11 1 =1!2= ••• = 11 h = al' 11 h +1 =l!h +2= ••• = 11 h +h =a2, ••• ' 11 d-h +1 1 1 1 1 2 n = Sd-h +2 = Bd-h +3 = • • • = Bd = an; then A admits a representation as . n

n

an upper triangular matrix of the form

A

0

Bd (with respect to a suitable ONB of ~d). It is not difficult to conclude from Theorem 2.1 that

T sim B = 0 )Jd

Moreover, the same result applies to any upper triangular representation of A. Hence, we have COROLLARY 2.31. Let A and T be as above; then dist[A,S (T)] ~ min

max

aEE(k) whe~e

E(k)

l~j~k

I JJ. - S (.) J

0

J

1.

(2.11)

denotes the set of aZZ permutations of k elements.

Unfortunately, the estimate (2.11) is very poor, in general. Name ly, if~ and Qk have the form of Proposition 2.28(i), then qk is cyclic, qk sim Qk (by Theorem 2.1) and

dist[~,S(qk)] ~ IILk-Qkll

< 5(1T/k)!oz

+

0 (k

+co).

However, sp (Lk) = 1 (-1 E a (Lk)) and a (qk) = {0}, so that the only information that we can obtain from Corollary 2.31 is that dist[Lk' S(qk)J s 1. PROBLEM 2.32. Find a formula for 36

dist[A,S(T)] (A, T

E

l(~d)).

We shall close this section with a partial answer to this problem. n

n

hk)

(o;k)

COROLLARY 2.33. If T=EDk=l ~k and A=EDk=l qk al'e finite rank opel'atol's, rank TJ =rank AJ fol' j = 1,2, ••• ,r and rank Tr+l < rank Ar+l fol' some r ;:.: 2, then 21/(r+l) -1

$

dist[A,S(T) J s 2 s([{r-1)/2]}.

PROOF. The lower estimate follows from Proposition l.lO(i). In order to obtain the upper estimate, we can directly assume without loss of generality that A, T E L(~d) (for some d, 0 < d < ~>. Then, our hypotheses and formula (2.2) imply that T· = o;. for j = 1,2, J J ... , r-1, but 'r < o:r· Since r > 1, this means, in particular, that T and A have exactly the same number of direct summands, which is equal toT=

Lj~l

'j"

After eliminating all common direct summands, we can directly assume (without loss of generality) that T =ED n q (Tk) and A=q (o:r> n {
q ED ( T)) n m

)

> r~ m

r

:'X

:X

r..._..... m

\

lrr

)

r

:X ~>r

:X: >r'-v-'

>

m

m blocks of length r

n" - - - - _ _ _ _ __,).,.........::.:...__:'O<"X--:::r·

'"V"""

~

n'

m

(where r < n" s n' s n and n' is the largest Jordan block of A), whence we obtain dist[A,U(qnED~(T))J s s(m). Similarly, we have dist[T,U(qnED~(T))J s s(m). Combining these two inequalities, we obtain dist[A,S(T)] s dist[A,U(T)] s dist[A,U(qnED~(T))] + dist[T,U(qnED~(T))J s 2 s(m).

0

2.4 on the distance from a compact operator to N(H)

37

j k-1 j j ••• ,ek can be chosen in such a way that llfj~fj- Jz}:r=l er~er+~l < lf + 2n/k. Define Gl Al2 Al3 G2 A23 G3 G 0

Gd k-1 where Gj = (;\/2) Lr=l

e;~e:+l

IIF&lO((k-l)d) -Gil

and Aij = aijfj_8fj; then

max{ll Gj- ;\jfj~fjll: < max{l;\jl/2:

1

~

j

1 ~ j ~ d} ~

d}+27r/k=sp(F)/2+27r/k.

Let K € K(HJ (H infinite dimensional) and let e: > 0 be given. Then there exists a finite rank operator F such that II K- F II < e:/3. £ £ Moreover, by the upper semicontinuity of the spectrum (Corollary 1.2),' F can be chosen so that sp(F ) < sp(K)+e:/3. £ £ Since F e: F I H) , there exists a finite dimensional subspace H of £ £ H, dim H£ = d ~ 1, such that H£ reduces F£ and F £ IHi£ = 0. Let F = F I"H , let M be a subspace of dimension kd containing H for some k £ £ £ £ large enough to guarantee that 2n/k < e:/3 and define G e: L (M ) as above £ and G e l(H) in such a way that G IM = G and G IMi = 0. £ kd £ £ £ £ Then G e F(H), G = 0 and £ e: IlK-Gil !> IIK-Fe:II+IIF'e:-Ge:ll < e:/3-+JIF-GII < e:/3+~ sp(F)+e:/3 < J;z sp(K)+e:.

Since e: can be chosen arbitrarily small, we obtain the following upper bound: PROPOSITION 2.34. If K e K(H) (H an infinite dimensionaZ space), then the distance from K to the set of aZZ finite rank nilpotent oper~ tors cannot e~ceed ~ sp(K). In partiauZar, every compact quasinilpotent operator can be uniformZy approximated by finite rank niZpotents.

2.5 Notes and remarks The problem of characterizing the closure of a similarity orbit in simple terms was raised by D. A. Herrero in [139]. This reference contains all the basic properties of the sets S(a) (for a in a Banach algebra A), the notion of asymptotic similarity, several properties of 38

the poset (A/i,<), the analysis of several relevant examples, etc. The results include the proof of the implications (ii) => (iv) => (i) of Corollary 2.3 (in the above mentioned more general setting [139,Proposi tion 1]) • Theorem 2.1 and Corollary 2.8 are due to J. Barrra and D. A. Herrero [43,Theorem 1.1], who also proved that (F(H)nN(H)/#,<) is a lattice (0 ~ dim H ~ oo). The concrete model (Ed,~) for this lattice (Theorem 2.7) is an unpublished result of c. Apostol and D. Voiculescu[34]. In [129], J. H. Hedlund tried to detetllline the eKaot value of 11 00 = dist[P(H),N(H)] and showed that 1/4 ~ 11 00 ~ n 2 =o 2 =12/2 (Proposition 2.15) . He also analyzed the norm of the difference Pn - Qn for several values of n ~ 100 and suggested that these operators could be used to prove that ok + ~ (k + oo).(Personal communication to the author.) The solution of the problem evolved as follows: 1) 11 00 =~ (D. A. Herrero [132,Corollary 9], by using infinite dimensional arguments; 2) ok does converge to~. ask+ oo (D. A. Herrero [133,Section 7(d)], modif~ ing the previous argument); 3) ok ~ ~+[l+(k-l)~J/2k ~ ~+1/2/k (N. Sal! nas, [18l,Lemma 3.3], by using an argument due toP. R. Halmos [125]): 4) ok >~and T)k < ~+(8 log k)/k (D. A. Herrero, [149,Proposition 6.5]) and 5) ok < ~+sin 1T/([ (k-1)/2]+1) (D. A. Herrero, [150,Corollary 5.2]). P. R. Halmos and L. J. Wallen called an operata~ T in L(H) a pown• parotiaZ isometroy if Tk is a partial isometry for all k <: 1 and pro~ ed that T has this property if and only if T ~ {ek:l qk(Tk)}es
<

10/log k,

for all k large enough. The results of Section 2.2.3 are unpublished observations of D. 39

A. Herrero (However, the fact that dist[ E(a:k) , N(a:k) J = 1/k was also i!!l dependently proved by B. Aupetit and J. Zemanek in [200,Example 2.4]), It is worth to remark that the proof of Proposition 2.17(i) actually yields the following stronger result: PROPOSITION 2.35. inf{I!T-QII:

T

but this infimum is

t:

Fo~

each k

~

2

L(O:k), a(T) ={0,1}, Q

neve~

t:

N(~)}

1/k~

attained.

The results of Section 2.3.1 are due to s. L. Campbell and R. Gellar [64]. That every compact quasinilpotent operator is the norm limit of finite rank nilpotents is an observation of R. G. Douglas [123,Problem. 7] and the improvement given here (Proposition 2.34) is due to D. A. Herrero [149,Proposition 6.6].

40

3 The main tools of approximation

In this chapter and the following one, we shall analyze the main "tools" of approximation. Some of these "tools" developed within the framework of Operator Theory: The Rosenblum equation AX- XB = C, Rota's universal model and its generalizations, the triangular representation of C. Apostol and the theorem of C. Apostol, C. Foia~ and D. Voiculescu on normal restrictions of small compact perturbations of operators. These, as well as some of their consequences, will be developed here. Two other important results (the Brown-Douglas-Fillmore theorem and Voiculescu' s "non-commutative Weyl-von Neumann theorem") , from the theory of C*-algebras, will be analyzed in Chapter IV.

3.1 The Rosenblum operator:

X

+

AX- XB

3.1.1 Linear operator equations Let A be a Banach algebra with identity. The mapping a + La, where La" L(AJ is defined by La(b) = ab (be A) is called the Pegutar Zeft ~ep~sentation of A. It is well-known that this mapping is an isometric isomorphism from A into L(A). Similarly, the regular right representation (defined by a + Ra " L(A), where Ra (b) = ba) is an isometric anti-isomorphism (i.e., ab + R R ) • In particular, o (L a ) = o (Ra ) = o (a) for all a in A. -b a Given a, be A, Tab" L(A) is defined by Tab(c) =ac-cb, i.e., r ab =La - ~. It is easily seen that La and ~ commute and that

()., ll "

a:) ,

so that

LEMMA 3.1. Let X be a aomplex Banaah spaae and ZetA, B"E: L(X) be

41

two operator>s such that AB = BA; then o(A+B)

a" o(A), B" o(B)}

o(A) +o(B) ={a+B:

c

and o(AB)

o(A)o(B) = {aB:

c

a " o(A), B " o(B)}.

PROOF. Let B c L(X) be a maximal abelian algebra containing A and B. Clearly, 1 f Band (A-A)-l ((~-B)-l) belongs to B for all A 1 o(A) (~I a(B), resp.). Let M8 be the set of all multiplicative linear functionals on B; then the Gelfand theory implies that o(A+B) = {cp(A+B):

cp

M8 J

E

= {cp(A)+cp(B):

cp " M8 J cr(A) +o(B)

and o(AB) = {cp(AB): c{cj>(A):

cp cp

COROLLARY 3. 2.

M8 J = {cp(A)cp(B):

E

E

M8 }.{cj>(B):

o (-r ab)

cp

E

cp " M8 }

M8 } = o(A).o(B).

0

cr (a) - cr (b) .

c

PROOF. Since La~ = ~La' it follows from Lemma 3.1 that a (La-~) c a (La) -a(~). On the other hand, a (La) = a (a) and cr(b), whence the result follows. We shall see later (Corollary 3.20) that when A sion is actually an equality.

L(H)

this inclu

3.1.2 Approximate point spectrum of a sum of commuting operators The appr>oximate point spectrum of A OTT(A)

~:

{A

E

{A

E ~:

E

L(X) is the set

A-A is not bounded below}

and

a 6 (A)

=

A-A is not onto}

is the approximate defect spectrum of A. It is completely apparent that op(A) c oTT(A) c oi(A) and a 6 (A) c or(A). Furthermore, if At denotes the Banach space adjoint of A, then it is not difficult to see that crTT(A) =·cr 6 (At) and o 6 (A)

=

crTT(At).

(If X is a Hilbert space, on(A) = oi(A) = crr(A*)* and a 6 (A) oi(A*)*, where O* = {~: A f Q} for each Q c ~.)

42

LE~~

3.3. Given any Banach space X, there is an isometric imbedding of X into a larger Banach space X', and a mapping A+ A' of L(X) int9 L(X') which is an isometric isomorphism such that every A' is an extension of A and a (A') =a (A') =a (A) • p 7r 7r PROOF. Let ~ 00 (X) be the Banach space of all bounded sequences of elements of X with the norm ll{x } : 1 11 = sup llx llx and let e (X) "' n nn n o {{xn} E ~ (X): II xn II + 0 (n + "')}. It is easily seen that e 0 (X) is a subspace of ~"'(X). We define X'= ~"'(X)/e (X) and the imbedding of X into X' by x + 0 [{xn}J (=the coset of {xn}), where xn = x for all n = 1,2, . . . • Clear ly, this mapping is an isometric isomorphism of X into X'. Similarly, given A in L(X), we define A' E L(X') by A'[{xn}J = [{Axn}]; then A+ A' defines an isometric isomorphism from L(X) into L(X') •

If A ~ a7r(A), then there exists a sequence {xn}n:l of unit vectors in X such that II 0 and for all x in X, then it can be easily seen that II
1T

1T

PROOF. (i) Let A E a (A+B); then, by Lemma 3.3, A € a ((A+B) ') 7r p cr (A'+B') and A'B' =B'A'. It readily follows that the subspace p M = ker(A'+B'-A) is invariant under A' and B', and B' IM = (A-A') IM. Thus, if a E a (A' IMl then A-a E a (B 1 1Ml and therefore 1T

A= a+(A-a)

E

a (A'IMl+a (B'IMl

1T

c

a (A')+a (B') =a (A)+cr (B).

1T 1T 1T 1T 7r 7r (ii) Take A € a (AB); then N = ker(A'B'-A) is invariant under A' 7r and B'. If A= 0 then a (A') or a (B') contains 0, so 0 € a (A)a (B). 1T 1T -1 7r 1T If A~ 0, then A' IN has the two sided inverse A B' IN. Choose a in 3cr(A'INl; then A/a E 3a(B'INl. But this implies that a € a7r(A' INl c a11 (A'), A/a E a (B' IN> c a (B'), giving in turn 1T 7r A = cdA/a) E a (A') .a (B') =a (A) .a (B). D 1T 1T 'If 7r

The duality between approximate point spectrum and approximate de feet spectrum mentioned above, gives the following 43

COROLLARY 3.5. If A, B € L(X) and AB (iJ a 5 (A+B) c cr 0 (A)+cr 0 (B). ( i i J a 0 (AB) c a 0 (A) • a 0 (B) •

BA, then

LEMMA 3.6. Let H be a Hitbert spaae and tet A (i) crn(LA) = cr 0 (RA) = at(A). (iiJ cr 0 (LA) = crn(RA) = ar(A).

E

L(H); then

PROOF. (i) Let A E a (A) =a. (A) 1 then there exists a sequence n ~ {xn}n=l of unit vectors such that 110.-A)xnll + 0 (n + ~>. It follows that II<X-LA) (xnexn) II= II (X-A)xn exnll = II<X-A)xnll + 0, so that A-LA cannot be bounded below. Hence,crt(A) c an(LA). Conversely, if A € an(LA), then there exists a sequence {Xn}n:l in L(H) such that 11Xn11 = 1 for all n = 1,2, ••• , and II<X-A)Xnll+ 0 (n _,. oo). For each n, choose yn E H such that 1 s 11Yn11 s 2 and 11Xnyn11 = llxnll = 1; then II<X-A) (Xn y n ) II + 0 (n + ~>, whence we conclude that A € a n (A). . It follows that at(A) = crn(LA). If X-RA is onto, then there exists B in L (H) such that (X-RA)B = B(X-A) = 1, so that XL cri(A). Hence, cr 0 (RA) ~ crt(A). Conversely, if X-A is left invertible and C(X-A) = 1, then (A-RA) ~

RC = R(X-A)RC = RC(X-A) = R1 = 1, i.e., ).-RA is right invertible, whence we conclude that a 0 (RA) c or (RA) c at (A) • Hence, a 0 (RA) =a R. (A) • The second statement follows by the same arguments. 0 From Theorem 3.4, Corollary 3.5 and Lemma 3.6, we immediately obtain the following COROLLARY 3.7. Let A, B € L(H); then (i) cr1T(TAB) c crt(A) -crr(B). (iiJ cr 0 (TAB) c crr(A) -crt(B). We shall see later (Section 3.1.4) that these inclusions are actu ally equalities.

3.1.3 Local one-side

resolvents in L(H)

LetT € L(H) and let n be an open subset of pr(T). A continuous function F:Q + L(HI satisfying the equations {;1.-T)F(A.) - 1 (A.

E

Q)

and F(A.)- F(Jl) = (Jl-A.)F(A.)F(Jl)

(A.,Jl

€

n)

will be called a right resotvent for T (defined on Q). It is completely apparent that such a function F is an analytic function defined on 44

derivative F' 0..) = -F 2 (A). Similarly, if t) is an open subset of p R. (T) and G: t) ... L (H) is a co~ tinuous function satisfying G(A) (A-T) = 1 (A E t)) and G(A)-G(~) = (~-A) G(A)G(~) (A,~ € t)), then G is called a left resolvent forT (defined on t1 clearly, G is analytic on t)). Moreover, if F(A) is a right resolvent forT* (defined on n t*), then G(A) = F(~)* is a left resolvent for T (defined on t)) • Q with

For any A {

€

~

pr(T),

E

pR.(T), we shall put

Rr(A,T) = (A-T)*[{A-T)(A-T)*J-l, RR.

(~

,T)

= [ (1J-T) * (~-T)] -1 (~-T) *.

(3 .1)

It is easily seen that Rr(.,T):pr(T) ... L(H) (RR.(.,T):p 1 (T) ... L(H)) is continuous on this domain and satisfies the equation (A-T)R (A,T) = r 1, >. E pr(T) (R 1 (~,T)(1J-T) = 1, ~ E p 1 (T), resp.). However, Rr(.,T) (R (.,T)) does not satisfy, in general, the resolvent equation, i.e., it is not a right (left, resp.) resolvent for T. THEOREM 3.8. LetT E L(H). Given E > 0, there e~ists a right resolvent for T defined on pF (T) npr (T) e~cept for an at most denumerable set S c pr(T) which does not accumulate in pr(T), such that

s

c

cap r (T)J £ ={A

€

cr:

distn,ap r (T)J s £}.

Applying the above theorem toT*, we obtain the following dual result. COROLLARY 3.9. LetT E L(H). Given E > 0, there e~ists a left re~ oZvent forT defined on pF(T)npR.(T) e~cept for an at most denumerable setS' c pR. (T) which does not accumulate in pR.(T), such that S' is included in [
LEMMA 3.10. Let T such that 0 closed.

€

€

L(Hl and Zet M be an invariant subspace ofT

Pre<TIM). Then ran TMi is closed if and only if ranT is

PROOF. Since 0 € Pre<TIM>, Pker(TIM>* will be a finite rank projection. Thus, if we write 45

N = {z < M.l:

Pker(TJM)*PMTz=O}

we haveN c M.l, dim (M.leN) <~and, consequently, ranT is closed if and only if T(M+N) is closed in H. This reduces our problem to showing that T(M+N) is closed

<=> 0

Observe that Pker(TJM)*PMTN we obtain that T(M+N)

TM+PM.1TN

c

On the other hand, given y

€

TM.1N

c

is closed.

implies that PMTM

c

TM, whence

TMM+TM.1N = TM+TM.1N.

M and z c N, we can choose y 0

€

M such

that TMyo = -PMTz, whence we obtain TMy+TM z = T(y+y 0 +z) € T(M+N). Henc~ we have the equality T(M+N) = TM+TM.1N, where the sum in the right side is orthogonal and TM is closed. It readily follows that T(M+N) is closed if and only if TM.1N is closed. 0 COROLLARY 3.11. LetT € L(H) be an operator with closed range and Zet M be an invariant subspace of T such that (TM) M and 0 belongs to pR.e(TM ). Then 0

€

pr(TM).

PROOF. Since T* has closed range and 0 € Pre(TM.L*), it follows from Lemma 3.10 that TM* has closed range. Moreover, since TM hasdense range, TM* must be injective, so that 0 € pR.(TM*); equivalently, 0 € Pr(TM). LEMMA 3.12. Let T

0 €

L(H) be an operator with closed range and Zet

M be an invariant subspace ofT such that ker T c M and 0 E pr(TM). Then 0 € pR. (TM.1l and the set {A € a:: ker(A-T) c M} is a neighborhood of the origin.

PROOF. Since ker T c M and TM is right invertible, 0 cannot be an eigenvalue of TM.l· Using Lemma 3.10 we see that ran TM.1 is closed and therefore 0 € pR.(TM.L) • For each A in the open set pR.(TM.L) and xA 0 = PM.L(A.-T)xA

€

ker(A.-T), we have

(A.-TN.L)PM.LXA.

Since A.-TM.L is injective, it follows that PM.1xA. = 0, i.e., xA. conclude that pR. (TM ) c {A. c «:: ker(A.-T) c M}.

46

€

M. \'le 0

\' n ~ n ~ l·k=l ker{~k-T) = ker ITk=l{~k-T) .

PROOF. The inclusion 'c' is obvious and the converse inclusion is trivial for n = 1. We shall proceed by induction over n. Assume that we have n mk n ~ Lk=l ker{~k-T) ~ ker rrk=l{~k-T) and let .

~n+l.: crp{T)\{~1'~ 2 ,

G~ven

(~k-T)~ (k

.• ,~n} and mn+l be a natural number. n+l ~ mn+l n x.: ker ITk=l{A.k-T) , we have that {~n+l-T) x <: ker ITk=l

and, by our inductive assumption, we can find yk.: ker{Xk-Tfk

= 1,2, ... ,n)

m

such that (A.n+l-T) n+l x = Lk~l yk. Define

~-l -i-1 .)m 1 {~k-T)~J n+ Yk' xk = { Li=l {~n+l-~k) k

1, 2, ••• , n, and

follows.

0

LEMMA 3.14. Let T .: l(H) and Zet x e H be such that the set of ze roes of the function ~ + Pker(A.-T)x• has an accumuZation point r;; e pr(T); then Pker(A-T)x = 0, foP alZ A in the component Qr, of pr(T) aontaining the point 1;.

PROOF. Let a= {~ e Ql;: Pker(~-T)x = 0} and assume that a ~ Ql;. Since a has an accumulation point in Q , we can find ~ e Ql;naa and a I; sequence {~k}k=l c a such that ~k ~ ~. lim(k .... oo) ~k = ~Observe that A + Pker (~-T) = 1-Rr (~ ;T) (A-T) (where Rr (~;T) !s defined by (3.1)) is continuous at~- Thus, if M = V{ker(~k-T)}k=l' then ker(~-T) c M and therefore ran(~-TM) is closed. But ran(~-TM) is obviously dense; hence~ e pr(TM). Let Q0 ={A e Ql;: ker(A-T) c M}. By Lemma 3.12, Q0 is a neighborhood of ~. Since x .;. M , we see that Pker ( ~ -T) x = 0 for all X e 1!0 , co!! tradicting the assumption that~ E aa. This implies that a Ql; andthe proof is complete. 0 00

47

LEMMA 3.15. LetT e L(H), x e

Hand~

e pr(T) ~e suah that

Pker(~,;-T)k

x= 0. for all k = 1,2, ••• , then Pker(A-T)x = 0 for aZZ A in the component 0~ of pr(T) containing 1;.

PROOF. Let M = V{ker(!;-T)k}k:l. Since 1,;-T is surjective, we see k+l that (1;-T)ker(~-T) c ker(~;-T)k (k = 0,1,2, ••• ), so that ran(~-TM) is dense. On the other hand, ker(~,;-T) c M. Hence, ran(~-TM) is closed andit follows that 1,; e pr(TM). Since X e M , it follows from Lemma 3.12 that the set no= {A e n~ : ker(A-T) c M} is a neighborhood of 1,;, and therefore Pker(A-T)x = 0 (A e n0 ). By Lemma 3.14, Pker(A-T)x = 0 for all A in 0~· 0 PROPOSITION 3.16. LetT e L(H) and Let a be a compact subset of pr(T)ncrp(T); then there exists x in H such that Pker(A-T)x f 0 for aLL A e cr. h

PROOF. Let {ni}i=l be the components of pr(T) having nonempty intersection with o and set cri =crnni. For each i (i = 1,2, ••• ,h), we choose ~i e cri and yi e ker(~i-T), yi f 0. Then it is possible to find non-zero complex numbers ai(i = 1,2, ••• ,h) such that the vector y = L·hl a.y. satisfies Pk ( T)y f 0, i = 1,2, ••• ,h. 1= 1 1 er ~iLet {).={A e cr: Pker(A-T)y=O}. If /:,.={6, we take x=y. If{). f {6, we can apply Lemmas 3.14 and 3.15 in order to show that /:,. is a finite set, /:,. = {Ak}k~l and that there exists natural numbers {mk}k~l such that On the other hand, since By Lemma 3.13, y n mk n ran{ II k=l (A k -T) } * is closed, we can find x e H such that y ={IIk=l (Ak-T)mk}* x. Assume that Pker(~-T)x = 0 for some ~ e crk then x = (~-T)*z for some z e Hand therefore y = (~-T)*{IIk~l(Ak-T) k}* x. But this is impo~ sible: if~ f Ak (for all k = 1,2, ••. ,n), we have Pker(~-T)y = 0; if ID'+l 1 } • In either ~=A· (for some j, 1 ~ j ~ n), then y e {ker(A.-T) J J J case, this contradicts our assumptions on y. 0 LEMMA 3.17. LetT e L(H), and Let n 1 be a component ofpr(T)such that nul(A-T) 1, A e n1 • Fo~ any £ > 0 there exists a right resoLvent R of T on n1 except for an at most denumerabLe set s1 , which does not accumuZate in n1 , and satisfies s1 c ran 1 J£.

=

PROOF. By Lemmas 3.14 and 3.15 and Proposition 3.16, there exists 48

a vector y E H such that Pker{A.-T)y f 0 for all A. E n 1 \s 1 , where s 1 is an at most denumerable subset which does not accumulate in n1 and such that sl c canlJ£. (Take 0 = {A. E Ql: dist[A.,an 1 J ~ £}, in Proposition 3.16.) Fix A. 0 E n 1 ,s 1 and let z be a non-zero vector in ker(A. 0 -T). If R0 is a fixed right inverse of A0 -T, then any right inverse of A. 0 -T is of the form Rt = R +zet, where t E H. Choose t

0

= -

-1

R~y

(Since Pker (A. a-T) y f 0, it readily follows

that f 0, so that t i s well-defined.): then =


{z~t) *y ,x> =




-1

R~y ,x>

0

for all x in H, so that y L ran Rt. For A E nl the following identity holds: (A.-T) Rt = ( {A.-A. 0 ) Rt+1) • -1

This shows that A.~ -(A.-A. 0 ) is a mapping from n 1 into the camp~ nent of pF(Rt) which contains the point at infinity. Also, for A. E n 1 , the Fredholm index of {A-A 0 )Rt+l is zero. Suppose thatker[{A 1 -A 0 )Rt+l] f {0} for some A. 1 E n 1 ,s 1 • Then it follows from the above identitytha~ for some x f 0, (A. 1 -T)Rtx = 0. In this case, Rtx E ker(A. 1-T) and since the last space is one-dimensional, ker ( A. 1 -T) c ran Rt. This contradicts ran Rt and Pker(Al-T)y f 0. It readily follows that the operator (A.-A 0 )Rt+l is invertible for all A. in n 1 ,s 1 • The operator valued function

y

E

R(A.) = Rt((A.-Ao)Rt+l]-1 is a right resolvent forT in n 1 \S 1 • This completes the proof. D LEMMA 3.18. Let T E L{H) and let Qn be a aomponent of pr(T) suah that nul (A.-T) = n, A. E Qn. Foro any £ > 0, there exists a right resolvent F ofT on Qn exaept foro an at most denumerable set Sn, whiah does not

accumulate on nn, and satisfies sn

c

cannJ£.

PROOF. We proceed by induction on n. The result is clear if n = 0, for then nn is a component of the resolvent set p(T) ofT (and F(A) = (>.-T) -l, A. E Qn) and the case n = 1 is contained in the preceding lemma. Suppose the result has been obtained in the case n = k-1. Let Qk be a component of pr(T) such that nul(A.-T) = k, A. E nk. It follows from Pr~ position 3.16 that for any £ > 0 there exists a vector y E H for which Pker(A.-T)y f 0, for all A E Qk\S', where S' is an at most denumerable set which does not accumulate in nk and satisfies S' c cankJ£. Let MA = ker(A.-T)n{y}L, for A E nk' and let M = V{MA}AEnk· Obviously, M is invariant under T and relative to the decomposition H = MGlM.L, 49

T [TM A ) • 0

T

1

M

It is easy to establish that A-TM is onto for A " Qk and clearly (A-TM ... ) is onto for A € Qk. It follows that for A " nk \S', nul ( >..-TM) k-1 and nul(A-TM.d = 1. By Lemma 3.17 and our inductive hypothesis TM has a right resolvent R(A) on Qk\S' and TM1 has a right resolvent G(>..) on Qk\S", where S" is a (possibly empty) finite or denumerable subset of Qk which does not accumulate on ~ and satisfies S" c [oQk]c· Define F (A) = (

R( >..)

0

(with respect to the above decomposition), where Sk = S'uS". It is eas ily seen that F(A) is a right resolvent for T on Qk\Sk' sk is at most denumerable, Sk does not have any accumulation point in Qk and Sk c rankJc·

o

PROOF OF THEOREM 3.8. Clearly, it suffices to define a right resolvent F on each component of pr(T), except for an at most denumerable subset S with the desired properties. Let n be a component of pr(T) such that nul(A-T) =n(n) ~ 0, A" n. I f n(Q) =0, then n c p(T) and the only possible definition for F is F(A) = (A-T)-l =the resolvent of T restricted to n~This is true, in particular, for the unbounded component of pr(T).) If 1 $ n(Q) < oo, then Q is a bounded component of pr(T). If Q intersects the compact set~= {A< pr(T)\p(T): dist[A,apr(T)] ~ c}, then we define F on Q\S(Q), where S(Q) is an at most denumerable subset of n which does not accumulate in n and satisfies S(Q) c Can> £ by using Lemma 3.18. If Qn~ = ~, then we can use the same arguments as in that lemma in order to construct a right resolvent F on Q\S(Q),where S(Q) is an at most denumerable subset which does not accumulate in Q (the condition S(Q) c Q c Can) is trivially satisfied in this case). £ It is completely apparent that this defines a right resolvent for Ton pr(T)\S, where S = u{S(Q): n is a component of pr(T)\p(T)} is an at most denumerable subset of pr(T) with the desired properties.o

3.1.4 The left and the right spectra of tAB THEOREM 3.19. Let A, B" L(H); then (i) a 6 (TAB) = crr(TAB) = crr(A)- aR, (B). (iiJ cr'lf(TAB) = aR,(tAB) = aR,(A) -crr(B).

50

PROOF. (i) By Corollary 3.7 and our observations at the beginmng of Section 3.1.2, cr~(TAB) c crr(A) -cr.R.(B) and cr~(TAB) c crr(TAB). Assume that \J E crr(A)- cr.R.(B) (i.e., \J can be written as \J =.a-a, where a E crr(A) and a E ot(B)) and that TAB-\J is onto. Then, given C ~ L(H), there exists X E L(H) such that (TAB-\J) (X) =TA-a,B-a(X) = (A-a)X -X(B-a) =c. Since ran(TAB-\J) is closed, there is a constant m > 0 such that II(TAB-\J) (Tlll ~ m dist[T,ker(TAB-\J) ], for all T in L(H). In parti£ ular, X= X(C) can be chosen so that (m/2liiXII :;;; IICII. Since a E or(A) and 8 E ot(B), we can find unit vectors x, yin H such that II = IICII; for this C and X=X(C) chosen as above, we have 0 < IICII = = <(TAB-JJ) (X)y,x> = <[ (A-a)X-X(B-a) ]y,x> s I<Xy,(A-a)*x>I+I<(B-S)y,X*x>l s (2/m>llciHIIylll < llcll, a contradiction. We conclude that o~(TAB) =or(A) -o.R.(B) c or(TAB). Assume that \J t ar(A) -a.R.(B); equivalently, or(A-j.i)nat(B) =,. Since crr(A-\J) and a.R.(B) are nonempty,compact, and disjoint, there exists Cauchy domains QA' n8 and Q such that crr(A-\J) c QA c (QA)- c Q, aR,(B) c n8 and (Q8 )-nQ- = ,. It follows from [1], [2] (see also [185]) that there exists an analytic function RA-\J (A) defined on pr (A-\J) such that (A-j.i-A)RA-j.I(A) 1 on this domain. (If ~\QA c pF(A-j.i), then we can choose RA -j.l (A) as the right resolvent constructed in Theorem 3.B.);sim ilarly, there exists an analytic function La(A) defined on p.R.(B) (If ~\QB c pF(B), then we can choose L8 (A) as the left resolvent given by Corollary 3.9.) such that L8 (A) (B-A) = 1 on this domain. Clearly, RA-\J and L8 are analytic in a neighborhood of an.Let n0 be a component of Q. Since aR, (B) nQ- = ,, it follows from Cauchy's theo rem that fan L8 (A) dA = o.

=

0

A fortiori,

Let a•n =
and 2;i

fan RA-~(A)dA

= 2;i

Given Yin L[H),

fa•nRA-~(A)dA

1.

let -1

~(Y) =X= 2 wi

f

an

RA-~(A)YLB(A)dA.

It is completely apparent that ~ defines a bounded linear mapping from L[H) into itself. l'le shall verify that~ is a right inverse for TAB-~· Indeed, . (TAB-~) (X) = (A-~)X-XB = -1

2wi

f an(A-~)RA-~(A)YLB(A)dA+

1

2wi fanRA-~(A)YLB(A)BdA

= -1 2wi f ao(A-~-A)RA-~(A)YLB(A)dA+ (TAB-~) o~

f aORA-~(A)YLB(B-A)dA

2 !i(fanRA-~(A)dA)Y = Y,

;;i YfaoLa(A)dA+ i.e.,

21wi

=

1 on L [H). Hence, ar (TAB) c ar (A) -a R. (B) and the proof of (i) is complete. (ii) By Corollary 3.7 and our observations at the beginning of Section 3.1.2, aw(TAB) c aR. (A)- ar(B) and aw(TAB) c aR. (TAB). Assume that~" aR.(A)-ar(B), i.e.,~= a-13 for some a ( aR.(A)and some f3 "ar(B). This means that, given € > 0, we can find unit vectors x, yin H such that II
= II TA-a,B-B (xey)

II

= II

(A-a) (x8y) - (xey) (B-13) II

II (A-a)x 8y-x8(B-f3)*yl! ~ I!
Since € can be chosen arbitrarily small, we conclude that TAB-II is not bounded below, i.e., JJ " aw (TAB). It readily follows that aw('A~ = aR. (A)- ar(B) c aR. (TAB). Now assume that aR.(A-~)nar(B) = 9 and let tA' tB and t be three Cauchy domains such that aR.(A-~) c tA c (tA) c t, ar(B) c tB and (t~­ nt- = 9. By [1], [2](see also [185], Theorem 3.8 and Corollary 3.9), there exist an analytic function Ra(A) defined on pr(B) and an analytic function LA-~(A) defined on pR.(A-~), such that (B-A)Ra(A) :1 for all A in pr(B) and LA-~(A) (A-A) 1 for all A in pR.(A-~). Proceeding as in the last part of the proof of (i), we can easily check that

=

1

~(Y) = 2wi

defines a left inverse of aR. (A)- ar(B).

fat

LA-~(A)YRa(A)dA

TAB-~·

It readily follows that aR.(TAB)

From Theorem 3.19 and its proof, we obtain 52

0

c

COROLLARY 3.20 (Rosenblum's theorem). If A, BE L(H), then (i) cr(TAB) = cr(A) -a(B); (iiJ If l.l I a (A) -a (B), then there ereists a Cauohy domain n suah that a(A-}.1) c n, a(B)nrl = ~ and

(TAB-}.1) By taking A obtain

-1

(X)

0 or B

0 in Theorem 3.19 and using Lemma 3.6, we

COROLLARY 3.21. If A E L(H), then ar(LA) = a 6 (LA) a 1 (RA) = cr~(RA) = crr(A). (ii) a 1 (LA) = cr~(LA) = crr(RA) = cr 6 (RA) = cr 1 (A). (i)

3.1.5 Rosenblum-Davis-Rosenthal corollary The following is the most important consequence of Theorem 3.19 for the purposes of approximation. COROLLARY 3.22. Let H1 and H2 be two Hilbert spaoes, Zet L(H 1 ), BE L(H 2 J and C E L(H 2 ,H 1 ) and assume that ar(A)na 1 (B)

A E =~.Then

the operators ( Ao

(aating on H

=

Be)

and

AIBB

H119H 2 .J are similar.

PROOF. Assume that H1 and H2 are infinite dimensional then we can identify them via a unitary mapping of H1 onto we can directly assume that H1 = H2 = H0 and H = H0 (2 ) • By (i), TAB is onto and therefore there exists X E L(H 0 ) such =-c. Then (:

and

(~ ~)

is invertible

~) (~ ~) ([~ ~)-l

Hence, AIBB =

spaces;

H2 , i.e., Theorem 3.19 that AX-XB

(~ ~) ~)

=

(:

=

(~ -~)).

(10 1X) -l (A0 BC) (10 X)1 .

If A acts on a finite dimensional space and B acts on an infinite dimensional space, define a = IIAIHIBI~l and consider the operators alBA e L!H 2tH 1 J, B and C'=(g) E L!H 2 ,H 2mH 1 ). It follows from the first part of the proof that 53

[ ~ ~ ~]

-

(aeA)eB

ae(AeB).

0 0 B

Assume that W

then {ae

(:~)}w-w[ae(AeB)J=~~~-

~ -:~~~-~ 1 ~<~•~>-

0 - --

-]=

l(o B)w21-w2la (o B)w22-w22(AeB)

o,

1

so that aw 12 -w12 (AeB) = 0 and r~ ~)w 21 -w 21 q = 0. Since cr (a) ncr ( (~ ~)) c cr (a) n[cr (A) ucr (B)] = cr (a) ncr (AeB)

= $§,

it readily follows from Corollary 3. 20 that w12 = 0 and w21 = 0. Hence W= w11ew 22 , w11 and w22 are invertible operators, and (consider the (2,2)-entry of the above 2 x 2 matrix!) w22-l (: i.e., (:

~)

~)

w22 = AeB,

is similar to A8B.

The cases when H2 is finite dimensional and H1 is infinite dimensional or both, H1 and H2 are finite dimensional spaces can be similar ly analyzed to reach the same conclusion: ( Ao

Be) - AeB.

0

3.1.6 The maximal numerical range of an operator It was observed in Section 3.1.1 thati~ABII s min{IIA->.If+liB->.11: ). E ~}. In order to complement our previous results about 'AB' it will be shown that this inequality is actually an equality. The concept of mazimaZ numeriaaZ range of an operator plays a central role here. Recall that the numeriaaZ range of T E LfH) is the set defined by W(T)

=

{:

x

E

H, llxll

= 1}.

The classical Toeplitz-Hausdorff theorem ~sserts that W(T) is a convex set [119,Problem 166]. The maximal numerical range of T is the set

54

LEMMA 3.23.

w0 (T)

is a nonempty, aZosed, aonvex subset of W(T)-.

PROOF. Everything but convexity is obvious. Let A, J..l

t'

w0 (T).

Wit~

out loss of generality, we can assume that IITII =1. Let {xn}n:l' {yn}n:l be two sequences in H such that II xn II =II y n Ii = 1 for all n = 1, 2, ••• , ll!'xn II .. 1, ltl'Ynll .. 1, .. A and .. J.l (n + "'). Consider Tn = Pn TPn' where Pn is the projection of H onto Vfxn,yn} and let n be a pomt of the segment [A,J..l] joining A and J..l. Then for each n, it is possible, by the Toeplitz-Hausdorff theorem, to choose an' f3n such that = -+ n and II unll = 1, where un = anxn +f3nYn· Observe t:hat I <xn,yn>J s e < 1 and therefore max{Janl ,JenJ} s M= (1-62)-!:1 (otherwise ll'hxn+ f3nyJI < 1) for all n sufficiently large. We have 1


Jl!'u n IJ2 = = JJun IJ2 - < (1-T*T) U n ,un >

Since 2

IJ(l-T*T)xn ll=llxn 112 -2JI!'xn II +IIT*Tx n 112 2 = (1-JtJ.'x n 112 >- 11 2 > s 1-JI!'xn II .. 0 (n .. "') and, similarly, 11<1-T*T)ynJI + 0 (n + "'), we conclude that ll!'unll + 1 and -+ n (n +"').Therefore, w0 (T) is a convex set. 0 0 s

If A € L (H) , the operator 15 A= T AA is the innel' del'ivation of L (H) induced by A. 3.24. Let J..l

LEMMA

w0 (T), then IJ15TJI

E


2(11Til2 -IJ..l! 2 >!:1.

PROOF. Note that IliST II= supfll TA-AT II: A € L (H) , II Ail= 1}. Since J..l E € H such that llxnll=l, for all n=l,2, ••• , ltl'xJI IITII and -+ J..l. Set Tx n =a nn x +a nn y , where y n is a unit vector

w0 (T), there exist xn -+

orthogonal to xn. Set Vn = xn8xn- yn8yn; then II
2:

2(!1TII 2 -JanJ 2 >!:1-e:n, where e:n + 0 (n + "').

Since an-+ J.1 (n .. "'), the proof is complete. PROPOSITION 3.25. Let T (i)

0

€

L(H); then the foZZowing al'e equivalent

w0 (T);

li~ll=2ltrll; IITII 2 + IAJ 2 s IIT+Ail2 fol' aZZ A IITIJ s IJT+AII for aZZ A E a:.

(ii) (iii)

(iv) PROOF. then 114r11

€

2:

(ii) =>

0

(i) => (ii)

€

a:;

It follows from Lemma 3.24 that, if 0

2]1!'1J. Therefore, 11~11= 2lh:'ll· (i) If 114rll=211!'11, then there exist xn

€

Hand An

w0 (T),

€

€

L(H) 55

such that llxn11=11An11= 1 and II
11,

I t readily follows that IIAnn x II-+ l=IIAn I!Txn II-+ IITII and !ITAnn x II -+ II T II· Passing, i f necessary, to a subsequence, we can directly assume that -+ ~ and -+ y (n -+ 00 ) . Since IIAnn X II-+ 1 =llxrJ'I· nn ,Ann i t is completely apparent that ll, y e w0 (T) • Moreover, since II
n

xn II -+ 211 T II and H is uniformly convex, the norm of z to

o,

n

= TA x +A Tx tends n n n n n

as n -+ ""· Thus, we have

= - +



= - + n n nn where En -+ 0, as n -+ ""·

= - + n nn n n

£

n

,

(Indeed, i t follows as in the last step of the

proof of Lemma 3.24 that IIAn*Anxn -xnll-+ 0, as n-+ oo.) Thus, y = lim(n -+ oo) = -lim(n -+ oo) = -ll·

Since both that

and -ll belong to

ll

w0 (T),

i t follows from Lemma 3.23

o "w 0 (T). If 0 e

(i) => (iii)

w0 (T),

then there exists a sequence {xn}n:l

of unit vectors such that

for all A e

a:.

(iv) => (i)

Assume that IITII s IIT+AII for all A "

By rotating T, we may assume that lle

w0 (T)

= 1, lle s c/2} and let M=supfiiTxll: Let ll = min{c/2,
I M,

"M,

a:,

but 0

I w0 (T).

~ c > 0. Let M={xeH:

llxll

x eM}; then M < IITII· then II
llxll= 1, let Tx = (a+ib)x+y, where y

.l

x; then

II ll > 0. Thus, II T-llll <

II T II,

contrary to the hypothesis.

Since (iii) => (iv) is a trivial implication, we are done.O THEOREM 3.26. LetT " L(H); then II6TII= 2 min{IIT-AII:

A "

PROOF. We have already observed that II6T II s 2 inf{ll T-A II:

a:). A " a:}.

By an elementary argument of compactness (observe that liT-All-+ ""• as IAI -+ oo), i t is easily seen that the above infimum is actually attained at some point ll " II
a:, " a:

i.e., this infimum is a minimum. But IIT-llll s implies that II6TII = 116 (T-ll)ll = 2 IIT-llll·

{iv) => (ii) .)

3.1.7 The norm of the operator TAB

56

(Use

0

LEMMA 3. 27. Let T continuous.

L (H) • The mapping X +

£

w0 (T+A) is

uppezo semi-

PROOF. We can assume, without loss of generality, that II T II = 1. Suppose that i.e w0 (T) s a and let e: > 0. Let f~ = sup{ll Ax II: II x II = 1, i.e ~ a+e:}; then M < 1. It is clear thatiiT+AII ~ 1-IXI. However, for y £ H, IIYII = 1 and l.e ~ a+e:, we see that II. 0 £ ~). Hence, w0 (T+X 0 ) coincides with the intersection of all the open halfplanes containing it. Thus, we can find finitely many open halfplanes sl' s2'"""' sm such that

w0 (T+X 0 )e:

m

nj=l Sj

c

c

w0 (T+X 0 ) 2e:.

By the first part of the proof, we can find o that

w0 (T+A)E

c

m

nj=l Sj

c

W0 (T+A 0 ) 2e:

provided I A - X0 I < o• Hence, X + w0 (T+X) is an upper semicontinuous mapping.

D

We define the nozomalized maximal numezoical zoange WN(T) of an ope~ ator T £ L(H), T ~ O, to be the set w0 (T/IITII>· From Lemma 3.27 we obtain the following COROLLARY 3.28. If IIT+AII is upper> semicontinuous.

LEMMA 3.29. Let A, B following azoe equivalent (i)

(ii)

E

~ 0

foro aZZ X, then the map X+ WN(T+A)

L(H) be two non-zeroo operoatozos; then the

11-rABII = IIAII+JIBI~ WN(A)nWN(-B) ~ ~.

PROOF. The proof is very similar to that of Proposition 3.25 (i) => (ii), and so we shall only sketch a portion. Let A£ WN(A)nWN(-B) and e: > 0; then there exist x, y £ H such that llxll = IIYII = 1 and = ~IAI~e:A and = -~IBI~e: 8 , 0 s e:A' e:B < e:. Since J11AII = -<sy,y>/IIBI~e:', it is possible to define an operator U of norm l+e:" which sends x to y

and -By/liB II to AxJ11AII<where e:', e:" are small if e: is small). The rest of the proof is virtually unchanged. D 57

LEMMA 3.30. Assume that A. B ar multiple of the identity. Then min{IIA-:l.II+I!B-:l.ll:

if and only if WN(A-).I)nWN(-(B-).1))

LIHI and neither A nor B is a saal

€

:l.

€

~

a:}= IIA-J.I!I+I!B-J.III 9.

PROOF. Assume that WN (A-J.I) nWN (- (B-Il) ) ~ 9. Then Itr"AB II = liT A-ll , B-).111 = II A-IJII + IIB-Illl • Since it is obvious that ltr" AB II cannot be larger than min{I!A-:l.II+I!B-:l.ll: ). € a:}, we see that the condition is necessary. In order to prove the sufficiency, we can directly assume that J.1 = 0. Thus, given ). € a: and £ > 0, there exist unit vectors x, y in H such that IIIA+:l.)xii+II(B+:l.)yjj /IIAII+/IIBII> s Kll:l.l 2+£), where K is a constant independent of). and£. Assume that WN(A)nWN(-B) = 9. Then, dH[WN(A) ,~7N(-B)] = cS > 0, and (by upper semicontinuity; Corollary 3.28) dH[WN(A+:l.) ,WN(-B-:l.)J > cS/2, for ). small. Thus, by convexity and upper semicontinuity, any choice of x, y which satisfies the above conditions, must satisfy the inequal ity j<(A+:l.)x,x>/IIAHII+<(BH)y,y>/IIBHIII:t cS/4 for:>.. small. But thenw~ are led to the inequality j:l.jcS/8 s K!:>..l 2 for a suitable choice of arq:>.. and l :>..I small, which is impossible. Thus, J.l = 0 was not minimal, which completes the proof. 0 THEOREM 3.31. Let A. BE L(H); then ltr" AB II = min{ll A-"11 + 1~-:>..11 :

:l. E a:}·

PROOF. Clearly, ltr"ABII s min{IIA-:l.H+IIB-:l.ll: :l. € a:}. I f A orB is a multiple of the identity, the rest of the proof is trivial. Let ll £ a: be any point such that the above minimum is attained at J.l• By Lemmas 3. 29 and 3. 30, liT AB II = ltr" A-ll ,B-ll II = II A-JJif-t- II B-J.III • The proof is complete now.

0

REMARK 3.32. It is completely apparent from Proposition 3.25(iii) that there exists exactly one J.1 E a: such that II T-JJ II = min £11 T- :>..II : ;\. E a:}. However, simple examples show that, in general, the J.1 of Lemma 3. 30 is not unique.

3.2 Generalized Rota's universal model Let T £ L (H) •. Since o(T) is a compact set, it has a fundamental system of open neighborhoods which are analytic Cauchy domains (i.e., Cauchy domains whose boundaries consists of pairwise disjoint regular 58

analytic Jordan curves: see definition in Section 1.1). Let r =an, where Q is an analytic Cauchy domain containing a(T), and let L2 (r) be the Hilbert space of (equivalent classes of) complex functions on r which are square integrable with respect to (l/2nrtimes the arc-length measure on r: M(r) will stand for the "multiplication by A" operator acting on L2 (r). The subspace H2 (r) spanned by the rational functions with poles outside n- is invariant under f1(r). By M+(r) and M_(r) we shall denote the restriction of M(r) to H2 (r) and its compression to L2 (r)eH 2 (r), respectively, i.e. M(r)

(

M+(r)

o

)H 2 (r) M_ L2 (r)eH 2 Z

( 3. 2)

with respect to the above decomposition.(Here and in what follows, we write

( 3. 3)

A

as an alternative way to indicate that A admits such a k x k operator k matrix decomposition with respect to the orthogonal direct sum H = ej=l Hj , where Aij:Hj ~ Hi' 1 ~ i,j s k. Such a decomposition is clearly unique.) It is well-known [36], [73], [74,Sections 9] that H2 (r) is areproducing kernel space and that it can be realized as a space of analytic functions defined on n: furthermore, it can be easily checked that a(M(r)) =aJM(r)) =ae(M+(r)) =ae(M_(r)) =r, a(M+> = a(M_(r)) = n-, nul(A-M+(r)) =nul(A-M_(r))*=O and ind(A-M+(r)) ind (A-M_ (r)) * = -1 for all A in

( 3. 4)

Q,

M(r) is normal and l!r(M(r))l! =llr(M+(r))ll =llr<~_(r))ll = max{lr(A)l: A e: for each rational function r with poles outside n-.

Q}

The Hilbert space completion H2 (r)8H of the algebraic tensor pro~ uct of H2 (r) and H can be regarded as a space of analytic H-valued functions defined on n, or as a space of (equivalent classes of) weakly measurable square integrable functions on r, in which case it will 59

also be denoted by H2(r:H). THEOREM 3.33. LetT E L(H) and 0 be as above, and let R =ran (M+(r)81-18T); then R is a subspaae of H2 (r:H) invariant under M+(r)&l, the restriation of M+(r)&l to R is similar to M+(r)&l and the aompression of M+(r)el to R~ is similar to T. PROOF. Since M+(r)el-l&T is equal to multiplication by X-T in 2 H Cr:H) and X-T is invertible on r, it is clear that R is closed. Mar~ over, the similarity of f1+(r)81 and M+(r>&ljR is implemented byH+(r)~n18T. All we have to do is find an operator L:H 2 (r:H) + H with range H and kernel R such that Lo(M~r)&l) = ToL. We shall define for f E H2 (r: H) the element Lf E H by 1

2 ~i

Lf

f ao(X-T) -1

f(X) dX.

It is easily seen that Lo(M+(r)&l) Lo(l8T) = ToL, so that R c ker L. On the other hand, if fx(A) = x (x E H), then Lfx x, so that ran L is equal to H. Let o 1 be an analytic Cauchy domain containing 0 such that the restrictions of functions analytic in a neighborhood of o 1 to r are dense in H2 (r) (By Runge's approximation theorem, it suffices to choose o 1 so that every bounded component of ~\0 contains a component of ~\ n 1 > and let e (X) = 1, e c H2 (r). 0 2 0 Lets E L(H (r:H)) be the operator defined by Sf= f-e&Lf. Since Lf = L(e9Lf), it follows that ran s = ker L. Thus, in order to complete the proof, it will be sufficient to show that S(flr> E. R for every Hvalued function f analytic in a neighborhood of o 1 . Because of the analyticity of (X-T)-lf(A) in a neighborhood of o;>n' we have Lf = 2 1~i It follows that for cs

I; E

f anl

(X-T) -1 f(X) d)..

0

1 2dfaol [ (A-1;)-l- (X-T)-lJf(X) dX

( r-T)

..

l

2~i

fan

(A-1;)-l(X-T)-l f(X) dX. 1

Since fanl (X-1;)-l(X-T)-l f(X) dX is an analytic function of 1; (1; c o 1 ), it readily follows that s = CM+Cr>el-l&T)( 2 fa 01 <x-c>- 1 <X-T>- 1 f(X) dx) cR. The proof is complete now. 0

!:L

REMARK 3. 34. According to the proof of Theorem 3. 33, the diagram

60

H -----=T:___ _ _ H

is commutative, i.e., T = (LIRL)-loT 1 a(LIRL). Since Lfx = x (where fx (A) :: x, x E H), it follows that Ll RL is bounded below by . inf{llfxll:

llxll =

1} =

([length(r)]/2-rr)~.

On the other hand, the Cauchy-Schwartz inequality implies that IlLII s ([ length(r) 1/2-rr) ~ max{ II< A-T) -11:

A

r},

E

so that (3.5) Let r be a rational function with poles outside Q

and let

B)R

T' RL'

where T' is similar to T1 then llrII s llr<M+(r)~H>II

=

llr<M+(r))ll

= max{lr(A)I:

A

E

r},

whence we obtain the following COROLLARY 3.35. (i) Given T E L(H) and a bounded open neighbo~­ hood ~ of a(T), ~ is a spect~aL set (in the sense of von Neumann) fo~ some operator T' - T, i.e •• llrII s max{jr()..)"l:

A

E

i!J-}

fo~ aLL rational functions r ~ith poles outside ~-.

(iiJ In pa~ticular, if a(T) {0} and e > 0, then '1'' - T such that !IT'll < E and T sim O.

the~e e~ists

~~ 3.36. Let H = H1eH 2e .•• eHk and assume that A E LIH) admits an upper triangular operator matri~ ~ith respect to this decompositio~ i.e .•

61

All Al2 Al3 A22 A23 A33

.Al,k-lAlk .A2,k-1A2k .A3,k-1A3k

Hl H2 H3

A 0

~-~k-h-l.k Hk-1 Hk ~k

then A s!m All$A22$A33e ••• e~-l,k-le~k· PROOF. Let wn = nen 2en 3e ••• enk-1 enk (w1.. th respect to the same decomposition). Clearly, W is invertible with inverse w -l=n- 1 en- 2en- 3 -k+l -k n n e ••• en en , and a straightforward computation shows that

IIWnAWn- 1 -·j~l Ajjll s [n- 1 +n- 2+ ..• +n-k+lJIIAII

+

0 (n +co). 0

It follows from (3.2) and Theorem 3.33 that MCr>el

=

(M+(r)91

o

Z91 )H 2 (r;H) M_Cr)el L2 cr>eHeH 2 Cr;H)

M+(r)&liR [

0

0

where

[:~)

z1 ]R T' z2 H2 (r;H)9R 0 M_(r)&l L2 (r)9HeH 2 (r;H)

B

= Z&l, M+(r)&liR - M+(r)&l and T' - T. Combining this obser

vation with Lemma 3.36, we obtain COROLLARY 3.37. Let T e L(H) and Zet n be an anaZytia Cauahy domain aontaining a(T) ~ith an = r; then MCr> s!m

M+(r)eM_Cr),

M(r)&l s!m (M+(r)el)e(M_Cr)el),

l4(r)91 sim (M+(r)9l)eTe(M_(r)91) and l1+(r)91 s!m (M+(r)8l)eT.

3.3 Apostol's triangular representation r

Given T e L(H), let H (T) = V{ker(~-T): ~ e Ps-F(T)}, let H.(T) r r ,. V{ker(~-T)*: ~ e ps-F(T)} and let H0 (T) be the orthogonal complement of Hr(T)+Ht(T). Denote the compression ofT to Hr(T), Ht(T) and H0 (T) by Tr' Tt and T0 , respectively. Recall that A e L(H) is a t~ian­ guZa~ operator if it admits an upper triangular matrix; i.e.,

62

all al2 al3 a22 a23 a33

A

(3. 6)

0

with respect to a suitable ONB. Apostol's triangular representation and its basic properties are established in the following THEOREM 3.38. (i)

Hr(T) is orthogonal to Ht(T), so that {3.7)

H = Hr{T)SH 0 {T)SHt{T).

(ii) Hr{T) and Hr{T)SH 0 (T) are invariant under T, so that Tadmits a 3 x 3 upper triangular operator matrix representation {3. 8)

T

with respect to the above decompostion, ~here (iii) Tr = TIHr(T) is a triangular operator, o{Tr) = ot{Tr) = aire{Tr)u[pr(Tr)\p{Tr)], a{Tr) is a_perfect set, every compon:nt of oire{Tr) intersects the set op{Tr) and op(T;> = ~. so that ps-F{Tr) ~and min.ind(A-Tr) = 0 for all A € ps-F{Tr)• and (iv) Tt is the adjoint in L(Hi(T)) of the triangular operator

T*IHt(T), o{Tt) = or{Tt) = otre{Tt)u[pt(Tt)\p(Tt)], o(Tt) is a perfect set, every component of otre{T{) intersects the set op(Ti)- and op{Tt) ~. so that p:_F(Tt) = 9 and min.ind{A-Tt) = 0 for all A € ps-F(Tt). Furthermore, (v) A~ Pker{A-T) (A € Ps-F(T)J is a continuous function for A € prs- F(T) and discontinuous for A € pss- F(T); ps-F{T) c pr{Tr)npt(Tt); (vii) Pi-F(T) c p{T0 ) ; (viii) p:_F(T) c o 0 {T0 ); (ix) a 0 lT) c p{Tr)np(Tt)no 0 {T0 ); (x) If A= {A 1 ,A 2 , ••• ,Am} is a finite subset of p:_F(T), then TN TASTA, ~here TA acts on a finite dimensional subspace, o{TA) =A (Vi)

and A

c

p~-F{TA).

We shall need several auxiliary results. LEMMA 3.39. If B

€

L(H), A

€

o{B) and II
Mllxll

for aZZ x

€

63

fl,

then IIA- Bll ;;, M for every triangular operator A.

PROOF. Assume that A has an upper triangular matrix of the form n n= n be the orthogonal projec tion of fl onto V{e 1 ,e 2 , ••• ,en} and let y be a unit vector orthogonal to ran(A-B) (so that (A-B)*y = 0); clearly, PnAPn = APn. Given e:, 0 < e: < 1, there exists n 0 = n 0 (e:) such that 11Pn (A-B) *y II < e: for all n;;, n 0 • Pn(A-B)*Pn can be regarded as an operator acting on the finite dimensional space Mn =ran Pn. It follows from the finite dimensionality of Mn that this subspace contains a unit vector zn such that II[P n (A-B)*P n J*z n II= liD l'"n (A-B)P n z n II< e:. ~1e have Pn z n = z n , and (3.6) with respect to the ONB {e } ~ 1 • Let P

IIA-BII;;, lim inf(n

~>

11<1-Pn) (A-B)Pnll

+ ~>

11<1-Pn) 0,-B)Pnll

+

2:

lim inf(n

2:

lim inf(n ....

ao)

2:

lim inf (n ....

ao)

?:

M-e:.

11 (1-P n ) (A-B)P n z n II

Since e: can be chosen arbitrarily small, we conclude that IIA- Bll 2:

M.

0

COROLLARY 3.40. If A is a triangular operator with matrix (3.6) with respect to the ONB {en}n:l. then (i) a(A) =aR. (A) = aR.re(A)uap(A); (ii) Every clopen subset of a(A) intersects d(A) = {ann}n:l and every component of a(A) intersects d(A)-; (iii) Every isolated point of a(A) belongs to d(A); (iv) If ker(A-A)* ~ {0}, then A " d(A). PROOF. (i) The equality a. (A)= a. (A)ua (A) is trivial. On the ... ..re p other hand (since A is triangular) Lemma 3. 39 implies that a (A) =a 1 (A). (ii) Let a 0 be a clopen subset of o (A) such that a 0 nd (A) = 11 and let Q be a Cauchy domain containing d(A) such that a0 nn- = 11. It is eas ily seen that, if 1

E = 2ni

fan

(A-A)

-1

dA,

then Een =en for all n = 1,2, •••• Since E is an idempotent and ran E => V { e n }n= "" 1 = fl, it readily follows that a 0 = 11. Since a(A) is a compact Hausdorff space, every component of a(A) is the intersection of all the clopen subsets containing it. Let a bea component of a(A) and let {a : a " A} be the family of all clopen su~ a sets of a(A) containing a. It follows from the first part of the proof that aand(A) # 11 for all a in A. A fortiori, and(A) = (nao;A aalnd(A)-= 64

na£A[aand(A)-J ~ 9 (by an obvious argument of compactness). (ill) This is a trivial consequence of (ii). (iv) Assume that (A-A) *y = 0 for some ). £ a (A) and some unit vector y £ H; then 0 = <{).-A)*y,en> = ' for all n = 1,2, ••• ,

so that y J. {).-A)Mn' where Mn = v£e 1 ,e 2 , .•. ,en}' for all n = 1,2, •••• If"'). I d(A), then {).-A)Mn=Mn (n = 1,2, ••• ) and therefore y J. V{Mn}n=l = H; so that, y = 0, a contradiction. Hence, ). € d(A). 0 PROPOSITION 3.41. LetT£ L(H) Q

and~

£ Pn-F(T); then

lim{). ~ ~) Pker{).-T)

exists (in the norm topoZogy). Q is the orthogonaZ projection onto ker (~-T)n{nn:l ran(~-T)n}.

PROOF. Assume first that ~-Tis onto. By Corollary 1.14(v), ).-T is onto for all ). in some neighborhood Q of ~· It follows that Pker {).-T) = 1-Rr 0. ,T) {).-T) (where Rr {). ,T) is defined by (3.1), ). £ g) converges in t~e norm to 1- Rr{).,T) {).-T) = Pker{).-T) as A ~ ~· Put M= nn:l ran (~-T) n. Because (~-T) is semi-Fredholm, M is closed (i.e., is a subspace) and (~-T)M = M. Hence (~-TM)M = M. For). ~~clearly (~-T)ker{).-T)=ker().-T) and so ker{).-T) c M. I t follows that p

=p

p

ker().-T) ker{).-TM) M and the first paragraph shows that this has the desired limit.

D

COROLLARY 3.42. LetT£ L(H) and~ £ Ps-F(T). The foZZowing cond£ tions are equivaZent (i) ~is a reguZar point of Ps-F(T). (ii)

Pker{).-T) is continuous

at).=~·

PROOF. (i) => (ii) If nul{).-T) is a finite constant in some neig~ DO borhood of~ then Corollary 1.14(vi) shows that ker{).-T) c nn=l ran {).-T)n. By Proposition 3.41, Pker{).-T) is continuous at ). = If nul{).-T)* is a finite constant in a neighborhood ollary 1.14(vi) shows that DO n ker(~-T)* c nn=l ran(~-T)* CLAIM:

Since

~· of~

then Cor (3.9)

is semi-Fredholm, (3.9) is equivalent to DO n (3.10) ker(~-T) c nn=l ran(~-T) • DO n Indeed, if ker(~-T) c nn=l ran(~-T) , then ~-T

65

1, 21 • • • •

then y= (~-T)*y 1 for some y 1 E H. There-1 [{y}] ~ ~. so that y = (~-T)* 2y fore ran(~-T)* J J (~-T)* 2 3 -2 for some y 2 E H and ran(~-T)* J ker(~-T)* J (~-T)* [{y}] ~ ~By induction, there exists y in H such that y = (U - T) * ny , n n n n = 1,2, .••. Hence, y E "n:l ran(~-T)*. This proves that (3.10) implies (3.9). The converse implication follows by taking adjoints. Thus, if y

E ker(~-T)*,

2 ker(~-T)*

Since (3.9) and (3.10) are equivalent, we conclude (as in the first part of the proof) that Pker(A-T) is continuous at A = U· (ii) => (i) If Pker(A-T) is continuous at A = u, then (3.10) holds. Hence (3.9) also holds and (i) follows from Corollary 1.14(vi). 0

Let H = ej~l Hj' where Hj ~ H for all j = 1,2, ••• ,k, and let A E LIHI with matrix of the form (3.3) with respect to this decomposition. Observe that A is compact if and only if Aij is compact for all i and j. On the other hand, since Hj ~ H (i.e., Hj is infinite dimensional) for all j, Hj can be identified with H via a fixed unitary map (j = 1, 2, ••• ,k).

Then L[HI can be identified with the algebra of all k x k matrices with entries in LIHI, KIHI can be identified with the ideal of all k x k matrices with entries in K[HI and (by taking the corresponding quotients) A(H) can also be identified with the algebra of all k x k matrices with entries in A(H) (k=l,2, ••• ). This identifications will play a very important role. The proof of the following elementary algebraic lemma is left to the reader. LEMMA 3.43. Let NK 2 (R) be the ring of all 2 x 2 matriaes with entries in a ring R with identity 1. Then

(i) If (~ ~) has a right inverse RJ. If q is 1-nvertible, then pa = 1. (ii) If (~ ~) has a Zeft inverse 1. RJ. If p is invertible, then dq

[~ ~) (~ ~)

in re 2 (R) • then qd

1 (in

in re 2 (R) • then ap = 1 (in

PROOF OF THEOREM 3.38. (i) By Corollary 1.14(vi), either ker r (A-T) c ran(A-T) or ker(A-T)* c ran(A-T)* for every A E Ps-F(T). If u E p~-F(T) and u ~ Ar then (A-T)ker(u-T) = ker(u-T) and so [ker(A-T)*J~ = ran(A-T) J ker(~-T). 66

By Corollary 3.42 JIPker(J.J-T)- Pker(A-T)JJ + 0 (J.J readily follows that ker(A-T) c [ker(A-T)*J~. Hence, Hr(T) = V{ker(A-T):

A~

r Ps-F(T)}

~

+

A), whence i t

r V{ker(A-T)*: A€ p 5 _F(T)}=HJI.(T)

and, a fortiori, H = Hr(T)$H 0 (T)$HJI.(T). (ii) The invariance of Hr(T) and Hr(T)$H 0 (T) =HJI.(T)~ under Tis immediate. Hence, T admits a matrix representation of the form (3.8). (iii) Let {A n }n= ml be a denumerable dense subset of pra- F(T). By Corollary 3.42 Hr(T) = V{Mn}n:l' where Mn=ker(An-T). It is completely apparent that Tr has an upper triangular matrix with respect to an ONB obtained by Gram-Schmidt orthonormalization of a union of ONB's of the subspaces {Mn}n:l and d(Tr) = {An}n:l in this representation (where d(Tr) has the meaning of Corollary 3.40). By Corollary 3.40, cr(Tr) =oii.(Tr) = crR.re(Tr)uop(Tr)' every component of o(Tr) intersects d(Tr)- c [p~-F(Tr)]- (Observe that A-Tr is o~ to for each A E prs- F(T)) and o p (T*) r c d(T r ) ={A n }n= 1 • Since {A n }n= 1 can be arbitrarily chosen (under the condition {A n }- = [ pra- F (T) J-, of course), it readily follows that op(T~) =~and, a fortiori, that p:-F (Tr) = ~ and min. ind (A-Tr) = 0 for all A E Ps-F (Tr). It is also clear that cr(Tr) does not have any isolated point, i. e., either H (T)={O} and cr(Tr)=~, or cr(T) is a nonempty perfect set r r + and cr(Tr) =cr ~re (T)ucr p (T) 0 r =on~re (T)u[p r (Tr )\p(T r )]. Since ps- F(T) r = pr(Tr)\p(Tr)' it follows that every component of oR.re(Tr) intersects [pr(Tr)\p(Tr)J-. In order to prove (iv), we only have to apply the arguments of (iii) to T*JHJI. (T) (Indeed, observe that the equivalence of (3.9) and (3.10) and Corollary 1.14(vi) imply that p~-F(T) =p~-F(T*)*)~ (v) is the content of Corollary 3.32 and (vi) follows from (iii), (iv) and 00

00

their proofs: If A E Ps-F(T)\pr(Tr)' then A E crR.re(Tr) c cre(T), a con tradiction. Hence Ps-F(T) c pr(Tr). A similar argument shows that Ps-F(T) c pli.(T 1 ). (vii) Assume that Hr(T) t- {0} and H1 (T) t- {0} ~ then Hr(T), HR. (T), Hr(T)$H 0 (T) and H0 (T)$HJI. (T) are infinite dimensional spaces. Suppose that A E: ps-F(T) and ind(A-T) > -m~ then A-T1 is invertible and A-T is right invertible. By decomposing H as [Hr(T)$H 0 (T)J$ H1 (T), A-T can be written as the 2 x 2 matrix A-T

[l•- [~Jll\_]= [P r). 0

A-T

R.

Oq

By Lemma 3.43(i), pis right invertible. Since

67

P =

* ) [ >..-T' 0 r >..-To '

it follows from the same lemma that >..-T0 is right invertible. Similarly, if ind(>..-T) < oo, we conclude from Lemma 3.43(ii) that >..-T0 is left invertible. Hence >.. E ps-F(T0 ) and therefore either >..-T0 is invertible, or nul(>..-T0 ) ~ 0 or nul(>..-T0 )* ~ 0. Thus, in order to complete the proof, it only remains to show that >.. ; ap (T0 )uo p (T*)*. 0 Assume that (>..-T0 )x= 0 or equivalently (>..-T)x 1 H0 (T). The invariance of Hr(T)$H 0 (T) implies that (>..-T)x E Hr(T). Since >..-T maps Hr(T) onto itself, there is some y E Hr(T) such that (>..-Tr)y= (>..-T)x. Since y- x E ker (>..-T) c Hr (T), x must be equal to 0. An analogous argument shows that nul (>..-T 0 ) * = 0 and so >..-T0 is invertible, i.e. >.. " p (T 0 ) . If either Hr(T) = {0} or H1 (T) = {0}, then the result follows by an even simpler argument. (viii) Let v" pss- F(T). If vis not an eigenvalue ofT, o then (vi) shows that we have ker (v-T) = ker (v-Tr) and (by Lemma 3 .12) we can find a neighborhood Q of v such that ker(>..-T)=ker(>..-Tr)' for all A

En.

This implies that v E p~-F(T), a contradiction. Hence, v E op(T0 ). On the other hand, the first part of the proof of (vii) indicates that v E ps-F(T0 ) and, by (vii) and Corollary 1.14 (v), v is an isolated point of a (T0 ) . Therefore, v E o0 (T 0 ) . (ix) This follows immediately from (vi) and (viii). (x) Let A be a finite subset of pss- F(T). It follows from (viii) that A c o 0 (T 0 ) and, a fortiori (since A is closed), A is a clopen sub set of a (Tr}, so that (by Riesz' decomposition theorem) To where

= [A T23J) 0

B

'

A acts on a finite dimensional space, a (A) = A and Ana (B) = ' . Therefore T admits the decomposition Tr Tl2 Tl3 Tl4 A T23 T24 0 T 0 B T34 0 0

where A= a (A)

0 c

0

TR.

pr (Tr) np 1 (T 1 ), i.e., or (Tr) noR. (A) =or (A) noR. (TR.) =a (A) n

a(B) = or(A)no 1 (CR.) = '· By Corollary 3.22 and its proof, there exist operators X, Y such that 68

and

Y) (A B12 ) [1 YJ-l 1

o c1 o 1

(A 0 ).

=

o c1

Hence

TA

A has the desired properties and A£ p~-F(TA)' where

Tr T'

A

*

= [0

B

0

0

0

From Theorem 3,38 and Lemma 3.36, we obtain the following COROLLARY 3.44. Let T (3.8); then

LIHI

£

with triangular representation

T slm Tr$T 0 eT 1 •

3.4 Correction by compact perturbation of the singular behavior of op-

erators The Weyl speatrum of T

£

LIHI is defined by

oW(T) = n{o (T+K):

K £ K (H)} •.

It is easily seen that ow(T) is the largest subset of the spectrum that is invariant under compact perturbations and that (by Theorem 1.13(v) ) ow(T) contains every complex X such that X-T is not a Fredholm operator of order 0. Furthermore, it is well-known that ow(T) actually coincides with this set, i.e., ow(T) ={X

£

a::

X-T is not a Fredholm operator of index 0}. (3.1])

(The inclusion 'c' will easily follow as a corollary of Theorem 3.48 below.) The Browder speatrum ofT is the complement of o0 (T) in the spectrum, i.e., oB (T) =a (T) \o 0 (T).

69

Clearly, a(T) ~ aB(T) aaW (T) c a ae (T) c a a tre (T) •

~

aw(T)

~

~

ae(T)

atre(T) and

'~

aaB(T)

c

PROPOSITION 3.45. Given T € L(H). there exists K E K(H) such that p:-F (T+K) = ' . a(T+K)= aB (T) and min.ind(T+K-).) = min.ind(T-).) foro all ). E Pi-F(T); moreover. K can be chosen equal. to a normal. compact operator such that

IITl'll r = max{dist[).,3ps-F(T)]:

).

s Ps-F(T)}.

€

PROOF. Let Tr T12 Tl3]Hr(T) T0 T23 H0 (T) 0 0 Tt Ht(T)

T = [0

be the triangular representation (3.8) of T and let {~n}lsn<m (0 s m s m) be an enumeration of the eigenvalues of a 0 (T). Then T0 admits the representation

* T

~

0

*

n +T n

0 •

- - - - - -

0

- - - - - -

I

-:~m-

Hm

where H1 , H2 , ••• , Hn•··· are defined so that ej~l Hj coincides with the Riesz subspace of T0 corresponding to the clopen subset {~ 1 .~ 2 , ••• , ~n} of a(T 0 ) (1 s n < m) and Hm=H 0 (T)e{elsn<m Hn}. It is not difficult to check that a (Tm) c a (T0 ) \a 0 (T0 ) = aB (T 0 ) . For each ~ n , let An E ap s- F(T) be any point such that 1~ n -A n I = dist[~ ,a(T)\p F(T)J. It is easily seen that ).n Ea... re (T). Thus, if n sT~ is the operator obtained from T0 by replacing ~n by An (1 s n < m), and T'=

[~r T~! :~!]. 0

0

T1

then it follows from Theorem 3.38 (Observe that (T') =T and (T'). = s r r "' ·T1 !) that ps-F(T') =,, a(T') =aB(T') =aB(T) and K=T'-T is a diagonal normal operators with eigenvalues ). 1 -~ 1 (with multiplicity equal to dim H1 ), ). 2 -~ 2 (with multiplicity equal to dim H2 ), ••• , ).n-~n (with 70

multiplicity equal to dim Hnl , ••• , so that

1 ~n<m}=max{dist [J.In•3Ps-F(T)J:

IIKII=max{j).n-llnl: LEMMA 3.46. LetT ~ank

exists a finite (i)

(ii) (iii)

IIKJ.III

<

r

'll

£,

L(H),

€

ope~ato~ €

'll

p:_F(T)\cr 0 (T) and E > 0. Then the~e

€

such that

K 1J

Ps-~(T+K'Il).

min.ind(A-(T+Kll.ll =min.ind().-T)k, A

If

\1

E

l~n<m}.O

€

ps-F(T)\{'11}, k=l,2, •••

~T EDT', whe~e T

p:_F(T)\{J.I} and T

aats on a fis F \1 E p s-

nite dimensional spaae, cr(T ) = {v} and \1\1 € " p r F(T'), " then \1 r s\1 (T+K ) and T+K ~ T EDT", whe~e \1 € p F (T") • 'll 'll " " s" PROOF. By Theorem 3.38, 1J

M the

ap((T 1 l*l* and denote by

E

crp(Trlucrp((T 1 )*)*. Suppose that'll

€

Riesz spectral subspace of T 0 corre-

H

sponding to cr(T 0 ) \{ll}• I f we put

'll

H

=

0

(T)9M then

Tl3 Tl4 Hr(T) T23 T24

T

M

TJ.I T 34 HJ.I

0

T1 H1 (T)

Let S ' --

(Tll T34) 0 T . R.

Since cr(T) = {ll}, we can find a ].1

(finite rank!) operator A

such that IIAII < E/2, cr(T +A) ={J.I} and nul(T +A-J.I) =1. ker(T +A-u)*, 'll

x

~

'll

].1

€

(Thus, i f x

L(H ) j.l

€

0, then x is a cyclic vector for T +A.) Define S"

(T~ +A

;J ~~

].1

(T).

By Corollary 3.22, S"- (Tll+A)EDT1 , i.e., S" has an invariant subspaceR complementary to HJ.I such that S"IR € T 1 • Let W € L(HJ.IEDH 1 (T)) be an invertible operator such that s 1 = WS"W-l= (T +A) E9T • 'll 1 E ker(Til+A-ll), 0 < llfll < e:/(2IIWII·IIW-11>, and let e £ ker(T1 1. Let s 2 =s 1 +e&f € L(HJJEDH 1 (T)), A E a:, y € H 1 (T) and z € H11 and assume that (s 2 -'lll (y+z) = 0; then (s1-~y+e+(~-~ z = 0. Hence

Let f

1J)*, llell

(T -A)Y+e = 0, (T +A-A)Z = 0. 1 j.l I f ). ~ 'll, then A ,l cr (T +A) = { 'll}. I t follows that z = 0 and therefore,

'll

(T 1 -A)y=O. Since crp ~ 0 and (T 1-J.JlY = -e ~ 0. But this is impossible because el ran(T 1-JJ). We conclude that crp (s 2 ) = jl.

71

Observe that 52=

[(T +A->.)* f@e ) (T +A 0 )H e~f Ti H~(T) , so that (S2-Al* = 0 (TP-A)* •

Since H is finite dimensional, it is not difficult to check that = ker(s 2 -~)*d(p) (where d(p) =dim HI!) and HP c v {ker(S 2 -Al*: A € Ps-F(T)}, whence we conclude that HR.(s 2 ) =HP8HR.(T) and (s 2 )R. = s 2 • Let s 3 = w- 1 s 2w and set

H~

S

=

Tr Tl2 8 13]Hr(T) [ 0 T0 1M s 23 M 0 0 s 3 HP8HR.(T)

where s 13 = (T 13 T14 ) and s 23 = (T 23 T24 > 1 then K = s- T is a finite -1

p

rank operator such that IIKPII s IIAII+IIWII·IIW ll·lleQfll < £/2+£12 = £1 Hr(S)=Hr(T) and HR.(S)=HpeHR.(T) and Si=s 3 , and H0 (S)=M and S0 =T0 1M· Now (i), (ii) and (iii) can be easily verified by using Theorem 3.38. If p £ ap(Tr), then we decompose T as

T

Tr Tl2 Tl3 Tl4 Hr(T) 0 Tp T23 T24 H p 0 0 (To)·M T34 M 0 0 0 TR. HR. (T)

where HI! is the (finite dimensional) Riesz subspace of T0 corresponding to {p} and M = H0 (T)9H p • Applying the above arguments to

(:r T~2r p

instead of S', we can find a finite rank operator K' 1.1 such that IlK' 11 < e::, and

£

L[H (T)8H l r 1.1

satisfies Hr(s 4 ) = Hr(T)8HI.I and In this case we define R14]Hr(T)8HI.I T34 M Ti HR. (T) 13 - (T where R13T23 ) and K'eo has the desired

I

It is easily seen that K

1.1

S- T "'

0

1.1

K[Hl,

72

PROPOSITION 3.47. Let T IIKII < e:: suoh that

£

L[H) and e:: > O; then there e%ists K

£

(i)

o 0 (T+K) = o 0 (T) and (T+K) IH(>.;T+K)

~

TIH(>.;T) for all >.

E

o0 (T);

(ii) p~-F (T+K) = Ps-F (T) \00 (T); (iii) min.ind(T+K->.) = min.ind(T->.) for aZZ >.

E

p~-F(T).

PROOF. Let 1:!. ={A" ps-F(T)\[o 0 (T)up(T)]: dist[A,Clps-F(T)] ~ E/2}. Clearly, 1:!. is a compact set and A= l:!.n[ps F(T)\o (T)J is finite. Appl~ so ing Lemma 3.46 a finite number of times (once for each point in A), we sati~

can find a finite rank operator K1 such that I!K 111 < E/2 and T+K 1 fies (i), (iii) and (iiI)

Now we can repeat the argument of the proof of Proposition 3.45 in order to "push" the points of p:_F(T+K 1 )\o0 (T) to Ps-F(T+K 1 J by means of a compact perturbation K2 , I!K 2 11 < E/2, which only affects the action of (T+K 1 ) 0

•

It is easily seen that K = K 1 +K 2 satisfies all our requirements.

0 THEOREM 3.48. LetT" L(H) and E > 0; then there exists K" K(H)

suah that IlK II < E+max{ dist[A, Clps-F (T)]:

>. " o 0 (T)}

andmin.ind(T+K->.)=0 foro all>." p 5 _F(T).

In partiauZar, o(T+K) ={A "

a::

>.-T is not a F1'edhoZm operato1' of index 0}

PROOF. By Propositions 3.45 and 3.47, we can find a compact tor K1 such that I!K 111 < E/2+max{dist[A, Clp 5 _F (T)]:

>.

E

oper~

o 0 (T)}

and the operator A= T+K 1 satisfies the following properties: p~-F (A) = r Ps-F (A) and min. ind (A-A) =min. ind (>.-T) for all >. " Ps-F (T). Thus, in order to complete the proof it suffices to show that there exists K2 " K(H), I!K 2 11 < r./2, such that min.ind(A+K 2 ->.J = 0 for all>. E ps-F(A) = p 5 _F(T). (Indeed, K= K1 +K 2 will obviously satisfy all our requirements.) Let 1:!. = {>." ps-F(A): dist[A,aps-F(A)] ~ E/6} and !:!. 1 ={A E /:!.: min.ind(>.-A) ~ 0}. By Proposition 3.16 we can find y,z " H such that Pker(>.-A)y ~ 0, Pker(>.-A)*z ~ 0, for all>. " /:!. 1 • Let Ca = ayez, a > O. i'le have min. ind (A+Ca ->.) =min. ind (A->.) -1, for all >. Since min.ind(>.-A) =0 for all>.

E

E

A1 , a> 0.

I:!.\A 1 , it readily follows that

73

6\6 1 that

c

p1 (A) pr(A). Therefore, there is a positive constant n 1 such 110..-A)x!l ~ n1 for all>. E p 1 (A)n(6\6 1 ) and 11<>.-A)*xll ~ n1 for all >. e pr(A)n(6\6 1 ),

for all x -.: H, II x II = 1. Thus, if 0 < a < min{e:/6,n 1 /2}, then IICall < e:/6 and min.ind(A+Ca1.) =0 for all>. e 6\6 1 • Now we can apply the same arguement to A+C (l , etc. After finitely many steps we obtain a finite rank operator c 1 such that 11c 1 11 < e:/6, min.ind(A+C 1->.) = 0 for all>.

E

6 and o(A+C 1 )

c

interiora(A)e:/ 6 • (3.12)

Let {Qj}l~n<m (0 s m s w) be an enumeration of the components of a(A+C 1 )np s- F(A) such that min.ind(>.-A) ~ 0 for all >. e nJ.• (Clearly, u. Q. c [ap F(A)] 13 .) A repetition of our previous argument shows J J se: that there exists a positive constant n < min{e:/6,n 1/2} such that the relations (3.12) remain true if A+C 1 is replaced by A+C 1 +B for any B in L (H) such that liB II s n • r If >. 1 e Ps-F(A+c 1 >no 1 , then the arguments used for the construction of C can be repeated here in order to find a finite rank operaa tor c 2 , 1 , II c 2 , 1 11 < n/2, such that min. ind (A+C 1 +C 2 , 1 ->. 1 ) = 0. By induction, we can construct finite rank operators c 2 1 , c 2 2 , ••• ,c 2 n•···• n

CO

I

I

IIC2,n11 < n/2 (n=1,2, ••• ) such tha; C2=~n=l c2,n min.ind(A+C 1 +c 2->.) =0 for all>. e Ps-F(A+C 1 +c 2 ), 6

c

r

Ps-F(A+C 1+c 2 )

and

cr(A+C 1 +c 2 )

Since max{dist[A,aps-F(A)]:

c

€

I

K(H), IIC211 < nr

interior cr(A)e:/ 6 • s

>. e Ps-F(A+C 1 +c 2 )} < e:/6,

we can apply the arguments of Proposition 3.45 in order to find a compact normal operator c 3 , IIC 3 !1 < e:/6, such that min.ind(A+C 1 +c 2+c 3->.)=0 for all>.

E

Ps-F(A). 0

An

operator T e L(H) such that min.ind(>.-T)

0 for all >. e Ps-F(T)

(3.13)

will be called a smooth operator.

3.5 Apostol-Foiaf-Voiculescu's theorem on normal restrictions of compact perturbations of operators 74

Given Tin L(H) we cannot expect, a priori, to find an infinite dimensional invariant subspace M such that TIM is normal. An important approximation argument (due to C. Apostol, C. Foia~ and D. Voiculescu) asserts that some arbitrarily small compact perturbation of T always has that property. We shall need some extra notation to make it more clear.

3.5.1 Schatten p-classes Recall that if K is compact, then (K*K)~ can be written as (K*K)~ A3

In:l An en@en with respect to some ONB {en}n:l' where ;~. 1 = IIKII 2! A2 2! ···~An 2! An+l 2! ••• ~ 0 and An+ 0 (n + ~>. The Schatten p-cZass cP(H) of compact operators is defined by

cP(H)

= {K

E

K(H):

2!

In:l A~ < oo}, 0 < p < oo,

and C (H) = K(H). The reader is referred to [77], [107], [183] for the properties of these ideals of operators. In particular, we have: (a) If K E cP(H), then K* E cP(H): (b) cP(H) (C 00 (H)) is a Banach space under the norm IKI = IK*I 00

00

p 1/p

p

p

(with respect to some ONB {fn}n:l of H. The result is independent of the particular ONB.): (e) c 1 (H) = the ideal of trace class operators is isometrically isomorphic to the dual K(H)t of K(H). This isomorphism is defined by C1 (H) :;; K ++4lK E K (H) t , where ~K(X)

= tr(KX), X E

K{H):

(f) c 7 (H)t = K(H)tt is isometrically isomorphic to L(H). This isomorphism is defined by L(H) :;; A++ ~A E C1 (H) t , where

~A(K) = tr(AK), K (g)

If A

E

E

c1 (H):

L(H) is a non-negative hermitian operator, then tr(A)

=

In:l (0 s tr(A) s

oo)

is well-defined and the result is independent of the particular ONB {fn}n:l of H.

75

(h) (C 2 (H), 1-1 2 > (=the ideal of Hilbert-Schmidt operators) is a Hilbert space. If A, B" c2 (H), then AB*" c 1 (H) and the inner product of c2 (H) is.given by = tr(AB*).

3.5.2 Normal restrictions With the above notation in mind, we have the following THEOREM 3.49. Given T" L(HI. a nonempty compact subset rt of ote E > o. there ezists an infinite dimensional subspace H c H 1 E and KE ,;: C (H) such that jK E 11 < E (so that IlK EII < eJ. (T-K E)H E c HE• T = (T-K ) IH is a diagonal normal operator of uniform infinite multi E E E plicity and o(TE) = oe(TE) rt. (T) and

PROOF. Let {An}n:l be a sequence of complex numbers such that {An}-= r t and (3.14) card{m: Am for each n= 1,2,... It is obvious that rt c ote(T) cot (T). Therefore, we can find a unit vector e 1 " H such that II
_ [Xl+tll

T -

1

T21 then [til]: v{e 1 } T21

+

T12)el 1

;

T22 {el}.i.

{e 1 }.L is a rank-one operator,

11[:~:]11

E/4 and <

Ti 2 :{e1 }.1. + V{e 1 } is also a rank-one operator. Clearly, we can find a finite dimensional subspace M1 => V{e 1 } such that IIT~ 1 1Mill < E/4. Let R1 be the orthogonal projection of H onto Mi and define c 1 by

0] [0

Ti2] Rl. cl = [til + 0 0 0 T21 It is easily seen that jc 1 11 <

l(:~~) j

1+1Ti21Mil 1 < E/2 and

where N1 = M1 n{e 1 }.1.. Clearly, o (T 1 ) = o (TY. Thus, we can apply the same argument te te to T1 in order to obtain a unit vector e 2 " M1 , a trace class operator c 2 such that 1c2 11 < £!4 and a finite dimensional subspace N2 ~ V{e 1 , 76

e 2 } such that V{el,e2} N2e V{e 1 ,e 2 }. A2 N2

E2

D2

By induction, we can find orthonormal vectors e 1 ,e 2 , .•• en•···• and trace class operators c 1 ,c 2 , .•. ,cn, ••• such that KE = Ln:l en € Cl (H), IKEil s Ln:l ICnll < E and

T-KE=[~E :):~ where HE = V{en}n:l and TE Ln:l An enQen (strong limit of the partial sums). The condition (3.14) guarantees that T has the desired properties. 0 L(H), nonempty compact subsets r~ c a~e (T) and rr c are(T), E > 0 and diagonal normal operators N~ and Nr of uniform infinite multiplicity such that r~ c a(N 1 ) = ae(N 1 ) c Cr 1 >E and rr c a(N r ) = o r (Nr) c Cr r ) E , there exists an operator L E L(H), COROLLARY 3.50. Given T

L '

€

[~·

:

such that liT- Lll < 2E. PROOF. Applying the inductive argument of the proof of Theorem 3. alternatively toT and toT*, we can find K € c 1 (H), IIKII s IKI 1 < r/2 such that 49

T-K

Mi * [0 A 0

0

where M1 and Mr are diagonal normal operators of uniform infinite multiplicity such that a (M1 ) = r 1 and o (Mr) = r r' resp •. Since r~ c o(N 1 ) =oe(Ni) c (fi)E and fr c o(Nr) =oe(Nr) c (fr)E, and N1 and Nr are diagonal normal operators of uniform infinite multiplicity, we can easily "spread" the spectral measures of M1 and Mr(CO!!, centrated in op(M1 ) and ap(Mr), resp.) in order to obtain normal oper~ tors NR_ and N~ such that NR_ "' NR., N~"' Nr, IIM1 - NR_II < 3E/2 and IIMr-N;II < 3E/2. The operator L obtained by replacing MR. and Mr by the normal operators NR_ and N;, respectively, satisfies all our requirements.o 77

3.5.3 Density of sets of operators with bad properties We shall say that a certain property (P) a Hilbert space H) is a bad property if:

(of operators acting on

(I)

If A has the property (P) , then n+BA has the property (P) for all a E ~ and all 6 ~ O, (II) If A has the property (P) and T - A, then T has the property (P), and (III) If A has the property (P) and cr(A)ncr(B) = ~. then A$8 has the property (P) • Examples of "bad properties" are frequent in the literature: nam~ ly, (1) T is not cyclic, (2) The spectrum of T is disconnected (or cr(T) has infinitely many components, or c components, where c is the power of the continuum), (3) crp(T) has nonempty interior, (4) The commutant of T is not abelian, (5) T is not similar to a normal operator, (6) T is not algebraic, etc, are examples of properties satisfying (I), (II) and (III). THEORm~

atoP A

~ith

3.51. If (P) is a bad pPopePty and thePe exists some the property (P), then the set {TE

L(H):

ope~

Tsatisfies (P)}

is dense in L(H).

PROOF. Let~ E acre(T)n3p(T). Given£> 0, we can find A E p(T) such that IA-~1 < £/2. By Corollary 3.50, we can find L E L(H) such that IIT-LII < £/2 and L =

[~

:)

with respect to some decomposition H = M$M~, where M and M~ are infinite dimensional subspaces and A I cr(B). (Observe that~ E acre(T) c crtre(T) and take r 1 = {A}ucr 1 e(T), rr = crre(T).) Let n = dist[A,cr(B)]. Clearly, 0 < n < £/2: thus, if 0 < 6 < n/(1-IJIAII>· then it follows from Corollary 3.22 that

R)

A+6A M= [ O B

-

(A+6A)$B.

Now (I), (II) and (III) imply that M has the property (P). Since II T-MII s II T-LII+~IAII < £, and £ can be chosen arbitrarily small, it read ily follows that T E {S E L(H): S satisfies (P)} • 0 An

78

inductive repetition of the same argument yields the following

COROLLARY 3.52. If {(Pn)}n:l is a denumerab~e set of bad properties, then (P) = un:l (Pn) is a bad property. FurthermoPe, if thePe ezist operators A1 , A2 , ••• , An•··· in L(H) such that An satisfies (Pn) (foP each n = 1,2, ••• ), then {T E L(H): T satisfies (P)} is dense in L (H).

3.6 Notes and remarks The operators TAB were first systematically studied by M. Rosenblum in [174]. Rosenblum's theorem has been proved there, except that the equality a (TAB) =a (A) -a (B) is replaced by the weaker statement a(TAB) c a(A)- cr(B). The fact that this inclusion is actually an equa! ity is due to D. C. Kleinecke (see [l57,IntroductionJ). Corollary 3.22, for the special case when cr(A)no(B) = ,, is usually called Rosenblum's corollary (see, e.g., [17l,Chapter 0]}. In [69], c. Davis and P. Rose~ thal re-analyzed Rosenblum's results and proved that cr 6 (TAB)= crr(A)oR.(B) and cr 11 (TAB) = crR.(A) -crr(B) and extended Rosenblum's corollary to Corollary 3.22. Their arguments include the use of the BerberianQuigley construction (Lemma 3.3; see [45], [l72,p.25J) in order to prove Theorem 3.4 as given here. Finally, L. A. Fialkow modified Rose~ blum's original argument (by using analytic one-sided inverses instead of resolvents) in order to show thatcr 6 CTAB) =crr(TAB) and cr'lf(TAB) = aR. (TAB) [87]. More precisely, Fialkow's results read as follows THEOREM 3.53. Let A, B

E

L(H); then the

fo~~owing

aPe equivalent

for TAB: {i) TAB is surjective; {ii) crr(A)naR.(B) = ,; {iii) ran TAB contains the minimal idea~ F(H); {iv) TAB is Pight invePtible in L(L(H)). Clear~y, TAB maps every norm idea~ {in the sense of R. Schatten [183]) J c K(H) into itse~hand each of the above conditions is also equivalent to each of the following ones {v) TABIJ is suPjective foP some norm ideal J; {vi) TABIJ is surjective for every norm ideal J; {vii) TABIJ is right invePtible in L(J) for some norm ideal J; (viii) TABIJ is right invertible foP evePy noPm ideal]. MoPeover, in this case ker TAB and ran TAB aPe complementary subspaces of L(H) and ker TABIJ and ran TABIJ aPe complementaPy subspaces of J for each norm ideal ).

79

THEOREM 3.54. The following are equivalent for TAB: (i) TAB is bounded below; (ii) at (A)nor(B) = ~; (iii) TAB!F 1 !HI is bounded below (where F 1 (HI is the set of all rank~one operators); (iv) TAB is left invertible in L(L(HII; (v) TABIJ is bounded below for some norm ideal J; (Vi) TABIJ is bounded below for every norm ideal ]; (vii) TABIJ is left invertible for some norm ideal J; (viii) TABIJ is left invertible for every norm ideal ]. Moreover, in this case ran TAB and ker TAB (ran TABIJ and ker TABIJJ are complementary subspaces of L(HI (of J for every norm ideal ], respectively). Furthermore, Fialkow also considered the operator TAB induced by TAB in the Calkin algebra, defined by ~AB(X) =AX-XB. His results for these operators can be summed up as follows (see [59], [86], [87], [99J, [89J, [9oJ).

THEOREM 3.55. The following are equivalent for TAB: (i) TAB is surjective; (ii) ~AB has dense range; (iii) ore(A)note(B) = ~; (iv) TAB is right invertible in L(A(H)). In this case, a right inverse for TAB is given by ~

-

-1

-

cJ>(Y) = 2ni Ian RA(>.)YLB(A.l d>.. where n is a Cauchy domain such that are(A) c n and ate(B)nn = ~. RA (>.) is an analytic right inverse of (>.-~) defined on a neighborhood of ~\n and Ls<>-> is an analytic left inverse of (A.-B) defined on a neighborhood of Q-. Moreover, ker ~~g and ran tAB are complementary subspaces of A(H). THEOREM 3.56. The following are equivalent for t~B: (i) ~AB is bounded below; (ii) ote(A)nore(B) = ~; (iii) TAB has a left inverse in L(A(H)J. In this case, a left in verse for TAB is given by

$
=

2!i {an Lx
d>..

where n is a Cauchy domain such that ate (A) c n and are (B) n n- = ~. ~ (>.) is an analytic left inverse of (A.-~) defined on a netghborhood of 80

~\G and Rs(A) is an anaLytic right inverse of (A-B) defined on a neig~ borhood of Q-. Moreover, ker TAB and ran TAB are compLementary subspaces of A(H).

THEOREM 3.57. (i) TAB is a Fredholm operator in L(L(H) l if and onLy if oe(A)no(B) =o(A)noe(B) =~; oe(TAB) =[o(A) -oe(B)Ju[oe(A)-o(B)]. (iiJ 'AB is a semi-FredhoLm operator if and only if o 1 e(A)no~B) o 1 (A)ncrre(B) =~or ore(A)ncrt(B) =or(A)ncrte(B) =16. Moreover, Fialkow also proved the following asymptotic version of Theorem 3.19(i) [90,Theorem 1.1]. THEOREM 3.58. The foLLowing are equivaLent (i) ran 'AB is dense in L(H); (iiJ 1) ore(A)note(B) = 16, and 1 2) There exists no nonzero X e C (H) such that BX = XA; (iii) Given Y e L(H) and E > 0, there exists X e L(H) such that tAB(X)- Y e K(H) and II
(~ ~) (acting on H

= H1eH 2 J

and

A$8

are asymptoticaLLy simiLar.

The construction of one-sided resolvents (Theorem 3.8 and Corollary 3.9) is due to c. Apostol and K. Clancey [15] (see also [14]). The proof strongly depends on a previous deep result of C. Apostol (Proposition 3.16 [lO]).This result extend to the case of an arbitrary compact set o the one obtained in [26] for the case when o is simply connected. The results of Section 3.1.6 and 3.1.7 have been taken from J. G. Stampfli's article [189]. Indeed, Stampfli proved that if Cis an irreducible C*-subalgebra with identity of L(H) (in particular, if C = l(H)), then II!STII = 2 min{IIT-AII: A e: ~}for each inner derivation 1ST inC (and the analogous result for the operators 'AB' A, 8 e C). Exte~ sions of Stampfli's results to derivations on certain C*-algebras and 81

W*-algebras were given by c. Apostol and L. Szid6 [31]. For the case of an inner derivation in the Calkin algebra, it is known that

{!IT-

THEOREM 3.60 (C.-K. Fong [99]). If T E L(H), then ilo:rll = 2 min A. II: A E a:}. Moreover, there exists an orthogonal projection X e

L(HI such that llo:rll = llo:r(2X-llll·

The interesbed reader will find a very detailed analysis of the spectra, left spectra and right spectra of large families of operators "somehow related" with the operators TAB (including tensor products of operators; the operator X~ Lj~l AjXBj, where {A1 ,A 2 , ••• ,An} and {B 1 , B2 , ••• ,Bn} are separately commuting families of operators in L (H),-etc), in [55], [79], [91],[92], [100], [187] and, very especially, in several papers by R. E. Harte [127], [128]. Very recently, the spectrum of the operator X ~ Lj~l AjXBj has been completely determined by R. Curto [68]. In [175], Gian-Carlo Rota proved that the backward shift of infinite multiplicity is a "universal model" for all operators T in L (H) such that sp(T) < 1 (see also [130,Lecture X]). Two (very close) universal models for an operator T with arbitrary spectrum and the spectrum of the model equal to a suitable neighborhood of o(T) were independently obtained by D. A. Herrero [142] and D. Voiculescu [l94,Prop£ sition 2.1]. The model given here (Theorem 3.33) is the one due to Vo! culescu. J. A. Ball obtained analogous models for commuting n-tuples of operators [41] and F. Gilfeather used Rota's original construction and an iterative argument to give a different (and independent) proof of Herrero's corollary (Corollary 3.35); moreover, Gilfeather [105] -1 actually showed that the invertible operator W such that T' =WTW satisfies the conditions of that corollary can be constructed in the C*alqebra C*(T) generated by T and 1. Theorem 3.38 (except for those details concerning the structures of Tr and Tt given by (iii) and (iv) ), Proposition 3. 47 and Theorem 3. 48 are due to C. Apostol [10]. A simplified version of Apostol's tria~ gular representation has been obtained by R. Bouldin in [52]. The version given here (Theorem 3.38) combines the arguments of [10], [52] and some improvements of D. A. Herrero (items (iii) and (iv)), based on Corollary 3.40 which, in turn, is a particular case of [144] (Theorem 1 or Theorem 2). However, the proof of Corollary 3.40(i) given here is not the same as the one given in [144]. This new proof has been based on Lemma 3.39 (due toR. G. Douglas and C. M. Pearcy [75]) which will play an important role in the characterization of quasitriangularity (see Chapter VI below). 82

In [190], J. G. Stampfli proved that, given T E LIHI there exists K € K(H) such that a(T+K) = aw(T) (without any estimate on II Kill; moreover (as observed by N. Salinas; see Remark on p.l74 of [190]), the fact that a(T+K) = a 8 (T) for some Kin LIHI is implicitly contained in that paper. A more general theorem, containing both results, has been obtained by C. Apostol, C. M. Pearcy and N. Salinas in [28] (see also Section 4.3 of Chapter IV below.) Theorem 3.49 is Theorem 2.2 of [23]. This clever approximation ar gument is one of the most important ingredients for the theorem on spectral characterization of quasitriangularity of c. Apostol, c. Foia~ and D. Voiculescu (obtained in that paper; see Chapter VI below). The result can be regarded as an asymmetric version of the analogous r~ sult of C. M. Pearcy and N. Salinas for semi-normal operators [162]. These results (and many other related ones scattered thorugh the lite~ ature; see, e.g., (3], [139], [163], [179] and, very especially, [56] and [57]) are based on an argument due to F. Wolf [199]. (See also the partially related articles [71],[95] and [96].) Corollary 3.50 appears in [27]. Theorem 3.51 is a mild generalization of Herrero's theorem for non-cyclic operators. Indeed, the same result is actually true for the Banach spaces tP (1 ~ p < ~J and c 0 (see [146,Theorem 4]). The underlying idea is contained in a paper of N. Salinas [178].

83

4 Two results borrowed from the theory of C* -algebras

The similarity orbit of T U(T)

E

L(H) contains the unitary orbit of T

= {UTU*: u

E

U(H)}

(where U (H) is the unitary group of L (H) ) , which plays a very important role in approximation problems. We shall see later that, since U(T) c S(T)-, a large amount of information about S(T) can be obtained from the structure of U(T)-. This structure is closely related with the C*-algebra C*(T) generated by T and the identity 1 E L(H). I~st of those results related with closure of unitary orbits (and many others) connected with approximation problems can be derivedfrom two deep results of the theory of C*-algebras. These results (the Brown-Douglas-Fillmore theorem and Voiculescu's theorem) will be stated here without proofs (suitable references will be given) • Several consequences of these two results will be completely developed her~

4.1 Essentially normal operators An operator T E L(H) will be called essentially normal if T is a normal element of the Calkin algebra or, equivalently, if the self-~ mutator [T* ,T] = T*T- TT* is compact. One way in which these operators arise is as compact perturbations of normal operators. The interestin essentially normal operators is due to two facts: a) Not all of them arise in that fashion; b) There is a rich family of operators of the form "normal+compact" whose structure is not obvious from their analytic expression. (This is, by far, the most important fact from our viewpoint.)

4.1.1 Brawn-Douglas-Fillmore theorem Essentially normal operators have been completely classified in the famous article [60] of L. G. Brown, R. G. Douglas and P. A. Fill84

more. Their main results will be quoted here in the most convenient form for our purposes. (The reader is referred to the above cited arti cle for proofs.) It is completely apparent from the definition that if T is essentially normal, then cr... e (T) = cr re (T) and therefore _,. < ind (A-T) < oo for all A £ Ps-F (T) , p s-F (T) = Pp (T) and cr R.re (T) = cr e (T) • THEOREl-i 4.1 (BDF theorem). If T1 and T 2 a1'e essentiaZly no1'maZ ope1'ato1's on H, then a neaessa1'y and sufficient condition that T1 be unita1'iZy equivaZent to some aompaat perturbation of T2 is that cre(T1) cre(T 2 J and ind().-T 1 ) =ind().-T 2 ) for aZZ). I cre(T 1 J. There is a special case of particular interest. COROLLARY 4.2 (BDF corollary). (i) If T is an essentiaZZy normaZ operator on H suah that ind ().- T) = 0 fo1' aZ Z A E ps-F (T) • then T is in NoJt(H)

+ K(H) =

{N+K:

is normaZ,

N

K

is aompaat}

In partiauZa1', eve1'y essentiaZZy normaZ operator T suah that oe(T) does not disaonneat the p Z.ane is in Noll ( Hl + K (H) • (iii) NOJt(H) + K(H) is a aZosed subset of L(H). (ii)

4.1.2 Berger-Shaw trace

inequal~

In order to make the above results useful for approximation purposes, a large family of essentially normal operators will be exhibited. We shall need some extra notation. Given T E L(H), let A(T) (= the weak closure of the polynomials in 1 and T) , Aa. (T) ( = the weak closure of the rational functions of T with poles outside o(T)), A' (T) ={A E L(H): AT=TA} (=the commutant ofT) and A." (T) = {B E L (H) : BA = AB for all A in A' (T)} (= the double commutant of T) be the four (weakly closed) sublagebras of L(H) naturally associated with T. It is apparent that A(T) c Aa.(T) c A"(T) c A' (T). The muZtipZiaity ~(A) of a (weakly closed, identity containing) subalgebra A of L(H) is defined by ~(A)

= inf{card(r):

H = V[Ax: A

E

A, x

E

rJ, r

c

H}

(4 .1)

(1 ~~(A) s ~>. Tis n-muZtiayalia (n-rationaZZy multiayaZ.ic) if 1dA(T)) =n (~(Aa.(T)) =n, resp.). An oper,ator T E L(H) is called hypono1'maZ if [T*,TJ is a positive semi-definite hermitian operator; T is called subnormal. if there ex-

85

ists a normal operator N acting on a Hilbert space R containing H such that His invariant under Nand T=NIH. It is not difficult to see that (Normal) => (Subnormal) => (Hyponormal) • None of these implications can be reversed; e. g., M+(r) of (3.2) are subnormal, but not normal. It readily follows from (g) of Section 3.5.1 that

the operators

tr[T*,T]=~n:l dT*T-TT*)fn,fn>=~n:l (jjTfnjl2 -jjT*fnlr>

(4.2)

(0 s tr[T*,TJ s ~) is well-defined for all hyponormal T. The main result of [50 J asserts that if Tis a hyponormal n-rationally multicyclic operator, then [T*,TJ e c7 (H); moreover, j[T*,TJj 1 =tr[T*,TJ s (n/-rr) meas 2 (a(T)),

( 4. 3)

where meas 2 denotes the planar Lebesgue measure. In order to prove this inequ~lity, we shall need an auxiliary result. LEMMA 4. 3. If ll (A a (T)) = n, then n s ll (A (T)) s n+l. PROOF. It is trivial that }.l(A(T)) ~ }.l(Aa(T)) = n. Assume that H = Vbxj: A e Aa(T), j = 1,2, .•• ,n} for a suitable finite family {x 1 ,x 2 , ••. ,xn} c Hand let {Akj}k:l (j=l,2, ••. ,n) ben pairwise disjoint dense subsets of p(T). If the positive constants {ck.}k~l ._1 2 , ••• ,n are suitably chosen, then l.J~;,~ 1 l~;k~lllck.(Ak·-T)-l J - ,J-, .J J xjll <~and therefore x

o

= L.j=ll.k=l ~; n ~; ""

( ) -1 ckj Akj-T xj

is a well-defined element of H. Now it is easily seen that, for suitably chosen polynomials pll,i' 1

n

2

Pll,i'pll,i'"""'pll,i' we have ll
-1

n

-1

so that (A 11 -T) x 1 e M=V{Axj: the same argument shows that (Ak.-T) J

-1

j

x 1 - {Pll,i (T)x0 +~j=l Pn,i (T)xj} 11

+

o

(i

+ <»),

A e A(T), j=O,l,2, ••• ,n}; moreover,

.

x. eM for all J=l,2, ••• ,n, and all k=l,2, ••• J

Hence (by Runge's approximation theorem), r(T)xj e M for all j l,2, ••• ,n and for all rational functions r with poles outside cr(T). It readily follows that M is invariant under Aa(T). Since M contains the vectors x 1 ,x 2 , ••• ,xn' we conclude that M=H. Hence, ll(A(T)) s card{x0 ,x 1 ,x 2 , .•• ,xn} = n+l. 0 THEOREM 4.4 (Berger-Shaw trace inequality). If T e L(H) is a hyp£ 86

normal n-rationally multiayalic operator, then [T*,TJ J[T*,TJI 1 satisfies the inequality (4.3). In particular, T is essentially normal.

£

C7 !HI and

PROOF. It will be convenient to split the proof in several steps. (I)

Assume that A is hyponormal and p-multicylic, i.e., k k k co H = V{A x 1 ,A x 2 , ••• ,A xp}k=O

for suitably chosen p vectors x 1 , x 2 , .•• , xp in H. Let {fk}k:l be the ONB of H obtained by means of the Gram-Schmidt process of orthonormalization applied to the sequence x 1 ,x 2 , .•• ,x ,Ax1 , 2 2 2 k k k p AX 2 , ••• ,Axp,A x 1 ,A x 2 , ••• ,A xp, ••• ,A x 1 ,A x 2 , ••• ,A xp•··· • It is easi ly seen that A admits the matrix representation all a21 a31 a41

A=

a12

a22 a32 a42

al3 a23 a33 a43

al4 a24 i34 a44

. ap2 apl ap3 ap+l,l ap+l,2 ap+l,3 ap+2,2 ap+2,3 ap+3,3 0

ap4 ap+l,4 ap+2,4 ap+3,4 ap+4,4

fp • fp+l fp+2 fp+3 fp+4

Hence, given m > p+2, we have (akr is defined equal to 0 for all

r s 0) ~ m 2 * 2 _ ~ m k+p 2 co 2 0 s Lk=l
s , m , k+pl 2 , oo 2 m k+p 2 Lk=tn-p+l
subnormal operator (e0 (X) = 1 is a cyclic vector forT~) and a(T~) = tji(D)-=0- [73]. I f {en}n:o is the canonical ONB of H2 (ao) (defined by en(;\)= Xn, n=O,l,2, ••• ), then ~DO 2 2 ICT¢,T4>] I 1 =tr[T;,Tcf>J=Ln=O(IITtjlenll -IIT$en11)

=

~n:o ( ll~m:o

~

DO

= Lm=l ml cml

IIIm~O

cmx n+mll 2 -

2

em An-mll 2 >

= (l/7T) f 0 14>' (X) I

2

dA = (l/7T)meas 2 Wl

(where dA denotes the planar Lebesgue measure) • Hence, lcT;,Tq,J1 1 = (l/1T)meas 2 W> = (l/7T)meas 2 [a(Tifi)J. Given

(III)

£

> 0, we can find finitely many simply connected

analytic Cauchy domains

o1 , o 2 , ••• , Oq such that

1) a(T)u(ur~l Or)- c 6. ={X: 2) 0 -nO - = {6, i f r '1- s;

r

lXI s IITIIl;

s

3) cr(T)n(ur!h Or)-={6; and 4) meas 2 (6.\[cr(T)u(ur~l Or)-]) < £. If cf>r:D + Or is a conformal mapping (r = 1,2, ••• ,q), k


1 and

A = [T$(61 q T ) (n)] (k) k r=l
L

E

Aa(T), j=l,2, ••• ,n}, it is not difficult to

check that

(H61[61r~l

H2 (3D) (n) ]) (k) =

V{Bk[yj61(61r~l e~~~)]:

i=l,2, ••• ,nk, 15 j so that

~

(IV) to A =

~,

nand j = i

5

Bk .: Aa(Ak),

(mod n)},

is nk-rationally multicyclic. By Lemma 4.3, Ak is (nk+l)-multicyclic. Thus, applying (I)

we obtain

ICA,,~JI 1 =ki[T*,TJI 1 +nk~r~ 1 1CT$ ,Tcf> Jl 1 r

5

(nk+lli1Akll 2 = (nk+l>IITII 2

r

= [ (nk+l)/7T]meas 2 (6.).

From this inequality, (II) and 1) - 4) of (III), it follows that ki[T*,TJI 1 < [(nk+l)/7THmeas 2 [cr(T)J+~r~l meas 2 COr)+d

-(nk/7T)Lr~l meas 2 (nr) = [(nk+l)/1TJmeas 2 Ca('I') J

88

Hence, I[T*,TJI 1

<

[(n+l/k)/irJmeas 2 [a(T)J-tjiTII2/k+nE.

Since k can be taken arbitrarily large and E can be chosen arbitrarily small, we conclude that I [T*,TJI 1 s (n/7r)meas 2 [a(T) ].

0

4.1.3 Examples of essentially normal operators The operator M+(r) of (3.2) is subnormal and rationally cyclic (the function e 0 (>.) = 1 being a cyclic vector for A4 (M+(f)).) Thus, by Theorem 4.4, M+(r) is an essentially normal operator. On the other hand, since 0 = [M(r)*,M(r)] = [

[M+ (r) * ,M+ ( r ) ] - ZZ* Z*M (r) - M (r) Z*

+

-

M+(r)*Z- ZM_(r)* JH 2 (r) Z*Z+[l1_ (r) * ,M_ (r)] H2 (r)l'

it readily follows from the same proposition that [M+(r) * ,M+ (f)]= ZZ* E C 1 (H 2 (r)). A fortiori, [M_(r)*,M_(r}]=-Z*Z € c 1 (H2(r)l), so that M_(r) is also essentially normal and Z is a Hilbert-Schmidt operator. (Indeed, Z is a finite rank operator.) From Corollary 4.2, we obtain the following COROLLARY 4.5. Let Q be an analytic Cauchy domain with boundary r and let M be a normal operator such that a (Ill) = ae (M) = Q-. Then MIBM+ (r) and .f.11BM_ (r) are unitar•iZy equivalent to compact perturbations

= an

of M.

A different kind of examples can be constructed as follows: Let be a nonempty bounded open set such that Q =interior Q- and let N(Q) be the "multiplication by >." operator acting on L2 (Q,dA). The subspace A2 (G) spanned by the rational functions with poles outside n- is inva~ iant under N(Q). By N+(Q) and N_(Q) we shall denote the restriction of N(Q) to A2 (n) and its compression to L2 (n)eA 2 (n), respectively, i.e., Q

N(Q) = (

N+(Q)

G )A2 (0)

0

N_(Q) A2 (Q)l.

(4 .4)

As in the case when U+ (Q) (Section 3. 2) , A2 (Q) is a reproducing kernel space and it can be realized as a space of analytic functions defined of Q: furthermore, 89

cr(N(n)) =cre(N(n)) =cr(N+(O)) =cr(N_(rl)) =cre(N_(O)) =n-,

o and ind <11- N+ (O)) = -1 for all ll € n. N(Q) is a normal operator, N+(n) is a rationally cyclic subnormal operator (Hence, N+(fl) is an essentially normal operator, by Theorem 4.4) and llr(N(fl)lll =llrCN+(O))II =max {lr(A) 1: A E an} for each rational function r with poles outside n-. ae (N+ Cnl) = an, nul <11- N+ Cnl) =

(4. 5)

Now it is clear that N+(O*)* is also an essentially normal operator such that nul Cll- N+ (O*) *) = ind C11- N+ Cn*) *) = 1 for all ].J € o and { llr (N+ (0*) *) II= max{ Ir (A: I : A E an} for each rational function r with poles outside n • Indeed, it can be actually shown that (see [SO], [Sl]) tr[N+(O)*,N+(Q)] s (l/n)meas 2 (0)

(not Q

!) •

Now it follows from the BDF theorem that COROLLARY 4.6. If N is a normal. operator suah that cr(N)

an, then N+(fl)$N+(O*)* is unitarily equivalent to a aompaat tion of N.

(=

cre(N))

perturb~

COROLLARY 4.7. If Tis an essentially normal. operator, nn={A € 11:: ind(A- T) = n} (1 s In I < oo) and N is an arbitrary normal. operator suah that cr(N) =cre(N) =cre(T), then Tis unitarily equivalent to a aompaat pertubation of L

= Ne{en:l [N+ Con*)* J Cnl }e{en:l N+ Cn_nl Cnl}

. d as t h e 0 operator ( wh ere N+ ( n) must b e ~nterprete

.

aat~ng

on

,.o . ~ ~n

the

aase when n = [6.)

Indeed, the conditions llr (N+ (fl) >II = llr (N+ Co*) *lll =max{ I r (A) I : X € iHl}imply, in particular, that IIEll-N+(O)J- 1 11= IIC].J-N+(Q*)*J- 1 11=1/dist [ll,rl-] and this suffices to guarantee that L is well-defined. Since tr[N+(On)*,N+(On)J s (l/~)meas 2 (nn), it is not difficult to check that L is essentially normal.

4.1.4 An application to approximation problems It is easily seen that if N is a normal operator, then so is ever 90

y operator M in the closure U(N)- of U(N). In this case, U(N)- admits a very simple characterization. The following result is straightforward.

LEMMA 4.8. If N

E

={ME L(H):

U(N)

L(H) is normal, then M is normal, a(M) =a(N),ae(M) =ae(N) and

nul(A-M) =nul(A-N)

for aU A£ a(N)\ae(N)}.

We have the following PROPOSITION 4.9. If M is a normal operator suah that cr(M) is a

perfect set, K is compact and cr(M+K) = a(M), then M

; M+K. s1.m

PROOF. I t is easily seen that ae (M) = ae (M+K) =a (U) • Let N be a diagonal normal operator such that a(N) = a(M), i.e., Ne =A e with oo oo n nn respect to some ONB {en}n=l and {An}n=l is dense in a(M). Assume,mor~ over, that all the eigenvalues of N have infinite multiplicity. By Lemma 4. 8,

U(M)- = U(N)-; furthermore, the BDF theorem implies that M

is unitarily equivalent to a compact perturbation of N. In other words, N can be chosen s.o that M- N is compact, and therefore M + K = N+C, where C = !M- N) +K E K (H). Clearly, a (N+C) =a (M+K) = a(M) = a(N). Since N " U(M) suf~icient

c

S(M)-, i t follows that

K s!m N. Hence, i t is

to show that N s!m N+C. Let Pn be the orthogonal projection

of H onto

y{e 1 ,e 2 , ••• ,en}; then IIC-PnCPnll + 0 (n +oo) (recall is compact), and therefore IICN+C)- [{1-P )N(l-P )+P (N+C)P Jll= n n n n CP II + 0 ( n + oo) • n By using this observation and the upper semicontinuity of spectrum (Corollary 1.2(ii)), given e: > 0 we can find m=m(e:) that lie- PmCPmll < e:/2 and max{disHA,a(N) J:

that C 1""-P ~ n the

such

A E a[Pm(N+C)Pm]} < e:/2.

(since a(N) is a perfect set) we can find an operator Fm E L(ran

Thus,

Pm) such that IIFm- Pm(N+C)Pmlran Pmll < e:/2 and a(Fm) is a set of m · t po1.nts · · . d J.StJ.nc ~l m ,~ 2 m , ••• ,~mm o f t h e po1.nt spectrum o f N. Let FE F(H) be the operator defined by:

Flran P =F

m

m

and

Flran (1-Pm) =0. Then II!N+C)- (N+Flll = llc-FII s IIC-PmCPnJ~!Fm-Pm(N+C) Pm11+11Fm- (F+N) Iran P~ll < e:, N+F

o:

N$Fm and N8Fm-N, whence the result

follows.

0

COROLLARY 4.10. If T E L(H),

boundary u (U) =

n-,

r = an

n

is an analytic Cauchy domain with

containinig a (T) and M is a normal operator such that

then

M s+J.'m M$M+(r) (oo)$T, M

+ M$M+(r) (oo), M + M$M_(r) (oo). sim sim

PROOF. By Lemma 4.8, Corollary 4.5 and Proposition 4.9, -t M(co) ; (M$M (f)) (oo) = t-!(oo) eM (r) (oo) T M$M (f) (co) S1m S1m + + S1m + and (by a similar argument) M -t M$M (f) Coo). S1m On the other hand, by Corollary 3.37, M+(f) (co) T M+(f) (oo)eT, (oo) s1m whence we readily obtain M s!m MeM+ ( r) $T. 0 .M

4.2 Matrix models for operators

LetT" L(HJ and let

~n=O.."

interior(~n>- (1 s lnl
pF(T): ind(A.-T) =n}. Define nn= (so that fln=interior(fln)-), and

N+(T) =en:l N+Cn_n) (n), N_(T) =en:l [N+(Sln*)*J(n), where N+(fl) is defined by (4.4). THEOREM 4.11. Given T" LIHJ, thePe sition H = H+eH 0 eH- suah that T+K "'

I

e~ist

K" K(HJ and a deaompo-

N+ (T) * 0 B 0

(4. 6)

0 N

pespect to that decomposition, whePe B is a smooth opePatoP and o(B) = oe(B) = oe(T).

~ith

PROOF. By Theorem 3.48, we can find K1 " KIHJ such that T+K 1 is smooth. Since 3ae(N+(T))u3ae(N_(T))=3(ulsjnj
fln)

c

a(ulsjnj
~n)

c

3oe(T),

which is contained in aire(T), it follows from Theorem 3.49 and Corollary 3.50 that

(with respect to a suitable decomposition of the space) , where N+ and N- are normal operators such that o(N+) =ae(N+) =oe(N+(T)) and o(N-) = a (N-) =a (N (T)) , respectively. e e Observe that lim(n-+ oo) {meas 2 (n n )+meas 2 (fl -n ) } = 0 and that the stronger form Qf the Berger-Shaw trace inequality for the operators N+ W_n) and N+ (rln *) * says that tr[N+(rln*)*,N+(fln*)J s and 92

(l/~)meas 2 (fln)

tr[N+{rl-n)*,N+(Q-n)] s (l/'lf)meas 2
It is not difficult to conclude that [N+(T)*,N+(T) ], [N_(T)*,N_ [N~(T)*,N~(T)] and [N:(T)*,N~(T) ], where

N' (T) =ED "" N (Q )(n) and N' (T) =ED "" [N (0 *) *] (n) n=l + -n n=l + n ' + are compact operators. Hence, by the BDF theorem (Theorem 4.1; see also Corollary 4.6), N+ (N-, resp.) is unitarily equivalent to a compact perturbation of N+(T)EDN~(T) (N~(T)EDN_(T), resp.), i.e., there exists a compact operator K3 such that

(T+K1 +K 2 )+K 3

0 N+ (T) 0 N: (T) 0 0 0 0 0 0

* * *

* * Bl 0 0

N~(T)

0

where

- [N~ 0(T)

B2 -

0

* * * 0 N (T)

*

:

0

N' (T)

B1

=

r··

0(T) B•2 *• J , 0 0 N_(T)

l •

+ Finally, by Theorem 3.49, there exists a compact operator K4 K4EDO (K4 acting on the subspace corresponding to B2 ) such that

~

OED

where B = B2+K4 is a smooth operator. In particular, cr(B) = cr'li(B) =cre(B)u{X

E

pF(B):

ind(X-B) "O}=cre(B)u{pF(B)ncr(B)}.

Let K = K1 +K 2+K 3+K 4 ; then (see observations after Corollary 3.41) ~

N+ ci> * [ 0 § 0

0

Since cre(N+(T))ucre(NjT)) c cre(T), we readily conclude that cre(B) must also be contained in a (T). Uoreover, since a (N+(T))ucr (N (T)) = e e e cse(N+' (T)) ucre(N'- (T)) c a e (B), it follows that a e (B) =a e (T). Thus, in or der to complete the proof it only remains to show that pF (B) ncr (B) = [1. Assume it is not true. Then X- B is a non-invertible Fredholm operator for some complex X; furthermore, by stability properties of the index (Corollary 1.14(i)), we can directly assume that X does not belong to the nowhere dense set ae(N+(T))ucre(N_(T)). By smoothness, ind 93

{).-B) =m~O. Now, i f A E p;{T) =p;(N+{T)), then A j. pF(N_.(T))ncr(N_(T)) and it is straightforward that ind(A- T) = ind(A- N+ (T)) +ind(A- B) = ind(A- T) +m, a contradiction. + (T) = pF+ (N_(T) ). A similar argument proves that A cannot belong to pF Since A j. p;{.T)up;(T), it follows that 0 = ind(A- T) = ind(A- B) =m, a new contradiction. We conclude that pF (B) ncr (B) =fl. Therefore, the compact operator K satisfies all our requirements. 0 4.12. In the above theorem (as in Corollary 4.7), we had to use the stronger inequality RE~~

tr[N+(!'l)*,N+(fl)J s (l/Tr)meas 2 (fl) (not fl

:)

in order to guarantee that N+(T) and N_(T) have compact self-commutators. Given e, 0 < e < 1, it is not difficult to construct denumerably many pairwise disjoint open connected subsets n 1 , n 2 , ••• , nn'''' of the unit square ({a.+ia: o s a.,a s 1}) with an =an 1 fol' aZZ n=l,2, .. n n . .. , such that meas 2 (!'ln) < e/2 , n = 1, 2,... • In th~s case, meas 2 meas 2 (ann> > 1-e for all n = 1, 2, ••• The following theorem provides a representation for T modulo compact operators, "symmetric" to the representation (4.6). Its proof fol lows by the same arguments as in Theorem 4.11 and will be omitted.

THEOREU 4.13. Given T E L(H), thel'e exist C sition H = H_IDH 0 1DH+ such that T+C "' [

N_(T) * 0 B 0

respect to that decomposition, a(B) =cre(B) =cre(T).

~ith

E

K(H) and a decompo-

(4. 7)

0 ~hel'e

B

is a smooth opel'ator and

REMARK 4.14. If T is essentially normal, then it directly follows from the BDF theorem that T- F "' N+ (T) IDBIDN_ (T) (for a suitable compact operator F), where B is a normal operator such that a (B) = ae (B) = ae(T).

4.3 Spectra of compact perturbations of operators 94

Now we are in a position to improve the qualitative parts of sition 3.45 and Theorem 3.48.

Prop~

THEOREf.1 4.15. LetT e: L{H). If {n} M is an enumeration of the n ne: bounded components of ps-F (T) (!JJhere either M = {1, 2, ••• ,m} or M is the set of aZZ natuPaZ. numbePs), {d } r• is a family of non-negative inten ne: ·• gePs, {Ak}ke:K (bJhePe either K = {1,2, ••• ,r} oP K is the set of alZ. nat!! raZ. numbePs) is an at most denumerable subset of p s- F(T) !JJith no accumuZation point in ps-F(T) and {Jk}ke:K is a family of operatoPs acting on spaces of positive finite dimension such that a(Jk) = {Ak} (k e; K), then thePe e~ists a compact opePatoP C such that (i) a(T+C) =al'l(T)u{Ak}u[u{On: dn > O}J; (ii) p:_F(T+C) = {Ak} and (T+Cl 0 IH 0 (Ak:T+C) ~ Jk foP all k e: K (whePe (T+C) 0 is the centPal opePatoP in the Apostol's tPiangulaP repPesentation (3.8) of T+CJ; and (iii) min.ind(T+C-A) =dn fop aZZ A e: On\{Ak}. COROLLARY 4.16. Let T e L(H) and let a be the union of aw(T) and some bounded components of ~\aw(T); then thepe e~ists K e: K!HI such that a (T+K) = a. In paPticular, thePe e~ists K0 e: K!HI such that ~\a(T+K0 ) is connected. PROOF OF THEOREM 4.15. By Theorem 3.40 there exists c 1 e: K{H) such that

where M is a diagonal normal operator such that M "' M(ao) and a(M) =a e (M) = a... re (T) , and a... re (B 1 l= a... re (T) • By Theorem 3.48, we can find c 2 e: K{HI (which only modifies the action of B1 ) such that :], and n B2 is a smooth compact perturbation of B1 . Clearly, T+C 1 +c 2 is also smooth. Let in= ind(T+C 1 +c 2 -A) = ind(B 2-A), A e: On (-ao s in s ao). Since the Ak's do not accumulate in ps-F(T), we can find points pk (k e: K) in ap (H) such that I Ak- Pkl + 0 (k + co). Let J+ = Gl{Jk_: Ak e pr(D 2 )} and J_ = 95

I9{Jk: Xk € p1(B2)}, where Jk - Jk is any operator such that IIJk- Xk II < 1/k for all k € K. It is easily seen that there exist compact pertur bations M+ and M- of U such that t-1+ = MIBJ+ and M- = Iii9J_, respectively. Now we can proceed as in the proof of Theorem 4.11 in order to ob tain a compact operator c 3 such that J o

o T+C 1 +e 2+c 3 =

0 IO * * * I N_ 1 o * * * -o- 1 N'_*_ '* I

*

-

* * *

I

0 0 o

0 1 0 B2 * 1 * * 0 1 0 0 N' I 0 0 o Lo- o -o±:N+_o_

o

o

o

o o 1o

J+

where N+ = IB{N+(gn) (dn): in~ 0, dn > 0}, N_ = 19{ N+(On*)* (dn): in < 0, dn > 0}, J_eN_eN: and N~IBN+IBJ+ are compact perturbations of ~1 and

B3

N'

= [0

0

l

* *

B2

* .

0

N~

Finally, we can use Theorem 3.48 (as in the proof of Theorem 4.1U in order to find a compact operator c 4 (which only modifies the action of B3 ) such that T+C 1+C 2+C 3+c 4 = [J-:N- : 0

:

l

(B is smooth).

0 N+IBJ+

Let C=C 1 +c 2+c 3+c 4 (clearly, C € K(H}). It can be easily checked that cr(J_) =crp(J_)- c {Ak}ucr.ll.re(T), o(J+) =crp(J+)- c {Xk}uo.ll.re(T), pss- F(T+C) c {Xk} and (since B is smooth) min.ind(T+C-X) = d n for all X € On\{Xk}, so that crW(T)u[u{Qn:

dn > 0}]

c

On the other hand, if Xk

P.~~,{[:

o(T+C) €

c

oW(T)u{Xk}u[u{Qn:

op(J_) then Xk

p.II.(B)np(N+IBJ+) =

€

N+:J+)) and Corollary 3.22 implies that T

dn > 0}].

~

Jki9Tk, where Xk

£

p~-F(Tk). A similar argument implies that the same result holds for Xk

crp(J+), whence we obtain (ii). Since (i) and (iii) follow immediately from (ii) and our previous observations, we are done. 0 £

4.4 Voiculescu's theorem

96

*:C

~

An abstract C*-algebra is a Banach algebra C with an operation C satisfying the conditions: (i) (a*)*=a, a € C; (ii) (a+b) * = a*+b*, a,b € c; (iii) (>..a)*=Xa*, >. € a:, a € c; (iv) (ab) * = b*a*, a,b € c; and (v) llall2 = lla*all• a E c.

(An operation *:C

~

C satisfying (i)- (iv) is called an involution.)

The most obvious example of a C*-algebra is L(H), with * equal to the usual'adjoint operation'. Given a (finite or denumerable) family {Tn} of operators in L(H), we shall denote by C*({Tn}) the C*-algebra generated by the Tn's and the identity operator, i.e., the minimal norm-closed algebra containing 1, T1 , T2 , ••• , Tn•··· and T!, T~, •.• , T*, •••• Every separable C*-subalgebra of L (H) (containing the identn ity) has this form. If {Tn} = {T} consists of a single operator, then we shall simply write C*(T) instead of C*({T}). A *-representation p of d C*-algebra C on a Hilbert space Hp is an operator p from C into L(H ) which preserves products (i.e., p is a p homomorphism of rings: p (ab) = p (a) p (b)) and involutions (i.e., p(a*) = p(a)* in L(H p )) • I f C has an identity e and p(e) =1 , then pis ca! led unital; p is faithful if it has a trivial kernel. It is well-known that a *-representation p is faithful if and only if it is an isometric mapping from C onto p(C) c L(HP) (see, e.g., [37], [72], [l76J). The following classical result can be found in any of these reference~ THEOREl1 4.17 (Gelfand-Naimark-Segal theorem). Every abstract C*algebra C with identity admits a faithful unital *-representation p in L(H ) for a suitable Hilbert space H , i.e., C is isometrically *-isoP p morphia to a C*-algebra of operators. Furthermore, if C is separable, then Hp can be chosen separable. Since we are only interested in separable C*-algebras, every Hilbert space (H, H , etc) considered here will be separable. p Two *-representations p 1 :C ~ L(H 1 ) and p 2 :C ~ L(H 2 ) are unitarily equivalent (or, simply, equivalent) if there is a unitary operator U from H1 onto H2 such that p 2 (a) =Up 1 (a)U* for all a E C; p 1 and p 2 are approximately equivalent (p 1 ~a p 2 ) if there is a sequence of unitary operators Un:H 2 ~ H1 such that p (a) -Unp 2 (a)U~ ~ K(H 1 ) (n=l,2, ••• ) and { 1 lim(n ~co) IIPl(a) -Unp2(a)U~II=O

(4. 8)

97

for each a

E

C. It is easily seen that

~a

is an equivalence relation.

THEOREM 4.18. (i) Let C be a separable C*-algebra with identity and let p:C + L(HP) be a unital *-representation of C. Let y be a uni! al *-representation of n•p(C) (c A(HP)) on a Hilbert space Hy. Then p

~a

p$yonop.

(ii) Let p 1 and p 2 be two unital *-representations of C on HiZbert spaces H1 and H2 , respectively, and let Hj = {pj (a)x: a E ker n• Pj' X E H.}-, j=l,2. The the following are equivalent:

J (1)

Pl "'a P2· There is a sequence of unitary operators Un:H 2 .•. ) such that (2)

lim(n (3)

II P1 (a)

+ oo)

- Unp 2 (a) U~ II= 0, for each a

E

+

H1 (n=l,2,

C.

There exist sequences of unitary operators Un:H 2

+

H1 and Vn:

H1 + H2 (n=l,2, .•• ) such that UnP 2 (a)U~ +

p1 (a)

and

Vnp 1 (a)V~ +

p 2 (a),

in the weak operator topology, for each a E C. (4) ker P1 = ker p 2 , ker nap 1 = ker 1fop 2 and the *-representations of C on Hi and induced by p 1 and p2 , respectively, are equivalent. (5) ker P1 = ker P2 , ker 11aP 1 = ker 11oP 2 and the *-representations of ker 71•P 1 ~m Hi and Hi induced by P_t and p2' resp., al"e equivalent.

Hi

Recall that a *-representation p of a C*-algebra C on HP is irreducible if there is no non-trivial subspace M of HP reducing all oper~ tors in p(C). COROLLARY 4.19.· (i) Every unital *-representation of a separable C*-algebra with identity is approximately equivalent to a direct sum of irreducible *-representations. (ii) Let p be a unital *-representation of a separable C*-algebra C on a Hilbert space Hp. In order that all unital *-representations of C appro:x:imately equivalent to p be actually equivalent to p it is neces sary and sufficient that p(C) be finite-dimensional. COROLLARY 4.20. Let p 1 , p 2 be infinite dimensional unital *-repr~ sentations of a separable C*-algebra C with identity; then (i) If ker p 1 =ker p 2 and neither p 1 (C) nor p 2 (C) contains any non-•ro compact operator, then P 1 "'a P 2 • (ii) If ker p 2 ~ ker 1r•p 1 , then p 1 ep 2 "'a p 1 • 98

4.5 Closures of unitary orbits Let T € L(H) and let C*{T) be the C*-algebra generated by T and the identity. Denote by p0 the identity *-representation of C*{T) on H ( p0 (A) =A for each A in C* {T)) and let S be any operator in U{T)-. It is easily seen that C' (T) = {L € L(H): {UnLU~}n:l is a Cauchy sequence in L(H) for each seque~ce {Un} in U(H) such that {UnTU~}n:l is Cauchy} is a C*-algebra containing C* {T) and therefore there is a (necessarily unique) unital *-representation p1 of C* {T) on H such that p1 {T) = S. {This representation is defined as follows: I f liS- UnTU~ II .... 0 {n + oo) for a suitable sequence {Un}n:l in U(H), then pl {A) = {norm) lim{n .... oo) un AU*n I A € C* {T) . ) Since T € U{S)-, it is not difficult to check that p1 is also faithful: indeed, p1 ~a p0 • Conversely, if p1 is a unital *-represent~ tion of C* (T) on H and p1 ~a p0 {clearly, this p1 is faithful), then p 1 {T) belongs to the closure of the unitary orbit of T = p0 {T). 4.5.1 Operator-valued spectrum and unitary orbits The above remarks and the results of Section 4.4 have the following direct consequences. PROPOSITION 4.21. Let T € L(H); then (i) S € U{T) if and onZy if there e~ist sequences {U0 }n:l and {Vn}n:l of unitary operators suah that Un TU~ -+ S (weakZy)

and

VnSV~

+

T (UJeak Zy ) .

(ii) If p is a unitaZ *-representation of C*{!) on H and p{T) A, then there e~ist s 1 , s 2 € U{T) suah that s 1 ~TeA, s 2 ~ T8A{oo}.(If p0 is the identity *-representation of C*{T), then p0 ~a p0 8p ~a p0 8 p {""). )

(iii) B= p(T) for some unitaZ *-representation p of C*{T) if and only if either T has a reduaing subspaae M suah that TIM ~ B or there exists a sequenae {T } ""1 in LIHI suah that T ~ B{oo)eT' (for suitabZe n n= n n operators T~, n=l,2, ••• J and IIT-Tnll+ 0 {n-+ oo}. (iv) If S € U{T)- and E > 0, there is a unitary operator U suah that S-UTU* € K(H) and IIS-UTU*II < E· (v) There exists S € U{T) suah that S is an infinite direat sum of irreduaibZe operators.

99

PROPOSITION 4.22. The follo~ing are equivalent forT in L(H): (i) U(T) is closed. (ii) C*(T) is finite dimensional. (iii) T "'AEBB(co) ~ ~here A E L(~)~ B E L(a:n) for some pair (m,n), 0 s m < co, 1 s n < co. COROLLARY 4.23. The set of all reducible operators is dense in Furthermore~ given Tin L(H) and E > 0~ there exists K E K(Hl~ < E~ such that T+K is reducible.

L(H). IlK II

REMARK 4.24. Given ann-tuple T= (T 1 ,T 2 , ••. ,Tn) er its unitary orbit {(UT 1U*,UT 2U*, ••• ,UTnU*)

E

L(H)(n):

L(H) (n), consid

E

u

E

U(H)}.

We shall say that Tis reducible if T1 , T 2 , ••. , Tn have a joint reducing subspace. Clearly, in this framework, the results of Section 4.4 imply that the analogues of Propositions 4.21 and 4.22 and Corolar y 4.23 hold for n-tuples. In order to determine U(T) in concrete terms we shall need the following definition: The operator-valued spectrum E(T) ofT E L(H) is the set of all those operators A acting on some subspace of dimension n, 1 s n
U(T)-, we shall say that T and S are approximately unitar£ ly equival~nt operators (T "'a S). It is easily seen that "'a is an equivalence relation and that T "' S implies that T 7 S and S 7 T. s~ s~ a With the above definition in mind, we have the following characterization of U(T)-: E

THEORIDt 4.25. Suppose

T~

S

E

L(H). Then T "'aS if and only if

I: (T) = I: (S) •

4.5.2 Concrete examples of closures of unitary orbits From Proposition 4.2l(iv), we obtain PROPOSITION 4.26. If N E L(H) is normal, L is a diagonal normal operator in U(N) and E > 0~ then there exists U E U(H) such that N - ULU* 100

E

K(H)

and

II N - ULU* II <

E•

Hence, every normal operator is equal to a diagonal operator up to a compact perturbation of arbitrarily small norm It is not difficult to check that if T € L(H) then H can be written as

HNor(T)eHAbnor(T), where HNor(T) and HAbnor(T) reduce T, TNor

= TIHNor(T) is normal (= the normal part of T) and TAbnor = TIHAbnor(T)

(= the abnormal part of T) has no nontrivial reducing subspace M such that TIM is normal; more precisely, HNor(T) =nn:l ker[T*n,Tn]. (Of course, HNor(T) and HAbnor(T)can be infinite dimensional, finite dimensional or even {0}.) PROPOSITION 4.27. If T, S c L(H) and Tis essentiaZZy normaZ, then T "' a S if and onZy if SAb nor "' TAb nor ,ae (S) =ae (T) and nul(A-S) for azt >. E(o(S)uo(T)]\cre('l').InparHcuZar,if K € K(H), then

= nul(A-T) U(K)- = {C €

Kfi,J) c

C "' KAbnoreod, wheztll Od is the 0 opeztato;ro acting on

a subspace of dimension d, 0 s d s ""• and d+dim HAbnor (K) = ""}.

PROPOSITION 4. 28. If T is hypono;romaZ and T = TAbnor• then U(T)

=>

{S c L{fl):

S"' TeN, N is no;romaZ and o(N) c oJT)}.

4.5.3 On normal and quasinilpotent restrictions Let T E L(H) and let {An} be a (finite or denumerable) sequence of distinct points of 01 e(T). According to Theorem 3.40, there exists a compact operator K1 such that - (Ni * ) Hl T+Kl- 0 Al Hl.l' where N1 € L(H 1 ) is a diagonal normal operator such that N1 "' (diag{Al' A2 , ••• ,An•···}l (co) and oe(A1 l =cre(T), ore
0

l~~ '

: Nr

H3

where Nr"' (diag{Jl 1 ,~ 2 , ••• ,~m'"""}) (oo). Let r = {A }- = cr (N ) = cr (N ) and r = { ~ } - = cr (N ) = cr (N ) and conin 1 et r m r er sider the (separable!) C*-algcbra C=C*({'f;ts1 ,ts 2 , ••• ,tsn, ••• ;it1 ,tt 2 , ••• , Rm, ..• }) generated by r, If and the images in the Calkin algebra of the orthogonal projections Pn (onto ker (An- N1 ) ; n = 1, 2, ••• ) and- Rut (onto 101

ker(l.lm- Nr) 1 m= 1,2, ••• ) • Let p:C ~ L(H ) be a faithful unital *-representation. Since

(1-~j~l Pn)(T+K)~j~l Pn=O, ~k=l~A(l-~k=ll\) =0, it is not difficult to conclude that

p(~)

(•) •

[~· ~ ~~-

By Proposition 4.21, we have PROPOSITION 4.29. LetT£ L(H) and let r 1 and rr be closed subsets of a. (T) and a (T), respectively. Then there exists S = T such ... e re a that

s •

TO[~• ~ ~rl·

~here

N1 and Nr are diagonal normal operators of uniform infinite multiplicity such that a (N 1 ) = ae (N 1 l = r 1 and a (Nr) = ae (Nr) = r r• respectively. COROLLARY 4.30. LetT£ L(H). If r is a closed subset of ae(T) and N is a normal operator such that a (N) = ae (N) = r, then there exists an operator A such that Ps-F (A) = Ps-F (T) \f > {

m~n.ind (A-A) k =min. ind (A-T) k for aZZ A

£

ps-F(A)

(4.9)

k-1,2, ••• , and

T slm NIBA. PROOF. Clearly r can be written as r = r 1 ur r' where r 1 c a R.e (T) and rr c are(T) are closed sets. Let S be the operator constructed in Proposition 4.287 then, by Lemma 3.36, we have T =a S sim TIB(N 11BBIBNr) = (N 119Nr)IB(TIBB). Clearly, a(N 11BNr) =ae(N 11BNr) = r, so that N119Nr =aN (Lemma 4.8). Let A be an operator unitarily equivalent to NIBTIBB. It is completely apparent that ps-F(A) c ps-F(T)\r. On the other hand, since p is fai~ ful, it readily follows that p s- F(B) =p s-F(B)up-s- F(B), p s- F(B) => p s- F (T)\f, p;:F(B) => p;:F(T)\r and p~-F(B)n(ps-F(T)\f) =~,whence we imme diately conclude that A satisfies (4.9). Since (N IBNr)IB(TIBB) ~a NIBA, we are done. 0 00

102

00

00

00

Applying Corollary 3.50 to the direct summand of S unitarily equivalent top(~)<~> (in Proposition 4.28), we obtain COROLLARY 4.31. LetT € L(H) and Zet rR. and rr be aZosed subsets of a... e (T) and a re (T). respeativeZy. Given e > 0 and normal operators NR. and Nr suah that rR. c cr(NR.) =cre(NR.) c (fR.)e and rr c a(Nr) =ae(Nr) c (r ) • there e:r:ists L € L (H) suah that r e:

L [~R. :1 : ]• 0

0

Nr

Ps-F(L) =ps-F(T)\[cr(Nr)ucr(NR.)J. min.ind(A-L)k=min.ind(A-T)k for all A € ps-F(L). k=l,2, ••.• and IIT-LII < 2e:. (Apply Lemma 4.8 to the result of the first modification.) LetT € L(HJ and assume that~ is an isolated point of ae(T). If e: > 0 is small enough to guarantee that {A: I A - ~I s e:} ncr e (T) ~},

={

then

1 I R = 2'1Ti IA-~I=e:

(A - T) -l dA

defines an idempotent commuting with T (in the Calkin algebra). This idempotent can be lifted to an idempotent R € L(H) such that ae(RTRI ran R) = {~} (RTR=RT=TR modulo K(H)) _ and a e ((1-R)T(l-R)) lker R) =a e 1 (T) \ { ~}. Let W € G (H) be such that WRW = P is an orthogonal projection and let L = WTW-l: then L = B$ (ll+Q) + (compact) for some B such that ll I ae(B) and Q is a quasinilpotent operator acting on the infinite dimensional reducing subspace H of L. ~ Let p:C*(~+Q) ~ LIH ) be a faithful unital *-representation of p C*(~+Q). Clearly, O$p is a (not faithful, unless B acts on a finite dimensional space!) unital *-representation of C*(L). By Proposition 4.2l(ii), L "'a L$(~+G)(co)' where G=p(Q). It readily follows that T ~ T$(~+G) (co) : furthermore, since p is sim faithful, we can use Corollary 1.8 to show PROPOSITION 4.32. Let T E L(H) and assume that cre(T) has an ted point ll· Then there e:r:ists a quasinitpotent G such that T

~

s~m

T$(l-[+G)

(co)

isoZ~

•

Moreover. if

103

{

(A -p) k in some open neighborhood of P.

0, in some open neighborhood of cre(T)\{p},

(k = 1, 2, ••• ) • then Gk = 0 if and onZ.y if fk (T) = 0.

4.6 Irreducible operators According to Corollary 4.23, every operator has an arbitrarily small compact perturbation which is reducible. If reducible is replaced by irreducible, the result is still true. In fact, in this case the answer is much stronger and the proof is definitely simpler. Hore precisely, we have LEf·if1A 4.33. The set of aZ.Z. irreduaibZ.e operators is dense in L(Hl. Furthermore, given Tin L(H), p > 1 and 0 < e < 1, there e~ists Kp in cP(Hl. IKPIP < e. such that T+Kp is irreduaibZ.e. PROOF. Let T = A+iB (Cartesian decomposition) • According to the Weyl-von Neumann-Kuroda theorem (see, e.g., [153, p. 525,Theorem 2.3]), there exists K1 E cP(Hl, IK 1 1 < e/3 such that A+K 1 is a diagonal herP ® ® mitian operator with respect to some ONB {en}n=l of H. Let {An}n=l be the sequence of corresponding eigenvalues (i.e., (A+Kl) e = A e , n = 1, . n n n 2, ••• ). It is easily seen that now we can find K2 E c1(1f), IK 2 11 <E/3. K2 an hermitian diagonal operator (with respect to the same ONB) such that (A+K 1 +K 2 )en=pnen' n=l,2, ... , and the pn's are all distinct. It is not difficult to check that A' (A+K 1 +K 2 ) =A (A+K 1 +K 2 ) = {B E L(H): B = ~n:lynen@len (strong limit of partial sums)} is the set of all diagonal normal operators with respect to that basis. In particular, a subspace M reduces A+K 1 +K 2 if and only if PM=~ {ejt!lej: j E N (M)}, where N (M) is a subset of the natural numbers (N (M) = { j: e. E J M}). Assume that j E N(M), but h t N(M) and let R E L(H) be such that either ~ 0 or ~ o. It is easily seen that R cannot commute.with PM' i.e., M does not reduceR. Let B = (b 'k) . k:l be the matrix of B with respect to the ONB J J, {e } : 1 and let 0 < n < e/6. Define K by n n2 3 n 4 n n /2 n /3 n /4 •

n2 /2 n3/3 n4 /4 n3/3 n4 /4 •

K

104

n

= n4/4

•

Then IKnll ~ ~n=l n(nn/n)=n/(1-n) < e:/3. Since the set {b J"k}.],k":.l is at most denumerable, there exists some n such that bjk+nj+k/(j+k) ;.o, for all j,k=l,2, •.•

( 4 .10)

Let K3 =Kn for some n• 0 < n < e/6, satisfying (4.10). Then B+K 3 cannot be reduced by any nontrivial reducing subspace of A+K1 +K 2 • Thus, T+(K 1 +K 2+iK 3 ) is irreducible and

THEOREf.f 4.34. (i) The set of aU ir:roeduCJibl.e operator'S is a GcS dense subset of LIH). (ii) The set of a1.1. redueibl.e operators is an F dense subset of a

LIHl.

PROOF. According to Corollary 4.23 and Lemma 4.33, it only remains to show that the set R(H) of all irreducible operators is an Fa. Let P be the set of all those hermitian operators H for which 0 ~ H ~ 1. Recall that P is exactly the weak closure of the set of all projections. Let P0 = P\ { ;>.,: ). " a:}. Since P is a weakly closed subset of the closed unit ball of L (H) , it is weakly compact, and hence the weak topology for P is metrizable. Since the set of all multiples of the identity is weakly closed, it follows that P0 is weakly locally compact. Since the weak topology has a countable base, the same is true for P0 , and therefore P0 is weakly a-compact. Let P1 , P2 , ••• , be weakly co~pact subsets of P0 such that un:l Pn = P0 • Let Pn ={A" L(H): AH=HA for A some H" P} (n=l,2, ••• ); the n spectral theorem implie~ that un:l Pn = R (H) • Suppose that~ " Pn (k = 1,2, ••• ) and II~- All-+ 0 (k -+co). For each k, find an Hk " Pn such ~Hk = Hk~. Since Pn is weakly compact and metrizable, there is no loss of generality in assuming that the sequence {Hk}k:l is weakly convergent to H " Pn. CLAnh

AH = HA.

Indeed, if x,y " H, then

~ 11~-AII·IIxii·IIYII+I<(~-H>x,A*y>l-+

o

(k-+ co).

Hence, ~Hk + AH (weakly, as k + ~>. By taki~g adjoints, we can similarly ch:ck that Hk~-+ HA and therefore A" Pn. Hence, P is closed (n=l,2, ••• ) and therefore R(H) is an F (in n a the norm topology) • 0 105

4.7 Notes and remarks The BDF theorem and the Berger-Shaw trace inequality have been pu~ lished in the "Proceedings of a Conference on Operator Theory, Halifax, Nova Scotia, 1973", Lecture Notes in Mathematics Volume 345, Springer Verlag, Berlin-Heidelberg-New York, 1973 (see [60], [50]). The interested reader can also consult [61], [62], [49] and W. B. Arveson's su~ vey article [38]. Lemma 4.3 is a portion of Proposition 2 of D. A. Herrero's article [146]. The present proof of the Berger-Shaw trace inequality (Theorem 4.4) combines four ingredients: (I) is an argument due to D. Voiculescu [196,Proposition 1]. This argument has been independently discovered by c. Berger (unpublished result; see also comments in [196]); (II) is a classical result (see, e.g., [SO,Theorem 1]); (III) is an "ad hoc" modification of the second computational lemma of C. A. Berger and B. I. Shaw [49]; and (IV) is an observation of D. A. Herrero. An extended version of the Berger-Shaw trace inequality can be found in the above mentioned article of D. Voiculescu. The reader is referred to [66], [167] and [168] for a detailed account of the theory of hyponormal operators. Proposition 4.9 is [133,Lemma 1]. The matrix models for operators are due to C. Apostol [6] and Theorem 4.15 is a consequence of those constructions and Apostol's triangular representation. Corollary 4.16 (in fact, a stronger form of it) was proved by C. Apostol, C. M. Pearcy and N. Salinas in [28]. The notion (4.8) of approximately equivalent *-representationsand all the results of Sections 4.4 and 4.5.1 are contained in the article [195] by D. Voiculescu, except for the equivalence between (1) and (3) of Theorem 4.18, which has been independently obtained by W. Arvesonin the above mentioned survey article (see [38,Theorem 5]) and by D. W. Hadwin [112,Corollary 4.2] (indeed, Hadwin established it in a slightly weaker form), and Theorem 4.25 is also due to Hadwin [112,Corollary 3.6]. Hadwin's reducing operator-valued spectrum E(T) has been modelled on J. Ernest's operator-valued spectrum [80] and the reducing essentiai matricial spectra of operators of C. M. Pearcy and N. Salinas. (See [160], [164], [165], [166], [179]. The article [63] of J. Bunce and N. Salinas is also relevant in this context.) The different pieces of E(T) in [112] correspond to the possibilities offered byProposition 4.21 (ill): namely: 1) A is in the assentiaZ Peducing .opePatoP spectPum of T, denoted by Ee(T), if A(•) E E(T); 2) If n is a positive integer, then 106

the n-dimensionaZ reducing operator spectrum ofT, denoted by En(T), is E(T)nL(~n) and the n-dimensionaZ essentiaZ reducing operator spectrum

ofT, denoted by En(T), is Ee (T)nL(~n); 3) The infinite-dimensionaZ e reducing operator spectrum ofT, denoted by E®(T), is E(T)nL(H), and the infinite-dimensionaZ essentiaZ reducing operator spectrum of T, de noted by r:(T), is Ee(T)nL(H). In [112] Hadwin also proved (Theorem 3.9) that U(T)- is always arcwise connected, thus extending the corresponding result for normal operators, previously obtained by R. Gellar and L. Page in [104] (see also [85]). Proposition 4.26 is essentially the I. D. Berg's extension to no~ mal operators of the classical Weyl-von Neumann theorem on diagonaliz~ tion of hermitian operators via small compact perturbations [47J.(This result has also been independently obtained by w. Sikonia in his doctoral dissertation; see [186].) Propositions 4.27 and 4.28 are essentially due to D. W. Hadwin [111] and to C. M. Pearcy and N. Salinas [162] (see also [160]). Proposition 4.29 was proved in [97] in connection with a problem of quasisimilarity of operators; see also [58], [160]. The problem of characterizing closures of unitary orbits has been developed by D. W. Hadwin in his doctoral dissertation [111] and in [1121. His ideas and techniques go beyond this particular problem. The interested reader can also consult Hadwin's papers [113], [114] and [54] (joint work with A. Brown and c.-K. Fang), and the closely related articles [101] and [191]. Lemma 4.33 is a mild improvement of a result of P. R. Halmos. That result and Theorem 4.34 are contained in [120]. The proof of Lemma 4~3 given here is due to H. Radjavi and P. Rosenthal [170]. It is convenient to remark that Voiculescu's Corollary 4.23 affirmatively answers in the most complete form Halmos' Problem 8: Is every operator the norm-limit of reducible ones? [123,p.919]. In [116], D. w. Hadwin extended Voiculescu's theorem (and many other related results) to cover the case of non-separable C*-algebras and operators acting on non-separable spaces.

107

5 Limits of nilpotent and algebraic operators

As a first application of the tools introduced in the previous two chapters, it will be shown that the closures of the set of all nil potent operators and the set of all algebraic operators can be characterized very simply in terms of the different parts of the spectrum. Essentially the same kind of arguments can be used to obtain two important results about closures of similarity orbits.

5.1 Limits of nilpotent operators THEOREM 5.1. The to~s

cZosu~e

of the set N(H) of aZZ nilpotent

one~a­

acting on H coincides with the set of aZZ those A in L(H) satisfy-

ing the conditions:

cr(A) is connected and contains the o~igin; ae(A) is connected and contains the o~igin; and (iii) ind(~-A) = 0 fo~ aZZ ~ € Ps-F(A). (i)

(iiJ

Since cr(Q) = cre(Q)= {0} for each 0 € N(H), the necessity of (i) and (ii) follows at once from Corollary 1.6(i) and (iii), respectively. Since ~-Q is invertible (hence ind(~-Q) = 0) for all ~ ~ 0, the necessity of (iii) follows from Theorem 1.13(iii). The sufficiency of condi tions (i), (ii) and (iii) will be proven in two steps. REMARK 5.2. Theorem 5.1 implies, in particular, that every quasinilpotent operator is a limit of nilpotent ones.

5.1.1 Normal limits of nilpotents We have already shown (Proposition 2.28(ii)) that N(H)- contains a normal operator t1 such that cr(M) = 0(0,1)-, where D(ll,r) = 0 € a:: 1~-lll < r} is the disk of center ll and radius r > 0. By Lemma 4.8,

U(M)

108

= {N

€

L(H):

N is normal and cr(N)

Since rQ € N(H) for each Q in N(H) and each r > 0, it readily fol lows that N(H)- contains every normal operator whose spectrum is a disk centered at the origin. LEl~~ 5.3. Let N be a normal operator such that cre(N) has an accumuLation point A0 and Let Q € N(H); then N sim N$(A 0 +Q).

PROOF. Let {An}n:l be a sequence of distinct points of a (N) such e (co) that An->- Ao (n ->-co). By Lemma 4.8, N ~a N1 = N$(diag{A 0 ,A 1 ,A 2 , ••_.}) Assume that Q € Nk(H): then Q admits a canonical k x k operator matrix representation of the form

0

012 0

Ql3 Q23 0

Ql,k-1 Qlk 02,k-l Q2k 03,k-l Q3k

ker ~leker Qo ker o2eker 01 ker o 3 eker 02 (5 .1)

Q 0

0

0

Qk-leker 0k-2 ok-l,k ker 0 ker Qkeker Qk-l

k

(Clearly, ker Q = {O} and ker Q =H.) Replacing, if necessary, Q by Q~£qk(co) .!given by (2.1): 0 < £ << 1) we can directly assume that ker QJeker gJ-l is infinite dimensional for all j = 1,2, ••• ,k. Given n ~ 1, let Rn be the operator obtained by replacing the diagonal entries A0 in A0 +Q by An+l'An+ 2 , ••• ,An+k. It is easily seen that

By Corollary 3.22 and the above observation, _ k (co) k (co) ~ Nl$Rn->- Nl$(Ao+Q) Nl- Nle(ej=l An+j) - Nl$[($j=l An+j) +Q] (n->- co),

i.e., N

~a

N1 sim N$(A 0 +Q) •

D

Recall that the Hausdorff distance in the family BP(X) of all bounded nonempty subsets of a metric space (X,d) is the pseudoMetric defined by (5 .2)

Moreover, the restriction of d 8 to the family Bc(X} of all cZosed bounded nonempty subsets of X is ametric space: (Bc(X} ,d8 ). Lemma 4.8 and the "spreading the spectral measure argument" (that w~ have already applied, e.g., in Corollary 3.50) applied to-normal o~ 109

erators with spectra equal to their essential spectra, yields the following simple (but very useful!) result: LEMMA 5.4. If M and N al'e nol'maZ. opel'ators such that o(M) o(N) oe(N) and dH[o(M),o(N)] < E, then dist[M,U(N)J < E,

dist[N,U(M)] <E.

COROLLARY 5.5. If M and N al'e nol'maZ. operatol's such that o(N) is a perfect set, o(M) ~ o(N) and each component of o(M) intersects o(N), then N s~m + M. PROOF. It is completely apparent that oe(N) = o{N) and cre(M) cr(M). Let E > 0 and let cr(M) c uj:l D(Aj,E/4) be a covering of o(M) by open disks of radii E/4 centered at points of cr(Ml. It is easily seen that if the A.'s are properly ordered, we can find points~· in J

J

D(Aj,E/4), j = 1,2, .•• ,m such that, if A1 = {~ 1 .~ 2 ····•~r 1 }, ~={~rl+l' ••.

,~

r2

}, .•• ,

A={~

p

+ , .•. ,~ }(r = m), then A1 rp-l 1 rp p

max{dist[~j,Ah-l]:

~j

£

Ah} < E/2, h

c

o{N), and 1,2, ••• ,p-l.

m

Let {Mj}j=l be a finite set ofmnormal operators such that o(Mj) cre(Mj) = D(Aj,E/2) and let {Qj}j=l be nilpotent operators. By Lemma 5.3,

rl N s!m Ne{ej=l (~j+Qj)} and, by using Propositions 2.28(ii) and our previous observations, we see that the oj 's can be chosen so that IIMj-(~j+O:fll is arbitrarily small for all j 1,2, .•• ,p1 . It readily follows that pl N s!m N${$j=l Mj) and, by an obvious inductive argument, that N s!m ME where ME is a normal operator such that o(M)

c

m

cr(Me:) = oe(Me:) = [uj=l

D(~j,e:/2)

J

c

cr(M) (JE/ 4 ),

so that ~(o{M),o(ME)J <E. By Lemma 5.4, we can find M~ ~Me: such that IIM-M'II < e:. Since E can be chosen arbitrarily small, we conclude e: that N sim

M.

PROPOSITION 5.6. If M is normal., then M £ N(H} o(M) is a connected set containing the ol'igin.

0

if and onZ.y if

PROOF. It was already observed that o(M) must necessarily satisfy the given conditions for each Min N(H}-. Assume that o(M) is connected and contains 0. Given e: > 0, it fol

110

lows from Lemma 5.4 that there is a normal operator Me such that a(tie) is the closure of a connected open neighborhood of the origin and IIM-M£ II <e. Clearly, it suffices to show that M£ e N(.H)-. Let n > 0 be small enough to guarantee that D(O,n) c a(M) and let N be a normal e: n operator such that a(Nn) = D~O,n)-. By Proposition 2.28(ii), Lemma 4.8 and Corollary 5.5, N e N(H) and N + M • n n s~m £ Since N[H) is invariant under similarities, it follows from Prop~ sition 1.15 that M£ e NIH)-. D

5.1.2 Spectral characterization of NIH) Recall that an operator R is called algebraic if it satisfies an algebraic equation. If R is algebraic with minimal monic polynomial p, p(A) = rrj~l (A-Aj)kj, then it is easily seen that a(R) = {A 1 ,A 2 , •••• , Am} and, by Corollary 3.22,

* (5. 3)

R

0

Am+0"''Il Hm

kl k. k. where H1 = ker(R-A 1 ) , and Hr= {ker rrj~l (R-Aj) J)e{ker rrj;;;f
r

Let Atg(H) denote the set of all algebraic operators ac'tinq on H. It is easy to see, by using the expressions (5.1), (5.3) and Corollary 3.22 that the algebraic operator R is the limit of a sequence {Rn}n:l of operators such that Rn is similar to a normal operator and a(Rn) consists of exactly Lj~l kj (= the degree of the minimal polynomial of R) points. LEMMA 5.7. Let N,T e L(H), a(N) is a perfect set and a(T)

~here

c

N is a normal operator such that a(N). Then N s!m N$T.

PROOF. Let £ > 0 and let n1 , n be analytic Cauchy domains such that a(N) c n1 c o1 c 0 c a(N)£/ 2 and each component of n intersects a(N). If M1 , Mare normal operators such that a(M 1 ) = n1- and a(M) = n~ and R is an algebraic operator such that a(R) c 0\0 1 -, then it follows from Lemmas 5.3 and 5.4, and Corollary 4.10 that

111

By Propositions 5.6 and 1.15, and Corollary 4.10, R can be chosen so that

where II S II < e./2. since o(R)no(M+(an 1 >(~))

c

(O\n 1->nn 1 -

~.

it follows from Corel

lary 3.22 that RGlM+ (ClQl)

(~).-

[RX

0 M+ ( ClS'Jl)

) (oo)

l

for any X. Thus, if we choose X= OGlOGlz(oo) (where Z is defined by (3.2)), then

MGlTGlRGlM+ ( Cl0 1 ) (oo) ~ MalTGl[M Gl M1Gl

[M- (zan(!)> ("")

o

(oo) +S$0]

M+ (00 1 )

MGlT[MGlM 1GlM(Cl0 1 ) ("")+S$0]. Since MGlMGlM 1 GlM(Cl0 1 ) ("") za M and dH[o(N) ,o(M)] < e./2, it follows from Lemma 5.4 that S(N) contains an operator unitarily equivalent to NGlT+S', where II S' II < e./2+11 S II < e.. Since e. can be chosen arbitrarily small, we conclude that N sim NGlT. D Now we are in a position to complete the proof of Theorem 5.1. Suppose that A € L(H) satisfies (i), (ii) and (iii) and let E > 0. Since o 0 (A) = ~, it follows from Theorem 3.48 and (iii) that there exists K1 € K(H), llx 1 11 < E/4, such that A1 = A-K 1 satisfies cr(A 1 ) = crtre(A 1 ). Clearly, A1 also satisfies (i). Let p be a faithful unital *-representation of C*(A 1 ) on H and ~ (oo) p le-t Ai = p (A 1 ). It readily follows that i f A2= (Ail , then a (A 2 ) =crtre(A 2 ) = o(A 1 ). By Proposition 4.2l(ii) and (iv), there exists x2 E K(H), IIK 2 11 < e./4, such that A3= A1 -x 2 a A1 GlA 2 • Finally, by Propositions 4.29 and 4.2l(ii) and (iv), there exists K3 E K(H), IIK 3 11 < E/4, K3 = x 3 , 1ex 3 , 2 , such that

A 1 -K 3 ,l~ (~ ~~)

and

A 2 -K 3 , 2 ~

where N is normal and cr(N) otre(A 1 ) Hence, if K = K1+K 2+K 3 , then

ll2

(: 2

= cr(A 1 ).

~2)

,

N 0 IC 1 0 I 0 _s 21 ~ _c~ 0 0 IBl 0 I 0 0 10 N

A-K

(5. 4)

Since a(B 1 )ua(B 2 l c a(N), it follows from Proposition 5.6, Lemma 5.7 and Proposition 1.15 that there exist nilpotent operators Q. such J that IIB 1EBN-Q 1 11 < £/4 and IINEBB 2 -o 2 11 < £/4. Hence, there exists R in l(H), II Rll < £/4 such that

~

A-(K+R)

Q = [: 2 k.

kl+k2

A straighforward computation shows that if~· J= O,then Q =0, J i.e., Q £ N(H) and therefore distrA,N(HJJ < IIKII+IIRII <£.Since£ can be chosen arbitrarily small, it readily follows that A£ N(H)-. The proof of Theorem 5.1 is complete now. 0

5.2 Closures of similarity orbits of normal operators with perfect spectra THEOREf.1 5. 8. Let N £ L (HI be a norma Z operator suah that a (N) is a perfeat set; then S(N) is the set of aZZ those operators A in L(HJ satisfying the following aonditions (i) a(A) ~ a(N) and eaah aomponent of cr(A) interseats cr(N); (ii) ae (A) ~ oe (N) (=a (N)) and eaah aomponent of ae (A) interseats cre (N); and (iii) ind(A.-A) = 0 for aZZ A.£ Ps-F(A). PROOF. Since o(N) oe(N) = a~re(N), the necessity of the conditions (i), (ii) and (iii) follows from Corollary 1.6(i) and (iii) and Theorem l.l3(iii), exactly as in Theorem 5.1. Assume that A£ l(H) satisfies (i), (ii) and (iii) and let£> 0. Minor modifications of the proof of Theorem 5.1 (formula (5.4)) indicate that we can find a compact operator K, IlK II < £/3 such that

A-K

::::!

N2 0 lc o I 1 0 s 21 o c2 0 0-~-B 1- 0 I 0 0 ,o Nl

-

-

ED

N4 0 IIC 3 0 0 8410 c4 o -o-i'i~ 3 -o- , I 0 0 10 N3

(5.5)

where Nj::::t Nand a(Bj) c a(Nj), j = 1,2,3,4. By Lemma 5.4 we can find an operator n such that N sJ.m "t M, a(N) c cr(n) = oe(M) c a(N)£, o(M) is the closure of a Cauchy domain and liN-Mil

113

L(H), IIRII < e/3, such that B =A(K+R) is the operator obtained from (5.5) by replacing the N.'s by nor J mal operators M. "'M such that IIN.-M.II < e/3, j = 1,2,3,4. J J J Since a(M) is the closure of a Cauchy domain, it is very easy to prove that M = A1eA2eA 3eA 4 , where the Aj's are normal operators such that a(A 1 lna(A 2 ) = a(A 3 )na(A 4 ) = ~. a(Aj) is the closure of a Cauchy domain and each component of a(M) contains a component of a(A.) forall J j = 1,2,3,4. By Corollary 5.5 and Lemma 5.7, N s1m U = (A 1eA 2 )e(A 3eA 4 ) and A. ~ M.$B., j = 1,2,3,4. Thus, we can find operators A! ~A. J S1m J J J J such that 11Mj$Bj-Ajll < e/3, j = 1,2,3,4. Since o(Ailna(A2) = a(Ajln a(A4l = ~. it follows from Corollary 3.22 that < e/3. A fortiori, we can find R

M

~

€

C

Hence, N s1m C and IIA-CII < IIA-BII+max{ 11Mj$Bj-Aj II= j

1 , 2, 3, 4} < II K II+ IIR ll+e I 3 < E •

Since e can be chosen arbitrarily small, we conclude that N s1m & 0

5.3 Limits of algebraic operators THEOREM 5.9. The c~osure of the set Aig(H) of a~~ algebraic oper~ tors acting on H coincides with the set of all those operators A in L(H) satisfying the following condition ind(). -A) = 0 for all A. e: ps-F(A). (5.6) PROOF. The necessity of (5.6) follows from Theorem 1.13(iii). On the other hand, if A satisfies (5.6) and e > 0, then we can proceed as in the proof of Theorem 5.1 in order to show that A-K has the form (5.4) for a suitable compact operator K, IIKII < e/3, where N is a normal operator such that a(N) =a (N) =a (A) = a(A-K) ~ o(B.), j = 1,2. e e J By Lemma 5. 4 we can find a normal operator l1 such that a (ri) is the closure of a Cauchy domain, a(N) c a(f1) c a(N)E/ 3 and liN-Mil < t./3. Thus, if B is a suitable operator in L(H) unitarily equivalent to the result of replacing the N's by M's in the 4 x 4 operator matrix of (5.4), then IIA-BII < IIKII+IIN-MII < 2e/3. Since r1 is normal and a(t1) has only finitely many components, it readily follows from Theorem 5.1 that M € Aig(H)-. Since a(M) is a perfect set and a(Bj) c o(M), j = 1,2, it follows from Lemma 5.7 that M 7 M$B., j = 1,2. Since Alg(H) is invar s1m J iant under similarities, it follows from Proposition 1.15 that M$Bj € Alg(H)-, j = 1,2. Thus, we can find algebraic operators R1 , R2 such that IIMeB 2-R 2 11

114

< £13 and !IB 1 et-1- R1 11 < c/3.

It is completely apparent that if R1 , R2 are algebraic operators, then so is

whence we deduce that

Since c can be chosen arbitrarily small, we conclude that A belongs to Alg(H)-. 0 Observe that the condition (5.6) is invariant under compact perturbations. By using this observation, our remarks at the beginning of Section 5.1.2 and Theorem 5.8, we obtain the following COROLLARY 5.10. Alg(H) Alg(H1-+K(H) ~ [Alg(H)+K(H)Jl(H): R is similar to a normal operator with finite spectrum} l(H):

indO.-A)

~ 0

{R e ~{A

e

fox• all). e Ps-F(A)}.

From Theorem 5.1 and Remark 5.2, we obtain COROLLARY 5.11. Suppose that A e l(H), ind(),-A) ~Ofor all A e and cr 0 (A) is a finite set and ae(A) has finitely many components a 1 , a 2 , •.. , a • Let A. be an arbitrary point of a., j ~ 1, 2, •.• ,m. m J ~ J Then there exists a sequence {Rn}n~l of algebraic operators such that a(R ) ~ {)..} .m 1 ua (A), R IH (a (A) ;R ) ~ AjH (a {A) ;A) for all n ~ 1,2, ••• n J J~ o n o n o and IIA-Rnll + 0 (n-+- oo). p 5_F(A)

COROLLARY 5.12. Let A be a finite nonempty subset of~ and let S(A) ~ {A e L(H): cr(A) c A}; then S(A) is the set of all operators T satisfying the conditions (ii Each component of a(T) or oe(T) intersects A; and (ii) ind{).-T) ~ 0 for alZ ). e Ps-F(T). Fur>thermore, S (A)-~ {A e S (A): A is algebraic}-.

5.4 Normal operators in closures of similarity orbits PROPOSITION 5.13. Let T e l(H) and Zet N be a normaZ operator such that o(N)~ o(T) and dim H(A;N) ~ dim H(A;T)for each isolated point X

of a (T). Then T

sim

N.

115

It is completely apparent that if T and N are related as in the above proposition, then cr o (N) = cr o (T) and cr e (N) = oB(T) = cr(T)\cr 0 (T). It will be convenient to provide a separate proof for the case when cr(T) is totally disconnected. LEMl1A 5.14. If T ~ L!H), cr(T) is totally disaonneated and N is a no1'maZ. operator suah that cr (N) = cr (T) and dim H ().;N) = dim H (;\ ~T) for eaah isolated point A of cr(T), then T sim N. PROOF. Let T and N be as indicated and let £ > 0. Then we can find a finite covering {Qj}j:l of cr(T) by pairwise disjoint open sets such that Q.ncr(T) ~~and diameter n. < £/4, for all j = 1,2, ••• ,m. J J Thus, i f Aj ~ Qj' then Qj c D(~j'£/2) (j = 1,2, •.. ,m). By Corollary 3.22, T ~ $. 1 T., where T.= TIH(o(T)nQ.;T). Let N. J= J J J J = NIHCcr(Tj)~N). It is easily seen that dim H(a(T)nnj~T) =dim H(a(T)nQj; N) for all j = l,2, ••• ,m. By Corollary 3.35, there exists T~ E J L[H(cr(T)nQ.;N)], T'. ~ T. such that IIT~->...11 < £/2. On the other hand, i t J J J J J is completely apparent that liN.-;\ ·II < £/2. m J J Hence, T ~ $j=lTj and IIN-$j: 1 Tjll

max{IINj-Tjll:

j = l,2, ••. ,m}

s max{IINj-).jll+ll).j-Tjll:

j = l,2, ••• ,m} < £,

0

whence we readily conclude that T sim N.

PROOF OF PROPOSITION 5.13. Let a denote the set of all non-isolated points of ae(T). Given£> 0, we can use Corollary 3.22 to obtain that T ~ A$B, where A= TIH
c

o(A). A fortiori, a(C)

c

o(M),

which is a perfect set. Given n > 0, let n 1 , Q be analytic Cauchy domains such that o(M) c n1 c n1 c Q c o(M) and each component of Q intersects a(M). Let M1 n be a normal operator such that o(M1 J = n 1 • By Lemma 5.4 we can find M normal such that o(M ) = a (~1 ) = Q- and IIM-M II < 2n and, by Coroln n en n lary 5.12 there exist algebraic operators R1 , R2 such that o(R 1 )uo(R2l 116

c fl\fl 1 -, a(R 1 )na(R2 ) M_ (ao)

II

=

fl, IIR1 -M11 1BM1 1BM+(afl 1 l(ao)ll < n andi!R 2-\eM1 e

< 11. By Lenuna 5. 3 and Corollary 5. 5,

A s!m M11 eR1 1BCIBR 2 • Since o(R1 ), a(R 2 ) and o(C) are pairwise disjoint, it follows from Corollary 3.22 that M11 1B(R1 eceR 2 ) - M11 1BL(C';x1 ,x 2 ,x 3 ), where M sim M

s!l'(l M11 1BR1 1BR 2 , so that

L(C';X 1 ,x 2 ,x 3 )

=

1:l :~ ::] , 0

0

R2

C and x1 , x2 , x 3 are arbitrarily chosen operators. It follows from (3.2), Theorem 3.23 and our choice of R1 and R2 that x 1 , x 2 , x 3 and C' can be chosen so that

C' -

II L(C' ;x1 ,x 2 ,x 3 )

- M1 <2 >IBM(an 1 > {ao)

II

< E.

Hence, S(A)- contains an operator A' such that IIA'-M IBM1 <2 >e M(an 1 > (ao) II< n· Since M11 eM 1 ( 2 )1BM(an 1 > (co) ::ta M11 (Lenuna and 11 can be chosen arbitrarily small, we conclude that A s!m U. D

4.ai

5.5

Sums of two nilpotents

LIHI; however,

THEOREM 5.15. N(H)-+[N(H)-nNo4(H)] +K(H).

1

t NIHI+NIHI

PROOF. LetT£ LIHI and~ £ aoe(T). By Proposition 4.29 there exist K £ KIHI and a decomposition H = H0 eH 1 (co) (H 0 , H1 are infinite dimensional spaces) such that T

=

AIBR(ao)+K, A

L {H 0

E

1, R

* R

= [:

B 0

L(Hl)'

E

A

"' T, R .., R{ca), and

T *

M~,

~ M3

where o(B) c ae(T). Let e: > 0 and let Mk be a normal operator such that cr(t1k) j;l.l s sp(T)+(l-2-k)E} (k = 1,2, ..• ). Define

{).:

Me:= OIB(OIB ••• IBOIBMiiBOIB ••• IBOIBMiiBOIB ••• IBOIBMk_IBOIB ••• ) (with respect to the decomposition H = H0 IBH 1 (co)) , where the direct sum mand Mk occupies the nk-th coordinate,

Mk

=

["' ~

0 0 0

·r 0

M2 , k Mk M3

1,2, ••• ,

117

and 1 s n 1 Let

<

n2

<

•••

< ~ <

••••

be the operator obtained from R by replacing the ~·s in the (1,1) and (3,3) entries by two copies of Mk• k = 1,2, .•. , let L be the operator obtained from AeR(~) by replacing R by Lk in the nk-th coordinate (k = 1,2, ••• ) and defineS = L+K. E It is completely apparent that T = S +(~-M ) . Since ~-M is nore: E E mal and a(~-ME ) = {A £ ~= jA-~1 ~ a+e:} is a closed disk containing the origin, it follows from Proposition 5.6 that M € NIH)-. On the E other hand, it is clear that a e (S e: ) =a... re (S e: ) = {A £ ~= jAj s a+c}. If a(S) =a (S ), then S € N(H)• (by Theorem 5.1) and we are done. E e E E Thus, in order to complete the proof of the first statement, we only have to show that if nk + ~ (k + ~> fast enough, then a(S) =a (S ). E e E Let Pn denote the orthogonal projection of H onto the n-th copy of H1 and let P0 be the orthogonal projection onto H0 • Clearly, if ~n-1

~n-1

Kn = K- (1-L.j=O Pj)K(l-L.j=O Pj), then IIKnll + 0 (n + co). By Corollary 1.2(ii) there exist constants &k' &1 > &2 > ••• > &k k > ••• > 0 and &k + 0 (k + ~>, such that a(Te[ej=l LjJeCk) c a(~~+ 1 ), provided II Ck II < &k (k = 1, 2, ••• ) • Let n 1 ~ 1 be the first index such that II Kn 1 II < &112 and define nk inductively so that 1. s n 1 < n 2 < ••• < nk < ••• and IlK II < &k/2 for all k = 1,2, •... Defl.ne S as above nk e: with this particular sequence. It is obvious that a(S) =a (S )ua (5) e: e o = a(M )ua (S ). Assume that a (S) ~~and let y E a 0 (S ); then jyj = E 0 E 0 E E a+e:+n for some n > 0. By Corollary 1.2(i) there exists & > 0 such that a(S +F)nD(y,n/2) ~~for all Fin L(H) with IIFII < &. Let h be large E enough to guarantee that &h < 2& and observe that (if n 0 is defined to be equal to 0) (n.-n. -j) n {(Ae[e.h 1 {R J J-l eL.}])+(K-K )jran~. hOPJ.} S -K L+(K-K ) J= . J nh J= e: nh nh (n.-n. 1 -J) co J Je[e._h+l{R eL.}J. J(n.-n. -j) J ~ J J-1 ) . a (M ) , we conSince IlK II < & and a<•j=h+l{R eLj} E nh elude that (n. -n. -j) n a(Ae[e.h 1 {R J J-l eL.}J+(K-K ) Iran l· ho PJ.) J= J ~ J= is not contained in a(M ) • E Since a(R) c a(Me:), it easily follows that if

118

V = (AGl Gl.h {R. J=l J

(n.-n. -j) n J J-l GlL.} +(K K ) I '\ h P.)GlR(oo), J - ~ ran l. j=O J

then o(V) is not contained in o(Me:). But, on the other hand, it is not difficult to check that v TGl[Glj=~ Lj)+Ch, where ch is compact and IIChll s 211Knh11 < oh. Hence, by our definition of oh, o(V) c o(~+l)

=

o(Me:), a contradiction. Hence, o(Se:) = oe{Se:) = o{ME), whence it follows that T = Sc+{~­ M) E: N(H)-+[N(H)-nNoll(H)J. E Finally, observe that if 1 = o1+o 2+c, where oe{Q 1 )uoeC0 2 l c D(O,~ and C is compact, then l-Q 2 = Q1 +C and therefore l-Q 2 = Q1 , so that oeC0 1 l=oefl-Q 2 l = oe{Q 1 lnoe(l-0 2 )cD(O,~)nD(l,~) =~.a contradiction. Hence, 1 ; N ( H ) +N (H) +K (H) • 0 A simpler construction yields the following PROPOSITION 5.16. For each T

E

L(H), dist[T,N(H)] s; sp{T).

PROOF. Let e > 0 and let {~n}lsnsm {1 s; m s; oo) be a dense subset oe(T) (c oR.re{T)). By Proposition 4.29, there exists K E K(H), IIKII < e/2, and a decomposition H = H0 GlH 1 {H 0 ,H 1 are infinite dimensional spaces) such that

of

T = AGlR+K, A ( L(H 0

),

R

E

L(H 1 ),

A~

T, o{R)

oe(T) and

where N {diag{~ 1 .~ 2 , ••• ,~n····l> {oo)= Gll~n~m ~nln (ln denotes the identity on a subspace Rn = H1 >. By Lemma 5.4 we can find normal operators Mn E L(Rn) such that o{Mn) = oe(Mn) = {t~n= 0 s; t ,;; l}e/ 4 and 1Jllnln-Mn11 < l>~nl+e:/2. Let M = Gll,;;ns;m Mn; clearly, o(M) = oe(M) = [ul~n~m o{Mn)]

= u>te:o(T) {ttJ:

Ostsl}e/ 4

is connected and contains the origin. Thus, if S is the operator obtained from R by replacing the two N's by M's, then o(S) oR.re(S) = o{M) and therefore (by Theorem 5.1) S e: N!H 1 )-. It is easily seen that E = o(A)\o(S) = o(T)\a(M) is a finite {eventually empty) subset {~ 1 .~ 2 , •.. ,~p} of a 0 {A). Hence, _ [AIH(E;A) A0

* ]H(E;A) A0 H0 eH(E;A)

where H(E;A) is finite dimensional and a(A0 ) c a(S). It readily follows that A IH ( E; A) = L+O, where L is a normal diaqonal opera tor such 119

that II Lll = max{ I Aj I: 1 ,; ; j ,; ; p} ,;;; sp(T) and Since cr(A0 ) c cr(S), it follows that cr ((A-LeO) es) = cr (Q) ucr (A0 ) ucr (S) N(H)

c

is a nilpotent.

Q

cr (S) •

Since S E N (fl}-, the above inclusion implies that Tl = (A-LeO) es (Use Theorem 5.1). On the other hand, by construction

€

IIT-T1 11 = max{IILIIriiR-SII +IJKIIl < sp(T)+e:. Since

£

can be chosen arbitrarily small, we conclude that dist[T,N(H)] s sp(T).

It readily follows from Theorem 5.1 that the upper estimate given by Proposition 5.16 is very poor, in general: namely, if a is hermitian, cr(H) = [0,1] and £ > 0, then there exist nilpotent operators g 1 and Q 2 such that IIH-Q 1 11 < £ and IIH-(l-Q 2 lll < r:. Hence, cr(l-Q 2 l = {1} :md sp(l-Q 2 l = 1, but dist[l-Q 2 ,N(H)] < 2£! Better estimates will be given in Chapter XI (second monograph) •

5.6 The Apostol-Salinas approach:

An estimate for the distance to

Nk(HJ LEMMA 5.17. LetT E N(H). Then fo~ eve~y n > 0, 8 > sp(T) and every positive integer k there existn Q € Nk(HeH) cuah that II- oil < aJITIJ+s+ IIT:l11 • nS PROOF. Let U and V be two isometries in L (H) such that VV*+UU* = 1. ·-1 we define for 1 s j s k the subspace M. of HeH given by M. = {TJ xe "l"L ] J nS]- vJ--ux: X € H}. (Note that X

€

H}

I

i.e., M. is the image under an isometry of the graph of the transpose of (1/n~j-l)Tj-l. Hence M. is closed, i.e., it is a subspace.) ]

It is easy to see that MjnMh = {0} i f 1 s j,h s k, j ,P h. This is due to the fact that the second components of the elements in M. and ~hare k orthogonal. Let M = '· 1 M. and let {y } 1 be a sequence in M such l.J= J m m= that lim(m ..,. coJIIY II= 0. Since M .nMh = {0}, j ,P h, we can write uniquely k m · J ·-1 ·-1 y = }". 1 y ., where y .= TJ-1x .$nf:\J vJ Ux ., for some x ·" 11, m J= m,J m,J m,J m,J m,J 1 5 j ;:; k, m = 1,2, •.•• Since lim(m ... co) IIY.mll = 0 I lim(m .. co) IlL .~ 1 nSj-lvj-lux .11 = 0. Then lim(m->-oolllvj-lux J-11 = 0, 1 < j ;:; k, Jm,J m, and hence lim(m->- coJIIx ·II= 0, 1-> j s k. Therefore, lim(m->-ooJIIY m, ]·II m,J 00

120

]

= 0, 1 s j s k, and hence the algebraic direct sum M is closed, i.e., it is a subspace. Now define Q £ l(HCBH) by .l k-1 k-1 QIMkeM = o, OILj=l Mj = (Tesv> ILj=l Mj. Thus, the representing matrix of OIM on Lj~l Mj is of the form 0

0

0

*

0

0

0

*

0

0 0 0

0 0 0

0 0 0 0 0 0 0 0 • • * 0 Therefore it is clear that Qk = 0. Let Pk be the projection onto ker V*k. Then [ (TesPkV) - QJLj~l (Tj-lxjeasj-lvj-luxj) k j-1 k-1 j-1 k j-1 j-1 [TLj=l T xj-TLj=l T xj]8[SPkVLj=l aS V Uxj k j-1 j-1 k k_k ~k-1 j-1 j-1 -SVLj=l aS V uxj] = T xk8[Pka8 v-uxk+~j=l aS V Uxj] k

T xkeo. Since 1 k-1--k-1 1 k j-1 j-1 llxkll=-"k-TIIaB v- Uxklls-"k-111/.j=laB v Ux.ll aS aS J 1 k j j-1 j-1 s -"k-1 lit j=l T xjeas v uxjll , aS k k-1 we conclude that IIC(TCBSPkV)- QJIMII s IIT-11/aS . Hence II(T80) -QII s II(T80)PM.iii+II[(T80) -QJPMII :s IITII.IIPHe{O}PM.ill k.

+II[(T88PkV)- Q]PMII+II08SPkVII s IITII.IIPHe{O}PM.iii+B+IIT Il/a8

k-1

•

1

Thus in order to complete the proof it suffices to show that IIPHe{O}PM±Ils a. Notice that M~ = {yez: +a = 0 for all x, H}. lienee M~ =- {(-aU*)zez: z, H} and therefore IIPHe{O}PM.lll = IIPHe{O}~M~II 1 s !X• 0 THEOREM 5.18. Let T r; l(Hl and suppose that 0 e a!re(T). Then for every a > 0, S > sp (T), y > 1 and every positive integer k there e~ists Q e N2k(Hl such that k (5.7) liT- Qll s y(ooiTII+s+ IITk~i>· aS PROOF. Let 0 < E < min{aSr (y-1)8/{!fa)}. Since 0 e cr~re(T), we can proceed as in the proof of Theorem 5.1 (formula (5.4)) in order to find K r; K(H), IlK II< E such that

121

o lc 1 o H2 R21o c2 0 - 0 R1- 0Hl I 0 0 10 0 0 0

T-K

where o(R1 l

u

o(R 2 )

c

' -.-

o(T). Furthermore, if£ is small enough, then

max{ll R1,I ,II R/ II} s II- ae, j = 1,2. Thus, if

then Q2k = 0 and IIT-QII s max{II
j = 1,2l+IIKII
The above theorem is especially useful to estiMate the distance from T to Nk(H) for the case when Tis a quasinilpotent. Namely, if o(T) = {0} and IIT,Il/k does not decrease very fast, then there exists a sequence {Q} ~lin N(H) where g has order k (k ~~,as n ~ ~> n n= n n k n such that the rate of decrease of the sequence II T n lf/kn is the Sat!IE! as the rate of decrease of the sequence II T-On II· COROLLARY 5.19. LetT € L(H) be a quasinilpotent ope'f'ato'¥' and let 6 > 1. Then there exists a sequence {Qk}k:l in N(H) such that Qk 2k = 0 and for some constant c = c(T,6) > 0, liT- Okll s c. PROOF. It is a direct consequence of Theorem 5.18: and 8 = 611 ~ If/k in ( 5 • 7) •

Take a

6-k 0

k ~ 2 and we choose 8 = [(k-ll 2 ltrii-IIT,If/(k+l) and a= 8[(k-l) IITIIJ-l in (5. 7), then the left side of this formula takes the form If

yc (k)
€

L(H) be a quasinilpotent operator; then

dist[T,N 2k(H) J s C(k) OITII.IITkllll/(k+l) for all k ~ 2. 122

In the converse direction, we have the following PROPOSITION 5.21. Let T £ L(H)kand suppose that there ezists a se quenae {Qn}n:l in N{H) such that Q n = 0, n = 1,2, ••• and n 1/k lim inf(n + ao>IIT-Qnll n = 0. Then T

is quasinilpotent.

PROOF. Since On

kn

= 0, it follows as in the proof of Proposition

D

REMARKS 5.22(i) Let T, ~. B and k be as in Lemma 5.17 and let R € Nm(H 0 ) , where H0 is an infinite dimensional space. Identify H1 "'H with ker R c H0 and let H2= H0 eker R. If 0

1 Rl,m-1 Rlm ker R eker Ro 2 Rl R2,m-l R2m ker R eker 3 R3,m-l R3m ker R eker R2

R12 Rl3 0

R23 • 0

=

Hl

R

0

and 0

€

Nk(HeH 1 ) is constructed as in that lemma, then R'

l,m-1 Rim HeHl 2 R ker R 9ker Rl 2,m-l 2m 3 2 R3,m-l R3m ker R 9ker R R

L =

. 0

where

Rb

(~lj)

0

1\n-J,m ker Rm-leker Rm-l 0 ker Rmeker Rm-l

maps ker Rjeker Rj-l into HeH 1 , is a nilpotent of or

der at most k+m-1 acting on.HeH II
<

and 0

k

all T I~B+ ~ aB

(ii) Assume that TIE L(O:d), 0 < d < oo, R € Nm(a:P), a > 0 and B > sp(T), and let k be a positive integer. Then we can apply the above construction to TeO and R(s)eo (where 1 s s < oo and 0 acts on an infi123

nite dimensional space). If s = (k-l)d, then the operator L admits are ducing subspace N of dimension [(k-l)p+l]d such that LIN~= 0 and max{ll x- PNxll :

d

x£a:,llxll=l}

is small enough to guarantee the existence of a unitary operator U map ping PN(a:d) onto a:d such that IIU -111 < a./2. I f M =UN, then M => a:d, dim M = [(k-l)p+1]d and there exists L' € Nk+m-l(M), L' ~LIN such that k II QEilR (k-l)- L 'II < 2a./2 II T If+- ( 1+2ai2Ha.ll T lf+-B+ II Tk!l1 } Furthermore, if nul R = n

~

aS

2, then M can be chosen so that

dim M = d+ [
5.7 Salinas' pseudonilpotents We shall say that T € LIHI is a pseudonilpotent operator if for every E > 0 there exists a decomposition H = 8lj~ 1 Hj of H into the direct sum of a finite orthogonal family of subspaces H1 , H2 , ••• , Hn s·uch that if Tl,n-1 Tln T2,n-l T2n T

Tn-:I.n-1 Tn-l,n Hn-1 Tn,n-1 Tnn Hn then the norm of

0

Tn-1,1 Tn-1, 2 Tn1 Tn2 •

Tn-J.,n-1 Hn-1 Tn,n-1 Tnn Hn

is less than e: • It is clear from the definition that a pseunilpotent operator is 124

an operator that can be "obviously" approximated by other hand, if Q is a nilpotent of order n, then it parent (see (5.1)) that Q admits a decomposition of 0. Thus, if we denote by ~(H) the set with 0 1ower patents, then

N(H)

c

~(H)

c

nilpotents. On the is completely apthe above type of all pseudonil-

N(H)

( 5. 8)

We shall see immediately that both inclusions are proper. By interchanging the subspaces H1 , H2 , ..• , Hn in the definition of pseudonilpotency it is easy to see that an opera tor T is in ~ (H) if and only if for every r > 0 there exists a decomposition H = ej~l ~ ·we"' t.hat the nor111 of Tupper is smaller than e:. (Tupper is the triang~ lar ooerator matrix with (i,i)-entries Tij for i > j and 0 for i s j.) It readily follows that ~(H) =~(H)*= (def) {T E l(H): T* E ~(H)}. THEOREH 5.23. Let A= H+K, wlzcY'f: H ·is he1•mitian and K is compact; thPn A E ~(H) if and on7y if A is a compact quaainilpotent. PROOF. Suppose that A= H+K <

r

~(H)

is not compact and let 0

<

e:

IIA!V3. Let H = E9j~l Hj be a_decomposit~on of the space with respect

to which IIAlowerll < e:: then 11Hlower11 = I!Alowerll s IIAlowerll < e:. Since H is hermitian, II Hupper II = II il 1 ower II < £ • Let Hd.1ag = H11e H~ 2 e ••• eHnn be the diagonal of H with respect to the above decomposition. It is easily seen that I!Hd.1ag II s llii 1 ower II < e: and, a fortiori, 11 1\li=lliill=llii lower +ii upper -ii. d1ag II< 3e:. Since e: can be chosen arbitrarily s!llall, we conclude that I!AII = 0, i.e., A is compact, a contradiction. On the other side, if A E K(H)n~(H), then A E N(H) and therefore it must be a compact quasinilpotent (Theorem 5.1). Conversely, if A is a compact quasinilpotent and {Pn}n:l is an i~ creasing sequence of finite rank orthogonal projections such that Pn ~ 1 (strongly, n • oo), then IIA-PnAII-> 0 (n..,. oo). From the upper semicontinuity of the spectrum (Corollary 1.2(ii)), given e: > 0 there exists a positive integer n 0 such that if n > n 0 , then cr(PnAPn) c D(O,e:/3). Lt;,t m > n o so that IIA-P mAll < e:/3. Since ran Pm is finite dil!lensional there exists a basis e 1 ,e 2 , ••. ,ek of PmH on which the representing matrix of the operator P111Alran Pm is in upper triangular form. Observe ti.at the diagonal entries of this matrix are in absolute value less than e:/3. Letting Hj = V (ej}, j = 1,2, ••• ,k, and definin~ Hk+l= ran (1-Pm), we deduce that with respect to the decomposition H = e~~I Hj, IIAlowerll < c. Since e: is arbitrary we conclude that A E ~(H). 0

125

It follows from Theorems 5.1 and 5.23 that if H is hermitian and a(H) = [O,lJ, then H E N(H)-\IJI(H). On the other hand, i f BE L(H) is the unilateral weighted shift, with respect to some ONB {en}n:l' defined by--Be =a e +l' n=l,2, ••• , a + 0 (n +co) then B is a compact n n n k n quasinilpotent, but B ~ 0 for all k = 1,2, ••• , so that BE IJI(N)\N(H). Hence, we have COROLLARY 5.24. N(H) ~ IJI(H) ~ N(H)-. The following result provides a simple way to construct operators in IJI(H) with positive spectral radius. LEMMA 5.25. Let T E L(Hk and suppose that there exists a sequenae {Qn} n:l in N (H) suah that Qn n = 0 and lim inf(n

+

co)/k n IIT-Qn II= 0.

Then T E IJI(H).

PROOF. Given e:: > 0, let n be sufficiently large so that lknll T-~11 For each j, 1 ~ j ~ k , let PJ. be the orthogonal projection of H . . 1 n onto ker QnJeker QnJ- • Also let x e H~ then <

E •

III~~ 1 I~~j

PiTPjxll

= III~~ 1 I~~j

Pi (T-~)Pjxll

~ L~~lii
Pi)

(T-~)Pj 11-IIPjxll

~liT-O--n III~n 11P.xll ~liT-~n !Ilk nllxll. J= 1 J Therefore III~~ 1 I~~j PiTPjll s lkniiT-~11 < e::, as desired.

0

COROLLARY 5.26. Let qk € L(~k) (k = 1,2, ••• ) be defined as in (2.1) and 1-et S be the unHatera1- shift of muZtipUaity one . If either a= 13 = co or a n ~ 0 ro ~ a n ~ co) for infinite1-y many n's, then the operator T = {e co q (an)}es n=l n is a pseudonHpotent with speatraZ radius sp(T) = IITII = 1. PROOF. It is clear that sp(T) = IITII = 1. On the other hand, by Corollary 2.24 (or minor modifications of its proof), there exist oper ators Qk "'qk(co)E9{$n~i 1 qn(an}} such that ILT-Okll < 4/k (k = 1,2, ••• }~ Thus lkltf -·Q~j + 0 Xk +co) and since als"O Qk = 0, Lemma 5.25 implies T e IJI(H). 0 PROPOSITION 5.27.(i) Let T e L(H) be suah that lim inf(n

126

+

co) kli Tk lll/k

=

0.

Then T is pseudoniZpotent and quasinilpotent. (ii) Howeve~, given a sequence t = {ck}k:l of positive ~eals inc~easing to ~ a~bit~arily sZowly, there e%ists a compact quasinilpotent unilateral weighted shift B = Bt in ~(HI such that lim(k

+ oo)

ckiiB~Il/k

PROOF. The first statement is an immediate consequence of Corollary 5.19 and Lemma 5.25. Define B as in the proof of Corollary 5.24 with weights a 1 = a 2 = =a 1, an 1+ 1= .•• = an 2= 1/2, ••• , a = 1/j, = ••• =a n1 nj-l +1 nj •.• , where {nj}j:l is a strictly increasing sequence of natural numbers. Given the sequence {ck}k:l' it is not difficult to define the S! quence {n.} .: 1 inductively so that lim(j + oo)c /j = ""· It follows J Jnj-1 that for j > 1, if n. 1 < k s n., then J-

J

ckllskll 1 /k > c

n. 1/n.

nj-1

liB Jll

Since B is a compact quasinilpotent, B lim(k 5. 28. Let B then A e: ~(H) and lim(k value of k. REMARK

+

+ oo)

J > c £

~(H)

nj-1

/j.

(Theorem 5.23) but

ckiiBklll/k = .. •

Bt be as in Proposition 5.27 and let A =B (oo), oo) ckiiA,Il/k = .. , but Ak is compact for no

Very little is known about the structure of the pseudonilpotent operators. In [149,Lemma 5.1], D. A. Herrero proved that if H is hermitian and cr(H) = [-1,1], then dist[H,Nk(H)J s 5/log k for all k large enough. On the other hand, Theorem 5.23 and Lemma 5.25 imply that dist[H,Nk(H)] ~ o(l/lk). But the gap between 0(1/log k) and o(l/lk) is still very large. (See also the results of Section 2.3.3.) The following proposition follows immediately from the definition of pseudonilpotency. PROPOSITION 5.29. (i) If A e: L(H) has the foPm of Lemma 3.36 and Ajj e: 'I' (Hj) for aZZ j = 1,2, ••• ,k, then A E ~(H). (ii) If A e: V(H), then U(A) c 'I'(H). 127

COROLLARY 5.30. If N is no~mal and cr(N) is equal to the closed unit disk, then N is a pseudonilpotent. PROOF. Let M be the normal operator defined in the proof of Proposition 2.2B(ii). Since the matrix of (Lk- Qk)upper has no more than four non-zero entries in each column or row, a suraightforward computation shows that if M= (M .. ) . . k 1 with respect to the decomposition k l.J l.,J= H = Qlj=l Hj, where Hj is the subspace defined by H. = ker[($ co

A Q) (co)]jeker[($ co A Q) (co)]j-l m=l m k m=l m k 2 (j = 1,2, ••• ,k), then IIMupper II < 20 (n/k) l/ , whence we conclude that M is a pseudonilpotent. Now the result follows from Lemma 4.8 and Proposition 5.29(ii) J

0

Theorem 5.23 and Corollary 5.30 suggest the following PROBLEM 5.31. Which normal operators are pseudonilpotents? PROBLEM 5.32. Let A € '!'(H). (i) IfWEG(H), is WAW-l € 'I'( H)? (ii) Is Ap(A) € '!'(HI for all polynomials

p?

PROBLEM 5.33. Is every quasinilpotent a pseudonilpotent?

5. B Limits of nilpotent and algebraic elements in the Calkin algebra Most of the results of Sections 5.1 through 5.7 induce analogous results in the Calkin algebra: namely, we have THEOREM 5.34. The cZosu~e of the set N(A(HII of aZZ nilpotent eZ~ ments of the Calkin aZgeb~a coincides ~ith the set of aZZ those A € A(H) satisfying the conditions: (ii'J cre(A) is connected and contains the o~igin; and (iii') ind(A-A) = 0 for aZZ A € ps-F(A). Moreover, N(A(H))- = n[N(H)J- = n[N(H)-J andN(H)-+K(H) [N(HI+ K(HIJ- = n-l[N(A(H) 1-J is closed in L(Hl. PROOF. The necessity of the conditions (ii') and (iii') exactly as in Theorem 5.1. On the other hand, if A satisfies (iii'), then it follows from Theorem 3.48 that A+K satisfies tions (i), (ii) and (iii) of Theorem 5.1 for some Kin K(H). A+K € N(H) and, a fortiori, A= A+K E n[N(H)-J c n[N(H)J c whence the results follow. 128

follows (ii') and the condi Therefore N(A(HII-,

0

Similarly, from Sections 5.2 through 5.6, we obtain the following corollaries.(All the proofs follow exactly the above scheme and will be omitted.) Of course, similarity orbits in A(H) are defined with respect to the invertible group G(A(H) )= {A: A £ L(H} is a Fredholm o~ erator}. THEOREM 5.35. Let N £ L(H} be a normal operator suah that ae(N) is a perfeat set; then S(N)- is the set o.f aU those A£ A(H) satis.fying the following aonditions: (ii')

a

e

(N);

ae(A)

ae(N) and eaah aomponent o.f

=>

ae(A) interseats

and

(iii')

ind(A-A)

=

0 for all A£ Ps-F(A).

THEOREM 5.36. (i) The alosure of the set Alg(A(H)) of all algebraia elements of A(H) aoinaides with the set of all those A in A(H} satisfying the aondition (5.6). (ii) Alg(A(H)) = {R £ A(H): R is similar to a normal operator with finite essential speatrum} = n[Alg(H)J- = n[Alg(H)-1. (iii) If A is a finite nonempty subset of~ and A(H;A) = {A £ A(H): oe(A) c A}, then A(H:A) is the set of all those T suah that eaah aomponent of ae(T) interseats A and T satisfies (5.6). Furthermor~ A(H;A) = {A £ A(H;A): A is algebraia}-. PROPOSITION 5. 37. If T £ A (H) and N is a normal operator- suah that ae(N) = aw(T), then T sim N. THEOREM 5. 38.

N (A (H))- +{N (A (H)) nn[NOII.( H) J}-

(i)

A(H); however,

I ' N(A(H) )+N(A(H)). (ii)

For eaah T

£

A(H), dist[T,N(A(Hl)] s sp(T).

It will be shown later (Theorem 7.2) that if Q £ Nk(A(H)), then Q can be lifted to an operator Q £ Nk(H). By using this result, we have THEOREH 5.39. If T

£

A(H) and 0

E

oi.re(T), then for every a. > 0,

S > sp(T), y > 1 and every positive integer k there existsQ £N 2k(A(H))

such that

Finally, observe that pseudonilpotency admits the followin9 expression in A(H): T E A(H) is a pseudoniZpotent if for every£> 0 there exists a decomposition i = Lj~l Pj, where {Pj}j~l is a finite 129

family of pairwise orthogonal hermitian idempotents, such that

II Lj~lLi~j i\TP j II

<

E.

With this definition in mind and using the well-known fact that every hermitian element H of A(H) can be lifted to an hermitian oper~ tor H0 in LIH) such that cre(H0 ) cr(H), we have the following THEOREM 5.40. (iJ

MIAIH))

~ ~(A(Hll ~ N(A(Hl )-. Indeed, ~IAIHl )\

N(A(H)) aontains quasiniLpotents as ~eLl as elements T suah that sp(T)

=IITII= 1; ifH is a non-zero hermitian, then HE: N(A(H))-\V(A(H)). ( i i ) If lim(n + oo)fk IIT-Qll = 0 for a sequenae {Qn}n:l c N(A(Hll _ kn n_ suah that Qn = 0, then T E: V(A(H) ). (iii) If lim(k + oo)lkJITklll/k = o. then Tis pseudoniZpotent and quasinilpotent. (ivJ Ho~ever, given a sequenae ~ = {ck}k:l of positive reaZs inareasing to oo arbitrarily slo~ly, there exists a quasinilpotent B in -k 1/k V(A(H)) suah that lim(k + oo),lckiiB II = 00 • PROPOSITION 5.41. If A

€

V(A(H)); then U(A)- c ~(A(H)).

5.9 On the spectra of infinite direct sums of operators

THEOREM 5.42. Let {Tn}n:l (Tn E: L(Hnl, n00 1,2, ..• ) be a uniformLy bounded family of operators and Zet T = •n=l Tn E: L{H), ~here H =

en:l H0 ; u0 :

then (iJ crp(T) = un:l op(Tn)• (ii) o(T) = [u 0 : 1 o(Tn)Juo (disjoint union), ~here o ={A I {ll(A-Tn)- 111}n:1 is not uniformly bounded}, 1 o(Tn): ~ (iii) m~reover, cr c o!re(T) and eaah aomponent of cr interseats

[un=l o(Tn)J • (iv) If there exists a funation

~(r)

defined on (0,6) suah that

IICA-Tn)- 1 11 s ~(dist[A,cr(Tn)])

(5.9) 00

-

for aZZ A E: o(T0 ) and aLZ n = 1,2, ••••• then o(T) [un= 1 o(Tn)] • (vJ The aondition (5.9) aannot be repZaaed by the ~eaker aondition (5.10)

for all A E: a(T0 ) and all n = 1,2, •.•• even if t(r) has the minimaL possible gro~th: t(r) = 1/r.

PROOF. (i) Let x = 130

s0 : 1 x0

E:

H, A

E:

~ and assume that (A-T)x

0.

Then 0 = (~-T)x = $n:l (~-Tn)xn so that either xn= 0 or ~ e crp(Tn) for each n = 1,2, •.. , whence the result follows. (ii)- (iii) The inclusion un:l o(Tn) c o(T) is obvious. If~ e cr, then there exists a sequence {x }k:l' x e H , llx II = 1, such that nk nk n~ nk 11<~-T )-lx II + 0 (k + 00 ) . Thus, i f yk= ($n~~l O)$x $($ _00 +l 0), nk nk nnk n-nk then IIYkll = 1 and 11<~-T)ykll + 0 (k + oo), Hence, o c crt (T). The same argument, applied toT* indicates that a c ot(T*)* = or(T). Since (by (i)) on[op(T)uop(T*)*J = ~. we conclude that o is ineluded in otre(T). Conversely, if 11<~-Tn) -~~ s; C (for so~e constant C depending only on ~), then it is easily seen that ~ e p (T). Indeed, 11<~-T) -lll s; c. Hence, cr(T) = [un:l cr(Tn)]ucr. Let r be a clopen subset of o and assume that rn[un:l cr(Tn)J- = ~­ Let !1 be a Cauchy domain containing r such that _!1-n[u n= ool cr(T n )]-= ~: 1 then there exists a constant C such that 11<~-T) II s; c for all ~ ~ an. A fortiori, 11<~-Tn) -lll s C for all ~ e n- and all n = 1,2, .•.• Since -1 (~-Tn) is an analytic function defined (at least) in some neighborhood of n-, it follows from the maximum modulus theorem that II
+

co)

{inf 11<~-Tn) xnll:

xn e Hn' llxnll

11<~-Nlxll:

llxll = 1} = 0,

= inf{

1}

so that sup{ 11<~-Tn) -~~: for all

~

e D(0, 1)

n

<:

1} =

oo

\[-1,1]. Hence, o(T) = cr(N) = D(O,l)

0

Now we are in ~ position to show that the result of Corollary 3.40 is the best possible. Indeed, we have

131

EXMWLE 5.43. Given a nonempty finite or denumerable bounded set d = {An}n:l and a compact set r such that every component of r intersects d-, there exists a triangular operator A such that d(A) = cr (A) p = d and cr(A) = r. Furthermore, if d is a perfect set, then A can be chosen so that nul(An-A) = 1 for all n = 1,2, •••• PROOF. Let N beanormal operator such that cr(N) = cre(N) = r. By Corollary 5.12 there exists a sequence {Aj}j:l of algebraic operators such that cr(Aj) = ~A 1 ,A 2 , ... ,Aj} (j 1,2, ••• ) and IIN-Ajll + 0 (j + oo). Define A $j=l Aj. By Theorem 5.42 (see, in particular, theproof of (v)) crp(A) = uj:l crp(Aj) = d and cr(A) = ducr, where cr ={A I d:

{jj(A-Aj)

-1

00

lllj=l is not bounded below}

cr(N)\d

r\d,

so that cr(A) = r. It is easily seen that A is triangular and that the only possible choice for d(A) is d(A) = d. This solves the first part of the problem for the case when d is denumerable. If d is finite, then the first statement follows from a trivial modification of the above argument. If d- is perfect, then it follows from the Bernstein-Schroder theorem (see, e.g., [122]) and Proposition 3.35 (and its proof) that A admits a compact perturbation A-K, where K £ K(H) is normal, such that A-K is triangular, cr (A-K) p d(A-K) = d, nul(A-K-An) = 1 for all n = 1,2, ••• , and cr(A-K) = r.O

5.10 Notes and remarks A classical example due to S. Kakutani (see rll9,Problem 87],[172, p. 282]) with weighted shift operators indicates that a limit of nilp2 tent operators can be an operator with positive spectral radius, so that neither the spectrum nor the spectral radius are continuous, in general. In his survey article "Ten problems in Hilbert space", P. R. Halmos raised the following PROBLEM 7. Is every quasinilpotent operator the norm limit of nilpotent ones? In view of Kakutani's example, Halmos remarked that the above question is "wrong" in the sense that the condition is already known to be not sufficient. The right question should be: "What is the closure of the set of nilpotent operators?" Can it be characterized in simple terms? In [129], J. H. Hedlund found several interesting examples of 132

operators with positive spectral radii contained in N(H)-, including the operator J = ek:l qk (oo) • This example was the key result for the first proof, due to D. A. Herrero [132], of Proposition 5.6. An alternative proof of this result was later given in [26]. (The proof included here is different from both of them.) N. Salinas [180] extended Herrero's result to certain classes of subnormal operators and C. Apo~ tol and D. Voiculescu affirmatively answered Halmos' problem in [32]. (See also [5], [11].) A different proof of thesame result was later given by C. Apostol and N. Salinas in [29]. The complete spectral characterization of N(H) (Theorem 5.1) was obtained by c. Apostol, C. Foia7 and D. Voiculescu [26]. Partial results in that direction were independently obtained by D. A. Herrero [133], [134]. The spectral characterization of Alg(H) is due to D. Voiculescu [194] and the results on closures of similarity orbits were obtained by D. A. Herrero in [140] and [141], respectively. Theorem 5.8 has been extended in [42] to cover the case when oe(N) = o(N) (N normal), but o(N) need not be perfect. The original proofs of Theorems 5.1, 5.8 and 5.9 and Proposition 5.13 do not depend on Voiculescu's theorem, so they are rather diffe£ ent than the ones given here; e. g. , the proof of Theorem 5.8 given in [140] strongly depends on D. A. Herrero's result on the existence of universal quasinilpotent operators [137]. In [135], [136], D. A. Herrero characterized N(A) and Alg(H) (in terms of parts of the weighted spectra; see, e.g., [78]) for the case when H is a non-separable Hilbert space. Theorem 5.15 improves two previous results [133,Theorem 7], [149, Proposition 6.2] of D. A. Herrero. Proposition 5.16 is due to c. Apostol and N. Salinas [29,Theorem 3.5] and the results of Section 5.6 are contained in the same article, except for some mild improvements in Corollaries 5.19 and 5.20, based on Voiculescu's theorem. The observations of Remarks 5.22 are due to D. A. Herrero; they are related with approximation problems in the class of quasidiagonal operators ([149, Theorem 5.11], [150,Section 2]; see also Section 6.2.1 below). The notion of pseudonilpotent was introduced by N. Salinas in his article [180], which also contains Theorem 5.23 (in fact, a slightly weaker form of the theorem; see comments at the beginning of tl50,Section 6]), its corollary and Problem 5.32. The results going from Lemma 5.25 through Remark 5.28 are due to c. Apostol and N. Salinas [29]. Proposition 5.29 contains two unpubliShed ~rvaticns of D. ~. Herrero. Corollary 5.30 is an unpublished result of I. D. Berg, who also raised Problem 5.31. The results of Section 5.8 are tacitly oontained in the corre133

sponding articles on approximation of operators. In particular, L. A. Fialkow proved in [83] that if N is normal, then

{WNV:

~

~

~-1

W,V are Fredholm operators and W

=

~

V}+K(H)

S (N) +K (H).

Finally, Theorem 5.42 is a particular case of [144] (Theorem 1 or Theorem 2; see also [84,Section 2]) and (147,Lemma 1], and Example 5. 43 is essentially due to N. Salinas (see [84,Theorem 3.2]).

134

6 Quasitriangularity

Our next step will be the analysis of the all important notion of quasitriangularity and its characterization in spectral terms. Several consequences of this characterization will be developed as well. An operator is quasitriangular if it can be approximated by tria~ gular ones. The most obvious examples of triangular operators are the algebraic operators and the operators M_(r) (defined by (3.2)). The main step of the characterization of quasitriangularity says that all ~uasitriangular operators can be constructed out of algebraic operators, the operators M_(r) and approximation. The introductory section of this chapter deals with a ~eneral argument of approximation of operators by operators with a very simple structure. Uost of the results of this chapter are "invariant under compact perturbations" and therefore they can be immediately "translated" to results about closures of subsets of the Calkin algebra.

6.1 Apostol-Morrel simple models We shall say that an operator S is a simple model if it has the form

s

(6 .1)

where (i) cr(S+)' cr(A) and cr(S_) are pairwise disjoint; (ii) A is similar to a normal operator with finite spectrum; (iii) s+ is (either absent or) unitarily e~uivalent to m (ki) ~i=l M+
135

1

:<;

h. ]

:<;

oo,

n

where {~j}j=l is a finite family of analytic Cauchy domains with pairwise disjoint closures. THEOREU 6 .1. The simp 1-e mode 1-s a1'e dense in L (H). Mo1'e pr•eaise Zy: Given T E L{H) and£ > 0 the1'e exists a simpl-e model S suah that o(S+) + c p F(T) c o(S+) • o(S) c p F(T) c o(S) • o(A) c o(T) • ind(A-S) = s£ : s- + - £ £ ind(>.-T) fo1' aZl >. E Ps-F(S+)ups-F(S_) and liT- Sll
+ m It follows from (3.4) that r = crte(S+) = ao(S+) = ui=l
s .

a Let r+, r be compact subsets of {r + ) and (r - ) , respectively, £ + £ + and let M+ , M- be normal operators such that o(M ) = r and o(M-) = r-, resp •• If S has the form (6.1) and II T- S II < £, then we can use our pr~ vious observations and Le~a 5.4 in order to show that there exists an operator S' such that liT- S'll < 2£ and S' admits a 3 x 3 matrix decomposition of the form (6.1) with S+ and S replaced by s+eM+ and s_eM-, respectively. Thus, we have the following.

COROLLARY 6.2. Given T S' "' [

c

L{H) and£

s+eM+

*

0

A

0

0

>

0, the1'r exists

s_~-l·

(6.2)

whe1'e (i) S+• A and S_ have the fo1'm of (6.1); (ii) M+ and M- a1'e no1'ma'l ope1'ato1's such that o(M+)no(S+) =

aa

(S+) and o(M_)no(S_) = acr(S_), 1'esp.; + (iii) ind(>.-S') = ind(>.-T) fol' aZl >. E ps-F(S+)ups-F(S_); + (ivJ a (S+eM ) ( a(s_eM ) J is the cZosu1'e of an anal-ytic Cauchy domain Q (~, 1'esp.) such that each POmponent of Q (~-. 1'esp.) inte1'sects o (S+) (o (S_) • 1'esp.) and o (S+) :: ;2 c o (S+) £ (o (S_) c ~ c

o(S_)£J; (v) 136

Q •

~

and o(A) a1'e pai1'wise disjoint sets;

(vi) (vii)

S'

~

(S+8M+)$A$(S_$M-), and

liT- S'll < 2e:.

Similarly, we can replace A by a more general kind of algebraic operator, or by an operator similar to a normal operator with "large" spectrum, etc , i.e. , Theorem 6 .1 must be regarded as an argument to construct very general distinct families of "models" dense in L(H). Each of the terms S+' A or s_ can be replaced by an operator in a very large class~ some other examples of this situation will be given late~ PROOF OF THEOREM 6.1. LetT£ L(H) and e: > 0.

If~

is an analytic

Cauchy domain such that otre(T) c ~ c otre(T)e:/ 8 , we can use Corollary 3.50 to obtain normal operators N1 and Nr such that o(N 1 ) = oe(N 1 ) = a (Nr) = ae (Nr) = ~- and an operator

., -[~· ~ :J such that min.ind(A-T 1 )k=min.ind(>.-T)k for all A ~ ps-F(T 1 ), k ••• , and liT- T 1 11 < e:/4. Clearly, o 0 (T 1 ) is a finite subset of o 0 (T) and therefore Tl =

1,2,

[col

where c 1 =T 1 1H(o 0 (T 1 )~T 1 ) and o 0 (L 1 ) =fl. By Theorem 3.48, there exists K1 e K (HI , K1 = 0$Ki (with respect to the same decomposition) ,

IIK 111 <

e:/4 and

where L 2 = L1 +Ki is a smooth operator. It is not difficult to check + + + -that Ps-F(L2)=ps-F(T2)=ps-F(T)W' Ps-F(L2)=ps-F(T2)=ps-F(T)\~ and ind(L 2 -A)=ind(T 2 -A)=ind(T-A) for all A e ps-F(T 2 ). Let n1 , n2 , ••• , Qm be the components of p;_F(T 2 )no(T 2 ) and let ~y t 2 , ••• , ~n be the components of p:_F(T 2 )no(T 2 ). It is easy to see that -. -. mnQinnh: fl_for all 1 ~ h, tjnth = fl for all J ~ h, (ui=l Gi) n(uj=ltj) = fl, nin~ =ani (i=l,2, ••• ,m) and ~jn~-=atj (j=l,2, ••• ,n). Dafine + (k.) m (ki) 1 , S+=$i=lM+(
-

n

M =l&j=l M(atj)

(h.)

n

J , S_=$j=lM_(atj)

(h.)

J ,

137

where h. = ind (A-T), A J

€

~j

I

T2 =

j

1,2, ••• ,n, and let

[~r

*

:J

To 0

+ u be the Apostol's triangular representation (3.8) of T 2 • Then o(M) o(M-)

c

3oe(T)

c

otre
can find K2 co K(HJ, such that K2 = KrE&OE&KR. (with respect to the same de composition as for T 2 ) , IIK 2 11 < e:/ 4 and

* * * *

M+

To

* * * *

0

0

B

0

0

0

R.

* M

where B0

=

[:r :

0

0

0

:

] •

BR.

and [:+B:J and [:R. :-) are smooth operators. According to (3.2), we have

T3

"'

s+ * * * (k.) * m 1 0 E&i=lM- (cHL) * * * l. 0 0 Bo n * (hj) * 0 E9j=lM+(34lj) 0 0 * 0 0 s 0 0

=

[s+ T12R Tl3] T23 0 0

s

0

-

where

m

R

(k.)

["'~' M~ un,l '

*

. l

Bo (h.) * n 0 ·j=l M+(3~j) J

I

(6.3)

•

=

It follows from our construction that o 0 (T 3 ) = o 0 (T 1 l a0 (B 0 ) = o 0 (R), T 3 jH(o 0 (T 1 );T 3 ) =TjH(o 0 (T 1 );T) =C 1 , B0 jH(o 0 (T 1 );B0 ) - c 1 and RjH(o 0 (;_); R) - c1 • Moreover, T 3 is a compact pertubation of T 1 , so that Ps-F(T 3 )

= p 8 _F(Tl)

and ind(A-T 3 ) = ind(A-Tl) for all A co p 8 _F(T 3 ). Assume that A co Qi and ind(A-T 3 ) = -ki is finite; then ind(A-S+) =

-k 1 and A-s_ is invertible, so that (A-S+) and (X-S_) are invertiblein the Calkin algebra. It follows that

(X-~)

must be invertible too, i.e.,

(X-R) is a Fredholm operator; moreover, ind (A-R)

= ind(A-T 3 )-

ind(A-S+)- ind(A-S_) = -ki+ki+O = 0.

If ind(X-T 3 ) = -ki =-co, then ind(X-S+) =-co and X-S is invertible. In this case, nul(X-R) = nul(X-M..(3Sli) (co))=co so that X co OR.e(R). On the 138

other hand, ~ € atre(T 1 ) and thereforem~ € atre(B~), by construction. Since ~-s_ is invertible and ~ e pr(et=l M_(aQt) ( t>),we conclude (Le~ rna 3.43) that~ e are(R). Hence~ e atre(R). Similarly, if~ e tj' then we conclude that either ind(~-T 3 ) =hj is finite and ~-R is a Fredholm operator of index 0, or ind(~-T 3 ) = hj = wand~ e aire(R). Hence a(R) is the disjoint union of a 0 (R) =a 0 (T 1 ),[a(R)nps-F(R)]\ m n ind(~-T 3 ) is finite} and a. (R) a 0 (R), Q = {A € (ui=l Q!) u (uj=l t.): J ~re m n ind ( ~ -T ±oo} , ) = = 'I' u{A € (ui=l Qi) u (uj=l tj): 3 R "' (col

Rl2) Rl - cleRl'

where a(R 1 )= a(R)\a 0 (R), ae(R1 )=atre(R1 )=a 1 re(R) and ind(~-R 1 )=0 for all~ e ps-FCR1 ). By Corollary 5.11, we can find an algebraic operator H1 such that a(H 1 ) =ae(H 1 ) c 'I' and IIR1 -H1 11 < E/4. A fortiori,

R12)

H __ (cl 0 H

is algebraic,

1

a (H) =a cc 1 ) ua (H 1 ) does not intersect a (S+) ua (S_) and

11[~ 1 =~ 2 )- [~ 1 =~ 2 )11=

IIRl-Hlll < E/4.

Furthermore, by using our remarks at the beginning of Section 5.1. 2, we can find an operator A similar to a normal operator with finite spectrum, such that a(A) c a(C 1 )Eu[a(H 1 )En'l'] (so that a(A)n[a(S+) u a(S_)J =~)and I!R-AII < E/4. Set

Then liT- Sll sliT- T1 11+11K1 11+1!K 2 11+11R- All< E· It is completely apparent that S is a simple model which satisfies all our requirements. 0 REMARK 6.3. It follows from (6.3) and the properties of the oper! torR that if E is an arbitrary perfect subset of'!'-, then R can beuni formly approximated by operators similar to c 1 eN, where N is a normal operator such that a (N) = E (use Theorem 5. 8) • This indicates that the central piece A of the model S can be replaced by many other operators (for the purposes of approximation~ see also Corollary 6.2 and remarks following it) •

139

6.2 Quasitriangular operators It is completely apparent that an operator A is triangular {see (3.6)) if and only if there exists an increasing sequence {Pn}n:l of finite rank projections such that P ~ 1 (strongly, as n ~ ~> and AP n n PnAPn= (1-Pn)APn=O for all n=l,2, •••• This formulation suggests an asymptotic generalization. An operator A is quasitriangular if there exists an increasing sequence {Pn}n:l of finite rank projections such that Pn ~ 1 {strongly, as n ~ oo) and such that 11<1-Pn)AP n II ~ 0 {n ~ oo). The set PF(H) of all finite rank orthogonal projections, ordered by range inclusion, is a directed set. Thus, given T in L (H), P -+ 11<1P)TPII is a net on that directed set and it makes sense to define q(T) = lim inf{P-+ 1) 11<1-P)TPII

(6.4)

(where P runs over PF(H); q(T) is the modulus of quasitriangularity). Let {QT) (or (QT) {H)) denote the class of all quasitriangular operators. It will be shown that q{T) is equal to the distance from T to the set (QT), so that an operator A is quasitriangular if and only if q{A) = 0. In fact, (QT) admits many other different characterizations, the most important one being the characterization in terms of the different parts of the spectrum. This can be summarized as follows: THEOREM 6.4. The following are equivalent for A E L(H). (i) A is quasitriangular. ( ii) 'I' here exists a (not neaessari ly inareasing) sequenae {P n} ; 1 in PF(H) suah that Pn + 1 (strongly) and 11{1-Pn)TPJI ... O, as n-+ oo. (iii) q(A) = 0. (iv) A is the limit of a sequenae of triangular operators. (v) A = T+K, where T is triangular and K is aompaat. (vi) Given E > 0, A aan be written as A = T +K , where T is tri E E E angular, KE is aompaat and IIKEII < E. (vii) p~-F(A) = ~. COROLLARY 6.5. (QT) is a aZosed subset of L(H), invariant under similarities and under aompaat perturbations. Furthermore, if A E {QT) and pis a polynomial, then p{A) E {QT). PROOF. Every triangular operator is obviously quasitriangular. Now the equivalence between (i) and {iv) indicates that {QT) = {T E L (H): T is triangular} is a closed set. The equivalence between {i) and (v) makes it completely apparent that (QT)+K(H) = {QT). On the other hand, since every operator similar 140

to a triangular operator is also triangular (to see this, we can use,e. g., the existence of a suitable increasing sequence of finite dimensional invariant subspaces), it readily follows that if A has the form of (v) and W is invertible, then WAW-l = WTW- 1 +WKW-l, where WT'I\1-l is tri angular and WKW-l, K(H). Finally, observe that p(A)

=

p(T+K) = p(T)+C, where p(T) is trian-

gular and C c K (H) •

0

REMARK 6.6. The proof of Corollary 6.5 only depends on the lence between (i), (iv) and (v) of Theorem 6.4.

equiv~

6.2.1 Equivalence between the formal and the relaxed definitions of quasi triangularity It is trivial that the definition of quasitriangularity implies the weakened form (ii) of Theorem 6.4. on the other hand, if q(A) = 0 and {en}n:l is an ONB of H, then there exists P1 c PF(H) such that e 1

E

ran P 1 and IIC1-P 1 )AP 1 11 < 1. Assume that we have used the conditionq(A) = 0 inductively to obtain a sequence P 1 $ P 2 s ••• $ Pn in PF(H) such that ej < ran Pj and IICl-Pj)APjll < 1/j for j = 1,2, •.• ,n. Then we can use the same condition in order to find Pn+l

c

PF (H) such that Pn s Pn+l'

en+l c ran Pn+l and 11<1-Pn+l)APn+lll < 1/(n+l). Hence, the condition q{A) = 0 implies that A~ (QT). (ii) => {iii)

Assume that 11<1-Pn)APnll-+ 0 (n -+ ao) for some se-

quence {Pn}n:l in PF(H) such that Pn-+ 1 {strongly) and let P c PF(H). Let E, 0

E < ~,

<

be given and let Rn be the orthogonal projection omo

Pn(ran P). Since rank P < ao and since IIPnx-xll-+ 0 {n-+ oo) for all x H, there exists n 0 = n 0 {£) such that liP nx- x II

ran P and all n ::- n 0

•

s {£/4 >II x II for all x

E

E

Let n ~ n 0 and let y" Pn(ran P), IIYII = 1, Y =

P x, x e ran P. Then llxll s llx-Pnx11+11Pnx11 -; (E/4lllxll+l so that llxll s n -1 (1-c) , and IIRny-Pyll

=

l[y-Py[[

=

llPnx-PPnxl[ s [[Pnx-Px[I+I[Px-PPnxl[

s [IPnx-xii+IIPII·IIx-Pnxll s (l+IIPIIl {c/4lllxll s 2(c/4)/(l-£) < £· Furthermore, if y ( ran P, II y II for n ::- n 0

,

IIRny-Pyii=IIPny-yll

<

£.

1, then Py

=y

and Rny

= P ny.

Hence,

Combining this with the previous

statement we h.ave that IIRny-Py II < £ for all y < (ran P) VP n (ran P) ,11 y II and all n ~ n 0 • If y E [{ran P) VPn(ran P)]L = {ran P)Ln[Pn(ran

=1

then R y = Py = 0. Taking the supremum (for each fixed n ~ n ) of n o IIRn y-Pyll over alliiYII = 1, we obtain IIRn -PII s £·Thus IIRn -PII-+ 0. . Define Rn so that ran Pn = Pn (ran P) EBRn and let Ln be the projec-

P) ] 1

,

141

tion onto ran P+Rn. Then Ln

E

PF(H), Ln

~

P and, since Rn is the pro-

jection onto P n (ran P) , it follows that liP n -Ln II = 11Rn -P II -+ 0 (n -+ co). Thus, since 11<1-Pn)APnll-+ 0 and 11Ln-Pn11-+ 0, we obtain 11(1-LnlALJI -+ 0. Hence, q(A) = 0. (iv) => (iii)

Suppose that {An}n:l is a Cauchy sequence of quas!

triangular operators and that II A- An II -+ 0 (n -+ co). Since (i) =>(iii), we see that given e: > 0 and P e: PF(H), we can find n 0 =n 0 ~ e:/2 and R

E

(e:)

so that

II A-An II 0

PFtR) such that R 2: P and ll
that tl (iv) and (vi) => (v) => (i) The first two implicationsare trivial. On the other hand, it is completely apparent that if K

E

K(H),

then lim sup(P -+ 1)

11<1-PlKII = lim sup(P -+ 1)

lim sup (P -+ 1)

IIK
II (1-P) KP II = 0.

Thus, if A has the form (v) and {Pn}n:l is the increasingsequence in PF(Hl naturally associated with the triangular matrix representation ofT, so that Pn-+ 1 (strongly) and (1-Pn)TPn=O for all n=l,2, ••• , then

Hence, A

E:

(QT).

0

6.2.2 Two lower estimates for the distance to (QT) LEMMA 6.7. Fo~ each ope~ato~ A

E

L(H) the~e is a sequence {Rn}n:l

of pai~wise o~thogonal finite ~ank p~ojections with Ln:l Rn = 1 ($ron~ ly) such that IIRn+lARnll-+ q(A)

(n-+

oo}

and Rn+kARn = 0 fo~ all k ~ 2.

PROOF. Let {en}n:l be an ONB of H and let e: > 0. Choose P 1 E PF(H) so that e 1 E ran P 1 and llran P 1 vran AP 1 V( V{e 1 ,e 2 , ••• ,e }) and n nnn 11(1-P lAP II< q(A)+e:/n. It is completely aPParent that {P } "" 1 is inn n n n= creasing and P -+ 1. Since q(A) s lim inf(n-+ oo) 11<1-P )AP !1, i t read(H)

n

ily follows that tl

n

n

(n ..,.· oo). Furthermore, since ran Pn+l

ran APn' it: follows that (1-Pn+l)APn = 0.

Define R1 = P 1 , Rn = Pn -P n-l for n ~ 2. Then the Rn' s are pairwise orthogonal finite rank projections, Ln:l Rn = 1 (strongly) and Rn+lARn = (1-Pn)APn and therefore IIRn+lARnll-+ q(A) (n-+ oo). Finally, observe that the equations (1-P n+l) AP n = 0 (n = 1, 2, ••• ) imply that Rn+kARn = 0 when k 2: 2. 0 142

PROPOSITION 6.8. For each A

more, there e:x:is ts T

E

E

l(H), dist[A, (QT)] = q(A). Further-

(QT) such that IIA- T II = dist[A, (QT)].

PROOF. I f q (A) = 0, then choose T =A. Assume that q(A) > 0 and choose {Rn}n:l as in the above lemma. Define s

=

n

l

q(A) if R AR ~ 0 11Rn+lARn11 ' n+l n , n = 1,2, ••• ,

1

and let Sn = snRn+lARn. Observe that sn -+- 1 (n -+- oo) and IISJI all n. Now define B

!>

q(A) for

= Ln:l sn (strong sum), T = A- B.

Since the Rn's are pairwise orthogonal,nwe2: S~Sn I ran Rn; therefore IIB*B II = sup{ IIS~SJI :

cla}n~w:i(Ate)2B,*Bso=•thn:atl 'a

IIBII~q(Al.

Let Pn = Ik~l Rk. It is immediate that (1-Pn)BPn = Sn = snRn+lARn,so that 11<1-Pn)TPnll = 11<1-Pn) (A-B)Pnll = (1-sn) IIRn+lARnll-+- O(n-+- oo). Hence T " (QT) and IIA- Til =liB II :s; q(A). Thus, in order to complete the proof, it is enough to show that II A- X II 2: q (A) for all X E (QT) • Assume that IIA- XII < q(A); then lim inf(P-+- 1) 11<1-PlAPil ~ lim inf(P !>

+

llni
IIA-xll+lim inf(P ... 1) 11<1-P)XPII :s; IIA-XII+qCX),

so that q(X) <: q(A) -IIA-XII > 0, proving that X fails to be

quasitria~

gular.

0

PROPOSITION 6.9. If A E l(H), i!CbPn)APn II + 0 (n + ..,·; for some increasing sequence {Pn}n:l in PF(H) such that Pn + 1 (strongLy, n-+- ooJ and E > 0, then there e:x:ists a compact operato:r> KE; liKE II < £,such that Ke: 2 = 0 and A- KE is trianguZar. PROOF. Start with the sequence {Rn}n:l of Lemma 6.7. Since q(A) = 0, we shall have liR2 n+lAR 2 n11 < e:/2n for all n R2n is a finite rank operator, we can define Ke --

Ln=l

=

1,2, •••• Since R2 n+lA

R2n+lAR2n « K(H).

(The sum converges in the norm.) The product of any two summands in the definition of K is 0; therefore K 2 = 0. The orthogonality of the Rn 's E e: implies that and therefore

n ;::

1} <

2

e: • 143

Hence, IlK e: II < e:. To see that T =A- K is actually triangular, observe that (with n e: e: Pn = h=l ~) (l-P 2 n)Ke:P 2 n = R2 n+1AR 2 n = (l-P 2 n)AP 2 n' so that

0

(l-P 2 n)T P 2n = 0.

The results of Section 6.2.1 and Proposition 6.9 imply that (i) => (vi) and therefore, that any two of the conditions (i) through (vi) of Theorem 6.4 are equivalent. This suffices, in particular (as observed in Remark 6.6), to complete the proof of Corollary 6.5. Given T E L(H), define the minimum modulus m(T) minimum modulus me(T)) ofT by

(the essential

m(T) =min{ A E o ( (T*T) !:;) } (m (T) =min{A E o (("l'*T)~)} =min{). E: 0 ((T*T)l:i)}). <6 • 6 ) e e Clearly, me(T+K) =me(T) for all K E K(H). By using the spectral decompositions of T*T and T*T and the Weyl-von Neumann theorem, it is not difficult to prove the following PROPOSITION 6.10. LetT E L(H); then (i) If T is not a semi-Fredholm operator of positive index, then me(T) = sup{m(T+K):

K

E

K(H)} = max{m(T+K):

K

E

K(H)}.

(ii) lim inf (n + co) II Txnll ~ me (T) for any ON sequenae {xn}n:l of Hand there exists an ON sequenae {yn}n:l suah that me(T) =lim inf(n +

IITYnll· (iii) If E(.) is the speatral measure of H = (T*T)~, then the smaZ Zest non-negative a suah that rank E ((a-£, a+e:)) is infinite, for every e: > 0, is a= me(T). (iv) Given E > 0 and p > 1 there exists KE ~ cP(H), IKe:lp < (, suah that nul(H+K e: -me (T)) = ""· (v) Tis left (right) invertible if and only if me(T) > 0 (me (T*) > 0, resp.). (vi) ran T is alosed and nul T < co (ran T* is alosed and nul T* 0 (me(T*) > 0, resp.). (vii) Tis Fredholm if and only if me(T) > 0 and me(T*) > 0. In that aase, me(T) = me(T*). (viii) If T1 is another operator, then

co)

lme(T) -me(T 1 ll s IIT-T 1 11 s IIT-T 1 11· (ix)

144

m (A-T) is aontinuous for A e

E

a:; moreover,

We have the following COROLLARY 6.11. Let A

£

L(H) and assume that

dist[A, (QT)] ~ max{me(A-A): PROOF. Let A

€

A

€

p~-F(A)

#

p~-F(A)} >

~; then

0.

p~-F(A). By Proposition 6.10(i) there exists KA

€

K(HJ such that me(A-A) =m(A+KA-A). By Lemma 3.39, the distance from A+

KA to the set of all triangular operators cannot be smaller than m(A+ KA-A). On the other hand, by Corollary 6.5, (QT) is invariant under compact perturbations. Since the triangular operators are dense in (QT) (equivalence between (i) and (iv) of Theorem 6.4) and the above observations hold for all A in p-s- F(A), we conclude that dist[A,(QT)J ~ sup{me(A-A):

A

€

p~-F(A)}.

Since me (A-A)= 0 for all A € 3p-s- F(A) c a... re (A) and (by Proposition 6.10(ix)) me(A-A) is a continuous function of A, the above supremum is actually attained for some A € p;_F(A). By Proposition 6.10(vi), this maximum is strictly positive. D

6.2.3 Spectral characterization of quasitriangularity Corollary 6. 11 implies, in particular, that p~-F (A) = ~ for all A € (QT). Conversely, if p~-F (A) = ~ and e: > 0, then according to Theorem 6.1 there exists a simple model se: = [:

B ) H1

S - H2 '

where R is alqebraic and S has the form of (6.1) (iv), such that IIA-se:ll + < e:. (If Ps-F(A) =II, then S is absent and Se: =R.) R is obviously triangular with respect to some ONB of H1 . On the other hand, if ~ is an analytic Cauchy domain, then either by a direct argument (based on Runge's approximation theorem [102,p. 28]) or by using Vitushkin's theorem [102,p.207J, we can easily see that the linear span of the functions {(~-A)-l: A I 3~} is uniformly dense in the space of all continuous functions defined on 3~. Since the mapping A + (~-A)-lis a norm-continuous mapping from ~\3~ into L 2 (a~), it is not difficult to deduce that if {A } : 1 is an arbitrary 2 2 n n-1 oo denumerable dense subset of~. then L (3~)9H (3~) = V{P_[(~-A) J}n=l' where P denotes the orthogonal projection of L2 ( 3~) onto L2 ( U!) eH2('an). Si~ce M_(3~) (P_[(~-A)-l]) =AP_[(~-A)- 1 J for all A € ~.it readily follows that M_(a~) is triangular with respect to the Gram-Schmidt or-1 thonormalization of of the sequence {P_[(~-An) ]}n=l" . Now it is easily seen that if {~k}k:l is a denumerable dense sub00

145

set of p:_F(S_) and {gkr} r:l is a denume:!:'able dense subset of ker(S_-\Jk) (k=l,2, ••• ), then s_ is triangular l·rith respect to the Gram-Schmidt orthonormalization of the family {qkr}k,:!:':l. Hence, R&S_ is triangular. Since cr(R)ncr(S_) =/I, Se: -R&S_(Corollal£_ y 3.22) and therefore S is also triangular. By taking e: = 1,1/2, ••• , e: 1/n, ••• , we conclude that A is the limit of a sequence of triangular operators, i.e., A satisfies (iv). The proof of Theorem 6.4 is now complete. 0

6.3 Biauasitriangular ooerators Let {QT) * = {T € L (H): T* € {OT)}. An operator A € L (H) is biquasitPianguZaP if both A and A* are quasitriangular. Hence, the class (BQT) of all biquasitriangular operators is equal to the set (QT)n(QT)t

6.3.1 Block-diagonal and quasidiagonal ooerators operator B€ L(H) is bZoak-diagonaZ if there exists an increasing sequence {Pn}n:l in PF(H) such that Pn + 1 (strongly) and PnB=BPn for all n = 1, 2,. • • • Clearly, this is equivalent to saying that B = en="" 1 Bn , where Bn = (P n -Pn- 1 >B I ran(!' n -P n- 1 ) = B I ran (P n -Pn- 1 > (P 0 = 0; n = 1,2, .•• ) • (It is obvious that the Bn's act on finite dimensional spaces.) An operator A is quasidiagonat if there exists {Pn}n:l as above such that II P n A- AP n II + 0 (n + oo). Let (QD) and (BD) denote the classes of all quasidiagonal and all block-diagonal operators, respectively. The main properties of the qu~ sidiagonal operators will be summarized in the following three theorems. (Their proofs are either very simple or minor modifications of the proofs of the analogous statements in Theorem 6. 4 and win be omitted.) An

THEOREM 6.12. The foZZowing aonditions aPe equivalent .foP A£ L(H): (i) A is quasidiagonaZ; (ii) ThePe exists a (not neaessa~iZy inaPeasing) sequenae {Pn}~1 in Pf(H) suah that Pn .... 1 (stPongZyJ and llPnA-APnll + 0 as n + oo; (iii) The moduZus of quasidiagonaZity qd(Al =lim inf(P

+

1) !IPA-APIL

(6.7)

where P runs over Pf(H), is equaZ to 0; (iv) A is the limit of a sequenae of bZoak-diagonaZ operators; (v) A= B + K, where B is btoak-diagonaZ and K is aompaat; (vi) Given e: > 0, A aan be written as A= Be: +Ke:' whe~e Be: is 146

bZoak-diagonal, KE is aompaat and liKE II < E.

THEORE."t 6.13. For eaah A q (A)

$

E

L Ul),

max{q (A) ,q (A*)}

Moreover, there exists C

E

$

qd (A) = dist [A, (QD)].

(QD) suah that IIA- Cll= qd(A).

TimOREM 6.14. (QD) is a alosed subset of L(H), invariant under unitary equivalenae and under aompaat perturbations. Furthermore, if A €(QD), then C*(A) c (QD). In partiaular, NoJt(H) +K(H) c (QD). But (as ~roven in [188,p.l4,Example 1.3]; see also [149],[150]) the classes (QD) and (BD) are not invariant under similarities and this makes it impossible to qive a spectral characterization of (QD) (analogous to Theorem 6.4(vii)). As we shall see immediately, such a characterization is actually possible for the larger class of all biquasitriangular operators. (It is completely apparent,from the definition, that (QD) c (BQT).)

6.3.2 Characterizations of biquasitriangularity THEOREM 6.15. The following aonditions are equivalente for A E L(H): (i) A is biquasitriangular; (ii) There exists a (not neaessarily inareasing) sequenae {Pn}~l in PF(Hl suah that P0 + 1 (strongly), ll(l-P 20 _ 1 )AP 2n-l11+ 0 and IIP 2nA(l- P 20 l II + 0, as n + co. (iii) Given E > 0, there exists an inareasing sequenae {P0 } 0 : 1 in PF(H) suah that Pn + 1 (strongly), 11<1- P 20 _ 1 )AP 2n_ 1 11 < E/4n and 11P 2nA(l-P 2 n>ll < E/4D for all n=l,2, ••• (iv) max{q(A) ,q(A*)} = 0. (V)

(vi)

A E Alg(H)-. A

E

Alg(H)-+K(H).

(vii) A E {R E L(H): R is similar to a normal operator with finite speatrum}-. (viii) A E {R E L(H): R - N, N is normal}-. (i:x:) A E {R E L (H): R- B, BE NoJt(H) +K{H)}-. (x) A E {R E L(H): o(R) is totally disaonneated}-. (:x:iJ A E {R E L(H): interior o(R) =II}-. (:x:ii) AE {RE L(H): R-C, CE (QD)}-. (:x:iii) p- F(A) = p+ F(A) =fl. ss(:x:ivJ (Stairaase representation) Given E > 0 there e:x:ists KE

E

K(H),

liKE II < E, and a family {Rn} n:l c PF (H) of pairwise orthogonal proj eations, suah that A- KE admits a matri:x: representation of the form 147

Al Bl cl

Rl R2

Dl A2 B2

R3 R4

0

c2 D2 A3 A-K

£

Rs (6. 8)

=

A B n n 0

cn

R2n-1 R2n

Dn An+l

R2n+l

where Rn =ran Rn, n = 1,2, •••• (%V) There e%ists a decomposition H nite dimensional subspaces such that

= H1eH 2

of H into two infi-

(6. 9)

where DJ. is a quasidiagonal operator acting on H., such that a (D.) = J e J ae(A), j = 1,2, and K is compact. (%Vi) Given & > 0 there e%ists K& £ K(H), IIKJI < &, such that A-K

( 6 .10)

£

where Bj is a block-diagonal operator acting on the infinite dimensio~ al subspace H., such that a (B.) =a (A), j = 1.2. J e J e (%Vii) Given E > 0 there e%ists Ke: £ K(H), I~JI < E, euch that

A-K

£

=

NSD1 * * * 0 NSD 2 * * 0 0 NSD 3 * 0 0 0 NSD 4

Ml M2 M, 3 M4

where N is a normal operator and the D.'s are block-diagonal operators J such ihata(N) =ae(N) =ae(Dj) =ae(A), j=l,2,3,4. PROPOSITION 6.16. (BQT) is a closed subset of L(H), invariant under compact perturbations and under similarities. Furthermore, if A £ (BOT) and pis a polynomial, then p(A), p(A*) £ (BQT). PROOF. 148

Since (BQT) =

(~)

n (QT)

*

and (QT) and (QT)

*

are closed, in

variant under compact perturbations and invariant under si~ilarities (Corollary 6.5), (BQT) has the same properties. The second statement follows from Corollary 6.5 as well. 0 PROOF OF THEORrui 6.15. The equivalence between any two of the con ditions (i), (ii), (iii), (iv) or (xiii) follows immediately from Theo rem 6.4. Let (BQT)v' {BQT)vi' ••• , (BQT)xii denote the families of operators described in (v), (vi), ••• , (xii), respectively. By Corollary 5.10 and the equivalence between (i) and (xiii), we have (BQT) v = (BQT) vi = (BQT) vii = {BQT) • Since {N e Nclt(H):

a(N)

is finite}

c Nolt(H)

c:

NOJL+K(H)

c

(QD)

c

(BQT),

we easily see that (BQT)vii = {BQT)viii = (BQT)ix = (BQT). Similarly, since (by Corollary 1.14(i) and Theorem 1.13) (BQT) = {BQT) vii

{BQT) x

c

= fl} = (BQT)

c

(BQT) xi

c:

{B .;: L (H) :

I

we see that (BQT)x = (BQT)xi = (BQT). Hence, any two of the conditions (i) - (xiii) are equivalent. (xvi) => (vx) => (xiii) The first implication is trivial. On the other hand, it is easily seen that if A satisfies (xiii), then A-K has the form (6.10) for some K in K(H). Since (BQT) is invariant under com pact perturbations (Proposition 6.16), it suffices to show that every operator of the form (6.10) belongs to (BQT). Since (BD) c (QD) c (BQT), it readily follows from the equivalence between (i) and (xiii) that a (B.) =an (B.) and ind(A.-B.) =0 for all e J ~re J J A e p F(B.), j=l,2. A fortiori, an (A-K) =a (A-K) and ind(A-K -A.) sJ ~re e: e e: e: = 0 for all A. E ps-F(A-Ke:). Hence A-Ke: satisfies (xiii). Since K£ is compact, it is completely apparent that A also satisfies (xiii) • (xiv) => (i) Ass~ue that A-K admits the representation (6.8) ne: (for some K e K(H)); then P =}:. 1 R € PF(H), P + 1 (strongly, as n e: n J= n n + oo), (l-P 2 1 ) (A-:< ) P 2 1 = 0 and P 2 (A-K ) (l-P 2 ) = 0, so that A-K ne: nn e: n e: and (A-K )*are triangular operators. Hence A-Y. e (BQT). A fortiori £ £ (since Ke: is compact) , A e (BQT) • (iii) => (:civ) Define P_ 1 = P 0 = 0 and Ke: = Ln:l { ( l-P2n-l) A (P2n-l-P2n-3) + (P 2n -P 2n-2) A ( 1 -P 2n)} •

(G

.ll)

Since (l-P 2 n_ 1 )A(P 2n_ 1 -P 2n_ 3 ) and (P 2n-P 2n_ 2 )A(l-P 2 nl are finite rank 149

operators and rmr..c{ ll
s: max{ ll
it is easily seen that the series (6.11) converges in the norm, KE

K(H) and I~JI

€

2 ~n=l E/4n < E. It is straightforward to check that A- K admits the matrix repr~ E sentation (6.8.). (Roughly: A- (l-P 1 )A(P 1 -P_ 1 l has O's everywhere in the firot column, except for the term A 1 : A- (l-P 1 )A(P 1 -P_ 1 l- (P 2-P 0 )JI. (l-P 2 ) has O's everywhere in the first two rows except for the terms A1 , B1 , c 1 , etc~) (i) => (xvi) Since A € (BQT), it follows that o .Q.re (A) = oe (A). By Proposition 4.29, there exists S"' a A, A-S € KIHl, IIA- Sll < E/2 and <

s where A' "' A and NR. and ur are unitarily equivalent diagonal normal operators of uniform infinite r.mltiplicity such that o(NR.) = oe(NR.) = o(N ) = o (N ) = o (A). Assume that N0 "' N "' (diag{A 1 , A. 2 , .•. }) (oo). Let r ere "r {e } ool be an ONB of H. n n= Now we proceed essentially as in the proof of Theorem 6.4(iii) => (i). First we find P1 € PFIHl such that ran P 1 contains e 1 and avector fi € ker(N.Q.-A.l), and ll
€

2

PF(H) such that ran P 2n+l contains ran P 2n' e 2n+l and vectors n+l

j

j

ker(N -A.j) and fn+l ~ran P 2 n(j n+l 2, ••• ,n+l), and ll
fn+l'fn+ 1 , ••. ,fn+l' where fn+l

1,

€

€

PF

1

(H) such that ran P 2n+ 2 contains ran P 2n+l' e 2 n+ 2 and vectors gn+l' 2 n+l j j g 1 , ••• ,g +l' where g +l € ker(N -A..) and g +l ~ ran P 2 n+l (j = 1, 2, ••• n+ n n r Jn+l n •• ,n+l), and 11P 2n+ 2s(l-P 2n+ 2 >11 < E/5 . ool satisfying By induction, we have constructed a sequence {P n }n= Define P1 = P 0 = 0 and (iii) with A replaced by S and E replaced by E/2. let

It follows from the previous step (see (6.11)) that CE "K(H), < E/2, K = (S-A) +C € K (H), IlK II < E and A··K admits a stair£ E E E representation of the form (6.8). Furthermore, by construction, 150

~

An is an eigenvalue of

and an eigenvalue of Ck for infinitely many

values of k, for each n=l,2, •••• liorc precisely, An" E

~

ap(Ck) for all k p,n}-=aR.re(A)

ap(~)

and An

n. Hence c

0

R.re( 61k:l

~)n°R.re(6lk:l

<6 • 12 )

Ck).

now observe that ~ 19_1 B

1(11 0 lo I 1 I I I I A2 olio I B2 10 I 10 1 0 IO I C1-I I A3 02 10 I e I() I 2-A4

A- Ke:

0

Rl

0

R3

0

R2

B__,

Rs

0

03

.

(6 .13)

R4 • R7 R6

e3 0

Since the column of A contains A in the main diagonal and O's n n in any other entry, it is easily seen that Al

Rl

Bl A2

01 B2 0

A3

R3 0

02 B3

Rs

0 00

A-K = e:

-·----·-

[61n=l

cl- -

R2

c2

R4 0

e3 0

B

An 00

61 n=l

rl

en

,

H2

R6

0

00

Clearly,

0

x1 = 6ln=l

00

An and

x2 = 6ln=l

en are block-diagonal operators

(acting. on H1 and H2 , respectivley) and ae(X 1 Juae(X 2 ) c ae(A) =aR.re(A). Combining this observations with (6.12) we conclude that oe(X 1 J =ae(X;l =oe(A). 00

00

(i) => (xvii) Let Ke:, x1 = 6ln=l An and x2 = 6ln=l en be the operators constructed in the previous step. Recall that An " ap(~)na 9 (Ck) for all k

~

n. Thus, if k

~

n we have

151

[ .\n 0

* F

)Mn,k n,k 1.J n,k

where Mn, k is one-dimensional, Fn,k R2kl9Mn,k and An E a p (F n, k). The same argument, applied to (A 2k_ 1 mA 2k)* shows that 1t

)

/.J n,k I

An M~,k, where M~,k is one-dimensional, Gn,k € L(N~,k), N~,k= CR 2 k_ 1 mR 2 k)9M~,k and An € ap(Gn,k) (k ~ n). Since we have infinitely many possible choices of k for each n, we can easily see that Xl = $n:l An= [$k:l (A4k-3$A4k-l) ]$[$k:l (A4k-2$A4k)] =

(mn:~

Gn :) m(: mrr.:l *FJ'

where N= (diag{A 1 ,A 2 , ..• })

the operators Gn ha•re the form G k for n, n a suitable subsequence {k n }n: 1 and the operators Fm have the form F m,km for a suitable subsequence {km}m:l' so that mn:l Gn and mm:l Fm (co),

are block-diagonal operators. Moreover, Ar € a p (Gn )no p (Fm) for all n and m large enough (depending on r) , whence we easily conclude that ae(mn:l Gn) =ae(mrn:l Fm) =o(N) =oe(U) =ae(A). Define o 1 = mn:l Gn' o 2 = mm:l Fm; then N 0 xl

0

0 X" 1 X' 01 1 0 = (N:Dl 0 N 0 N:DJ •

0

0

= (01 Xi) (N Xi) = o N m o o2 0

0

02

Similarly, we can obtain _ (N$0 3 x2- o N:DJ and, a fortiori, N$0 1 A- K = (Xl * ) e: o x2

0

* N$0 2

0

0

0

0

* * N$0 3 0

* * * N$0 4

(xvii) => (xiii) Observe that a (N$0.) =a {N$0.) = o (A·-K ) = e J ~re J e e: a~re(A-Ke:) =oc(A) =a.tre(A), j = 1,2,3,4. Now the result follows exact152

ly as in (xv) => (xiii) • The proof of Theorem 6.15 is complete now.

D

6.4 On the relative size of the sets (QT), (QT)*, (BQT), [N(H)+K(H) and N(H) THEOREM 6.17. Eaah of the N(H)

c

fo~~owing

[N(H)+K(H)J

(BQT)

c

ina~usions

(QT)

c

c

is proper:

l(H).

Moreover, eaah of these subsets (exaept L(H)J is nowhere dense in the next one. Furthermore, the same is true for the ahain of inaZusions

N(H)

c

[N(H)+K(H)J

(BQT)

c

(QT)*

c

c

L(H).

L(HJ.

and (QT)n (QT)* is nowhere dense in

PROOF. Consider the first chain of inclusions. Since each set in the chain is closed, in order to prove that a set is nowhere dense in the next one it suffices to show that its complement is dense in the containing set. (i) Observe that "T J (QT)" is a hbad property" (as defined in Section 3.5.2) and the unilateral shift is a concrete example of an non-quasitriangular operator. By Theorem 3.51 L(HI \(QT) is dense in L (H).

:ii)

LetT " (BQT) and E

>

0. By Proposition 4.29, there exists

T1 " (BQT), liT- T1 11 < d2 such that

Tl =

(~

;J

for some A. c acre(T)n ap(T), where cr(T 2 ) =cr(T). tTake 1' 9, ={A.}, rr=fl.) Proceeding as in the proof of Theorem 3.51, we can find T

= [A.+n+oS* E:

0

* ) T2

I

where E > 0 and S* is a backward shift of multiplicity one, such that o(>..+n+oS*)ncr(T 2 ) ={6, T 2 is biquasitriangular and IlT-TEII < c• Clearly, T " (QT')\ (BQT) • Hence, (QT) \ (BQT) is dense in (QT) • E: (iii) LetT" [N(H)+K(H)] and c > 0. I f JJ " a e (T) E 1 L... \a e (T), then ll 'I 0 and it follows from Theorem 5.34 that there exists Q" N(H)+K(H) such that liT- (JJ+Q) II< E· Clearly, 11+Q" (BQT)\[N{H)+K(H) J and therefore (BQT)\[N(H)+K(H) 1 is dense in (BQT). (iv) If T € N(H)-, E > 0 and J1" acr(T)ncr~re(T), then we can use 153

the argument of the first part of the proof of Theorem 3.49 inorder to obtain a finite rank operator F , !IF 11 < £• such that 00 (T+F ) nO()l,e:) e: e: e: # ~. It is easily se~n that T+Fe: € fN(H~K(H)J \N(H) , so that [N[H)+ KtHJJ-\N(H)- is dense in fN(H)+KfH)J-. By taking adjoints, we conclude that (BQT) is nowhere dense in (QT)* and (QT)*.is nowhere dense in L(HJ. Hence, both (QT) and (QT)* are nowhere dense in L (H) . A fortiori, so is their union. 0 REMARK 6.18. Theorem 6.17 indicates that N(H)-, [N(H)+K(H) ]-, (BQT), (QT), (QT)* and (QT)u(QT)* are "very small" subsets of L(H).Ho~ ever (as proved in Theorem 5.15), even N(Hl is large enough to guara~ tee that L (H) = N ( H l - +. N (H) - • On the other hand, we also have {L(H)\[(QT)u(QT)*J}+{L(H)\[(QT)u(QT)*J} = L(H). Indeed, by Proposition 4.2l(ii), given T E L{HJ there exists K E K(HJ such that T-K=AEilB (A, B acting on infinite dimensional spaces). I f s is the unilateral shift of multiplicity one and r > 21!T II• then T = T1 +T 2 , where T1 = (A+2r+rS)Eil(B-2r-rS*)+K

and

T 2 = (-2r-rS)Eil(2r+rS*).

It can be easily checked (use Theorem 1.13(iii) and (v) that T1+2r and T2-2r are Fredholm operators of index 1, and T1 -2r and T 2+2r are Fredholm operators of index -1. Hence, T1 , T2 E L(H)\[ (QT)u(QT)*J.

6.5 A Riesz decomposition theorem for operators with disconnected essential spectrum The classical Riesz decomposition theorem asserts that if o(T) is the disjoint union of two nonempty clopen subsets o 1 and a 2 , then the space can be written as the algebraic direct sum H = H1+H 2 of two invariant subspaces ofT, H]. and H2 , such that o(T!Hj) =oj' j = 1,2. (See Chapter I.) This result is false, in general, if we merely assume that the essential spectrum splits into two clopen subsets. Namely, if T = M__,(ClQ), where Q is the annulus {).. E a:: 1 < I AI < 3}, then such a deco!! position is impossible: If H = H1+H 2 , where Hl and 11 2 are invariant under T, oe(T!H 1 ) ={A: !AI= 1} and oe(T!H 2 ) ={A: !AI= 3}, then ind (A-T! H1 ) = 0 for all A in Q and therefore ind(A -T! H2 ) = -1 for all A in Q. Since oe(T!H 2 )nD(O,l)-=~, we conclude that ind(A-T!H 2 ) =-1 for all A in 0(0,3). Therefore, ind(A-T!H 1 ) = ind(>l-T)- ind(A-T!H 2 ) = +1 for all A E 0(0,1) and this implies that 0(0,1) c a (T), a contradiction. p

154

A mo=e careful analysis indicates that oe(M+(iH"l)+A) splits into two nonempty clopen subsets o 1 and o 2 , but M+(il!:l)+A does not have two complementary invariant subspaces, H1 and H2 , such that oe(M+(iln)+AjH1 l = o 1 and oe(M+(anl+AIH 2 l = o 2 , for any A c: L(HJ such that JJAJI < l . However, it readily follows from the BDF theorem that H+(anl is unitarily equivalent to a compact perturbation of 35$5*, where S denotes the un! lateral shift of multiplicity one. Clearly, oe(3S) ={A: oe(S*)={A:

1

A 1 = 3} and

IAI=l}.

Consider the bilateral shift B defined by Ben= en+l for n

~

-n

0 and

Ben= 2 en+l for n > 0 with respect to an ONB {en}-M.-B) = nul(A-B) = 1 for all A E D(O,l) e \{0}. Either by general results about the structure of bilateralweigh~ ed shifts (see, e.g., [103], [131], [184]) or by a direct argument, it can be shown that B does not have any pair of complenentary invariant subspaces. However, if Bm is the result of replacing the m-th weight by a 0 (i.e., Bm=B-2-mem+leem) for some m > 0, then Bm admits t"''o co~ 00 plcmcntary reducing subspaces H1 = v {e } n n~m and H2 = v {e n } n=m+l such that oe(Bm I H2 > = { 0} and oe(Bm J H1 > = an (0, 1) • (Indeed, Bml H2 is a compact quasinilpotent and Bm I H1 - S*!)

These two examples illustrate both the possibility of restoring the validity of Riesz decomposition theorem for some snall compact

pe~

turbation of T (where T c: L(H) has a disconnected essential spectrma) and the "index obstructions" that make it an impossible attempt. PROPOSITION 6.19. Let T c: L(H) and Zet

set of

o~re(T) oo

r be a cZopen nonempty sub

contained in a unique component n of Ps-F(Tlur such that > inf{ind(>.- T):

>.

E

nnps-F(T)} = n > 0

(6.14)

and ind (.A-T)= n for every A in a aomponent i!> of Q\r whose boundary

()4>

intersects an. Given e: > 0, there exists K

E

K (H), II KJI < e:, such that

T- K - A$R,

where R is a smooth operator, o~re(R) = r, o(R) = ru{A € nnps-F(T): ind(A-T) >n}, oe(A)nn =~and nul(A-A)=nand nul(>.-A)*=O for aU A €

n. PROOF. Proceeding as in the proof of Theorem 4.11, we can find CE

K (H)

such that

T-C where V "' N + W 0 ) * (n) ,

= (v0

*)

B'

(interior Sl-)*, o(B) =ruo 0

,

OR.re(B) =fuo 1 , 155

o 1 nQ = f1 and o 0 nn ={A E Qnps-F(T): ind(J.-T) - n, for all A E o 0 nQ).

ind(>.-T) > n} (Obviously, ind(>.-B) =

It readily follows that B is the algebraic (not necessarily

orth~

gonal) di=ect sum of an operator G whose spectrum does not intersect Q and an operator J' such thatoR.re(J') =rand o(J') = ru(o 0 nnl. Furthermore, J' E (QT) because ind(>.-J') rem 6.4.) Hence,

2:

0 for all A

ps-F(J'). (Use Theo-

E

T-C

where J - J' and

Let

then

Since J

(QT) and c 33 is compact, J+c 33 also belongs to (QT) and therefore (by Theorem 6.,:,4) J+c 33 =J0 +C 0 , where (l-Pn)J0 Pn=O for ani~ creasing sequence {Pn}n=l in PF(H 3 ) such that Pn ~ 1 (strongly, as n ~ ~> and C is compact. Furthermore, by Proposition 3.45 and its proof, 0 is compact, the operator c 0 can be chosen so that o 0 (J0 ) =fl. Since C 3o E

0

Kl (n) = ( ( 1-PO) C (1-P ) (J +C )P n 3o n o o n

) H0

H3

•

is compact and IJK 1 (nlll ~ 0 (n ~ ~> Let A4 (n) be the restriction of T- K1 (n) to its invariant subspace H4 (n) =H 0 eran Pn and let M(n) be the compression of J 0 +C 0 to the subspace H5 (n) = ran(l-Pn). Clearly, there exists n 0 such that if K1 = K1 (n 0 ) , A4 = A4 (n0 ) and M=M(n0 ) , then JJK 1 11 < F:./2 and o 0 (M) c: nnrE 12 • By Proposition 3.45 and

K4 and K5 such that JJK 4 JI < < e/2, A=A 4 -K 4 has the desired properties and o 0 (l·1-K 5 ) =f!, so that R = U- K5 also has the desired properties.

Theorem 3.48 there exist compact operators

e/2,

IIK 5 11

Thus, i f

K2 = K4 eK 5

and K = K1 +K 2 , then K e: K(H), T- K

=

< e and

(~ :) .

Since o(R) is contained in Q and Q

156

IlK II

c:

Pr(A), it follows from Coroz·

lary 3. 22 that T- K

~

AEBR.

0

REMARKS 6.20. (i) By taking adjoints, it is easily seen that Pro position 6.19 remains true if (6.14) is replaced by -oo

<

sup{ind(A-T):

A£ nnps-F(T)} = n

<

0

(6

.15)

and "o(R) =ru{A • Qnps-F(T): ind(A-T) > n}, oe(A)nn=!l and nul(A-A) =n and nul (A-A)*= 0 for all A E Q" is replaced by "o(R) = ru{A € Qnps-F(T): ind(A·-T) < n}, oe(A)nn = 11 and nul(A-A) =0 and nul(A-A)*=~n for all A ( s-2".

(ii)

especially important case of this proposition is the one to the case when r is a singleton (or, more generally, a finite subset of oire(T)). This case will be analyzed with detail in An

correspondi~g

Chapter VIII.

6.6 Notes and remarks The concept of quasitriangularity plays a central role in the proofs of the Aronszajn-Smith theorem [36J on the existence of invariant subspaces for compact operators, and in the proofs of several gen eralizations [39], [51], [118]. However, this important concept stayed hidden for many years until P. R. Halmos explicitly stated it (and some of its related properties) in his article [121]. The notion of quasitriangularity immediately drew the attention of many operator theorists who used it to extend the previously known results on the existence of invariant subspaces for operators "somehow related to compact operators" (see, e.g., [9], [70], [158], [161], [192], [193]) or worked on the structure of the quasitriangular operators and its i~ plications. The most important result in this line is the spectral characterization of quasitriangularity (i.e., the equivalence between (i) and (vii) in Theorem 6.4). The equivalence between any two of the properties (i) - (vi) in Theorem 6.4 is due toP. R. Halmos [121J (see also [123,Problem 4]), who also proved that (QT) is closed, invariant under compact perturbations and closed under the formation of polynomials. (The simplified proof of the step (ii) => (iii) given here is due to G. R. Luecke [154].) That (QT) is also invariant under similarities is an observation of w. B. Arveson [12l,p.291J; the first proof of this fact (very different from the present one) appears in [75]. In [12l,Theorem 3], Halmos proved that the unilateral shift of m~ltiplicity one is not quasitriangular. The argument of the proof was 157

used by R. G. Douglas and ( i)

=>

c.

M. Pearcy in

[75,Theorem 1] to show that

(vii) •

The final step, (vii) => (i), proved to be the most difficult one In [75], Douglas and Pearcy proved that every operator with a finite spectrum is quasitriangular. Halmos extended this result to operators T with the property that cr(T) has analytic capacity zero [124]. (The reader is referred to [102] for the definition of analytic capacity. In pa=ticular, if cr(T) has analytic capacity zero, then cr(T) is totally disconnected and has planar Lebesgue measure zero.) C. Apostol proved in [4] that every decomposable operator (in the sense of C. Foia~: see [67])is quasitriangular, thus extending the previous re&Ws, and introduced the notion of "modulus of quasitriangularity" (q(T) ). In [19] (see also [21]), C. Apostol, C. Foia~ and L. Szid6 obtained several partial results about non-quasitriangular operators. Finally, the goal was attained by C. Apostol, C. Foia~ and D. Voiculescu in a series of papers published in the Revue Roumaine de nathematiques Pures et Appliquees: [ 21], [ 22 J, [ 23 J, [ 24], [25 ]. (The main result was previously announced in [2QJ.) In fact, a first proof of the impl! cation (vii) => (i) was obtained in [23] and a second one, including a stronger statement, was obtained in [25]. Simultaneously (and independently), L.A. Fialkow obtained several relevant results on the subject, including a proof of the equality q(T) = dist[T, (QT)]. A simplified proof of the main theorem of C. Apostol, C. Foia~ and D. Voiculescu was given by R. G. Douglas and C. 11. Pearcy in [74]. The four proofs of this result (i.e., the two obtained in [23J and [25], Douglas and Pearcy's version and the one given here) strongly depend on constructive techniques of approximation. The present proof (largely based on Voiculescu's theorem, which was unknown at the time of publication of the other three proofs) provides a more natural approach. The relevance of the achievement of C. Apostol, C. Foia~ and D. Voiculescu goes far from the significance of their main result. It is worth mentioning, at least, two other consequences: a) The introduction of a large number of constructive techniques that have become standard tools since then (including Theorem 3.49 of Chapter III~. b) From a heuristic viewpoint, the result strongly suggested that the only real obstructions for many problems of approximation are thosg derived from the stability of the semi-Fredholm operators and the upper semicontinuity of separate parts of the spectrum. (This is, precisely, the aspect of the Approximation of Operators that has been continuously emphasized throughout this monograph.) 158

Finally, it is convenient to recall that the notion of quasitria~ gularity also admits several other alternative definitions; namely, R. G. Douglas and C. M .Pearcy [75] have also shown THEOREl-1 6.21. (Addendum to Theorem 6.4) The are equivalent for A <

fo~lowing

aonditions

L(H)

A is quasitrianguZar. (viii) There exists e: > 0 and a sequence {F n} n:l of finite rank operators converging strongly to 1 such that each Fn is bounded below by (i)

e: on (ker Fn).l and (i:r:)

11<1-Fn)AFnll+ 0 (n +

00 ) .

There exists a sequence {Kn}n:l of positive semi-definite

compact operators that converges strongly to 1 and satisfies 11(1- Kn) AKn

II

+

0 ( n + oo) •

Furthermore, if (FC+,~) denotes the directed set of all positive semi-definite finite rank contractions acting on H (partially ordered by the relation: A1 ~ A2 if and only if ~ for all x < H, as usual) and qa (T) =lim inf (A + 1) (A

E

11<1-A) ~TA ~II

FC+), then we have [2l,Section 3] THEOREM 6.22. If T ~ LIHl. then (i) qa(T) =qa(T+K) =qa(T$0) =q(T$0) for aZZ K (ii) ~qa (T) ~ qa (T+A) ~ q(T) for all >. E a:. (iii) q(T) =lim(>.+ oo) qa(T+A).

< K(H).

In [192J,[l93J, D. Voiculescu analyzed certain notions of "exten!! ed quasitriangularity" in Hilbert spaces (where the role of K(li) is played by a Schatten p-class cP(H)). On the other hand, P. MeyerNieberg defined quasitriangularity in arbitrary Banach spaces (the definition coincides with the usual one in case the underlying Banach space io a Hilbert space) and proved several invariant subspace theorems in this new setting [158]. c. Apostol [7], [8] and C. Apostol and D. Voiculescu [33] analyzed the properties of this newly defined quasi triangularity. Among many other results, the last paper extends the ApOstol-Foia~-Voiculescu's spectral characterization of quasitriangularity to the case when the Banach space is either e 0 or !P, 1 ~ p < ""· The introductory result of this chapter (Apostol-Uorrel simple models) is one of the most important "tools" of approximation. Theorem 6.1 has been proved by c. Apostol and B. B. Morrel in [27] where ithas been applied to obtain (among other results) a spectral characterization of the closure of the set of all operators with spectrum contained 159

in a fixed nonempty subset of ~. (This result will be proved and analyzed in the second monograph.) The existence of quasitriangular approximants (an operator T in (QT) is a quasitriangular approximant for A E L(H) i f IIA-TII = dist[A, (QT) ]) is due to C. Apostol, C. Foia~ and D. Voiculescu [2l,Theorem 2. 2]. The proof presented here (Lemma 6. 7 and Propositions 6. 8 and 6.9) is a refinement obtained by R. A. Smucker in his doctoral dissertation [188]. (Proposition 6.9 refines Halmos' result in [123,p.905].) Thee~ sential minimum modulus me(.) has been analyzed by several authors; n~ mely, in [95] the main theorem states that seven conditions are equiv~ lent, and aach of these conditions is equivalent tom (T) =0. Proposie tion 6.10 (essentially equal to [53,Theorem 2]) sums up the most rele· vant properties of the essential minimum modulus and Corollary 6.11 is just a quantitative version of [75,Theorem 1]. Theorems 6.12 and 6.13 are due toP. R. Halmos [123,Problem 4]. Halmos' article and the doctoral dissertation of R. A. Smucker are the basic references for quasidiagonality from the viewpoint of ap~roxi­ mation (The interested reader can also examine [149], [150], [155] and [197,Chapter XIV].) In particular, Theorem 6.14 is also due to R. A. Smucker [138,Theorem 5.5]. The equivalence between any two of the conditions (i) - (xiii) of Theorem 6.15 is either an immediate consequence of the Apostol-Foia~­ Voiculescu result on spectral characterization of quasitriangularity or a consequence of Voiculescu's theorem on algebraic limits of algebraic operators (Theorem 5.9). The equivalences between (i) and any of the conditions (xv) or (xvi) are due to C. Apostol and C. Foias [17]. I The staircase representation (xiv) is a result of C. Foia~, C. H. Pearcy and D. Voiculescu [98] and the representation of (xvii) is one of the steps of Voiculescu' s proof of Theorem 5. 9 [ 194, Proposition 3..2]. C. Apostol, C. Foia~ and D. Voiculescu [2l,Theorem 2.8] proved that (QT) is nowhere dense in L{H) (and, a fortiori, that (QT)u(O.T)* is nowhere dense). In [155], G. R. Luecke proved that each of the sets of the chain No~(H) c No~(H)+K(H) c (QD) c (QT) (except (QT)) is nowhere dense in the containing set and D. A. Herrero [149,Theorem 3.1] proved that the same is true for the chain (QD)

c

(BQT)

c

(QT).

Hence, Theorem 6.17 admits the following companion result. THEORID~

160

6.23. Each of the

foZZo~ing

inaZusions is proper

NOir.(H)

c

NOil(H)+K(H)

c

(QD)

c

(BQT)

MoreoveF, eaah of these subsets the

ne~t

c

(QT)

(e~cept

c

(QT)u(QT)*

c:L(H).

LIH)J is nowheFe dense in

one.

Finally, the results of Section 6.5 are due to J. Barr!a and D. A. Herrero [42].

161

7 The structure of a polynomially compact operator

An operator T is called potynomiatly aompaat if p(T) E K(H) for some monic polynomial p. Every algebraic operator is polynomially com-· pact (p(T) =0) and so is the sum of an algebraic operator and a compact one. Indeed, this is the most general example of a polynomially compact operator (Theorem 7.2 below). The structure of these operators did not play any special role in the previous chapters. The intrinsic reason was that in those chapters we considered the set of alt algebraic operators (or att nilpotent operators) and the structure of the whole set "hides" the structure of many interesting subsets. In order to analyze several more specific approximation problems, we shall need a finer analysis of these operators. This is the content of the present chapter, as a preparation for the results of Chapter VIII on closures of similarity orbits.

7.1 Reduction to theh (essentially) nilpotent case The following auxiliary result will help to reduce ~any problems about polynomially compact operators to the case when the essential spectrum is the singleton {0}, i.e., to essentially nilpotent operators. (T is essentially nilpotent if T is a nilpotent element of the Calkin algebra.) LEMMA 7.1. Let T E L(H) and assume that cr(T) is the disjoint union of finiteLy many atopen subsets cr 1 , cr 2 , ••• , crm. Let Hj = H(crj;T) (j = 1,2, ••. ,mJ; then H = H1 +H 2 + ••• +Hm and (i) S(T) ={A € L(H): A~ ED.ml A., A.~ TjH., j = 1,2, ••• ,m}; J= J J J (ii) If A E S(T) and cr(A) is the disjoint union of finitely ma~ y alopen subsets cr 1•, cr 2•, ••• , cr' suah that cr! ~cr. for each j = 1,2, ••• m m _] . J •• ,m, then A~ EDj=l Aj, where Aj E S(TjHj) (J = 1,2, ••• ,mJ. (iii) If T is algebraic with minimal monic polynomial p, p(A) = II.m1 (A-A.)kj (A.;. A. if i;. j), then cr(T) is the disjoint union of J= J 1 J m alopen (finite or denumerable) subsets cr 1 , cr 2 , ••• , crm such that crjn 162

{A.} for eaah j = 1,2, ..• ,m. Furthermore, i f A J o(A) = o(T) and oe(A) = cre(T) = {A 1 ,A 2 , •.• ,Am}· o (T)

e

=

(iv)

E

S(T)

then

If a(A) = cr(T) for all A in S(T)-. then ={A

(v)

If T is polynomiaZly aompaat,

E

L(H}:

A~ IBj:l Aj' Aj

S(T)

E

S(TIHj)-, j = 1,2, •.• ,m}.

T has

minimal monia polynomial

p (as in (iii)) and TIHJ. = L.+K., where L. E L(H.}, (A. -L.)kj = 0 and J J J J J J E L(H.} (for eaah j 1,2, •.• ,m), then T = L+K, where L E L(H},p(L) J = 0 and K E K(H}.

K. J

PROOF. (i) Since {crj}j:l is a finite family of pairwise disjoint compact sets, it follows from Corollary 3.22 that T ~ Ql.m 1 T., where J= J m Tj E L(Mj)' Tj "'TIHj' j = 1~2, •.. ,m. (Observe that cr(A) =cr(T) =uj=l cr .• ) Assume that W= (W .. ) . . 1 , where w .. :M. ~ N. (i,j;: 1,2, ..• ,m),is J l.J l.,J= m l.J J m 1 -1 m an invertible operator such that IBj=l Aj =W(IBj=l Tj)W ; t!'J.en (IBj=lAj) W = W(IBj:l Tj). Proceeding exactly as in the proof of Corollary 3.22, we conclude that W.. = 0 for all i I' j and W.. is invertible for all j l.J JJ -1 1,2, ••• ,m. Therefore M."' N. and A. =W .. TW.. T. "'TIH· for each J J J JJ JJ J J j 1,2, .•• ,m. (ii)

Proceeding as in (i), we can write T

~

m

IBj=l Tj' Tj

E

L(Mj),

m

and A~ IBj=l Aj' Aj E L(Nj}' where cr(Tj) =crj c crj=o{Aj), j=l,2, ••• ,m. Clearly, it suffices to show that A. is unitarily equivalent to an op·· J erator in S(T.) (for each j = 1,2, ••• ,m). J m I f W ={W .. ) .. :M. ~ N1. , i,j=l,2, ••• ,m; n=l,2, ••• )and (w .. n l.J,n l.,J= 1 l.J,n J II~Bj:l Aj -Wn(IBj~l Tj!.wn-lll ~ 0 (n ~ oo), then we can combine the argument of the proof of Corollary 3.22 (as in (i)) with a limit argument in order to show that II W. . II ~ 0 ( n ~ oo) for all i I' j , M . o: N • (so l.J ,n J J that we can directly assume that M. =N.) and IIA. -w .. T.W .. -111 ~ 0 (n J J J JJ,n J JJ!n

~

oo).

(iii)

Clearly, oe(T) =o(T) = {A 1 ,A 2 , ••• ,Am} and o(T)\cre(T) can on ly accumulate at oe(T), whence we immediate deduce that o(T) is the dis joint union of m clopen subsets cr 1 , o 2 , ••• , om such that crjncre(T) = {A . } ( j = 1, 2, ••• , m) • J

If A

E

S(T)-, then

A

E

S(T)- and therefore (by Proposition 1.7)

p(A) =0.

Hence oe(A) c cre(T) ={Al'A 2 , ••• ,Am}. It follows that o(A) is a finite or denumerable set and therefore i t is a totally disconnected set. By Corollary 1. 6, oe (A) = oe (T) and o (A) =a (T). (iv)

This is a corollary of (ii).

Let W be an invertible operator such that WTW-l =Ill .m 1 T., J= J where T. o: TIH .• Then there exists operators L! "'L. and K! o: K. such J J J J -1 Jm J that T. =L!+K! (j=l,2, .•• ,m). Then T=L+K, where L=W ($._ 1 L~)W sa! J J J -1 m JJ isfies the polynomial equation p(L) =0 and K=W (IBj=l Kj)W E K(H). (v)

0 163

7.2 The structure of a polynomially compact operator THEOREM 7.2. Let T € L(H) and assume that p(T) = 0 foP some poZym k· nomiaZ p, p().) = Jlj=l ().- Aj) J. Then thePe e:r:ists a aompaat opePatoP K suah that p (T-K) = 0. LmR1A 7.3. Let P, R

E

L(H) be two orthogonaL projeations with

ao~

paat produat PR; then for evePy E > 0 there e:r:ists a finite Pank projeation F s R suah that IIPCR-Flll < E. ~ROOF. Let {Fn}n:l be an increasing sequence of finite rank projections converging strongly to R. Since PR e: K(H), IIPR(R-Fnlll .... 0 (n .... "'). Thus, for a sufficiently large n, IIPCR-Fn)jj = IIPR(R-Fn)jj <

£.

0 LEMMA 7.4. Let H be a positive opePator, 0 s H s C (where C is a

aonstantJ; then thePe e:r:ist an invePtibte opePatoP Wand a sequenae {Pn}n:l of pairwise oPthogonaZ projeations suah that <; "' -n (i) WH = HW = Ln=l 2 Pn; (ii) P W = WP and P H = HP for aZZ n = 1,2, ••• ; n n n n "' (iii) ran P H = ran P c ran H c Gln=lranPn= (ranH) - (m=l,2, .•• ); (ivJ IIWII s ~/C, Jlw-lt~ 2C.

PROOF. Let H =

f ).

dE ().) be the spectral decomposition of H and set -n - (n-1) "' C]) (n=1,2, ••• ). The sequence {Pn}n=l P 0 =Pker H' Pn=E((2 C,2 is pain.rise orthogonal , HP n = P nH for all n, and it is also clear that the Pn's satisfy (iii). Define W=ho,cJf().) dE(A), where 1, if ). = 0, { 1/(2 n A) , if A

f ().)

E

(2

-n - (n-1) C,2 C),

n=l,2, ••••

f [O , C)

-n n- 2 P, n WP n =P n W for all n=l,2, •.• , IIWII =sup()." [O,CJ) f(A) ~ ljc, tl is bounded below by inf -1 -1 -1 (). E [O,CJ) f().)=(2C) tsothatwisinvertible) andN =J[O,CJ f(A) dE().). Hence, IIW-lll s sup(). e: [O,C]) f().) -l = 2C. 0

Then WH = HW =

COROLk~RY

Af().)

7.5. Let A, B

thePe e:r:ists a projeation P aompaat.

E E

dE().)=

<; "' L -l

L(H) and assume that AB e: K(H). Then L(H) suah that AP and (1-P)B aPe both

PROOF. Let A= UH 1 and B* = VH 2 be the polar decompositions of A

H1 H2 = U*(AB}V is compact and this reduces our problem to show that there exists a projection P such that H1 P and (l-P)H 2 are both compact. Hence we can directly assume that A= H1 and B = H2 are and B*. Then

positive seui-definite hermitian operators.

164

Two projections M and N will be constructed so that AM, NB, and (1-M) (1-N) are compact. Then (1-M)B will also be compact, since (1-M)B= (1-M)NB+ (1-M) (1-N)B. By applying Lemma 7.4 to A and B, we obtain invertible operators w, v and two families of pairwise orthogonal projections {Mj}j=~ (commuting with A) and {Nk}k:l (commuting with B) such that

'{I~A=L;:l2-jMj'

rj:lMj= 1-Pker A'

BV=Ik:l2-~k

and Ik:lNk= 1-Pker

B'

Thus, we can write _ ~ "' -m-n _ -j -k M.WABVNk -M. (l. M N )Nk- 2 -~.Nk. J J m,n= l 2 m n J Therefore MjNk is compact for each j, k = 1, 2, • • . • Thus the product (Lj~l Mj)Nk is also compact. By applying Lemma 7.3 to this product, we obtain a finite rank projection Fk s Nk such that I: k -2k llll < 2 I k=l,2, ... . II Mj (Nk -F k) II < 2-k-j , J· s k, k = 1, 2, . • . • In part1.cular, Similarly, there exist finite rank projections G. s M. such that ' J' = 1 , 2 , • • • • Th us f or a llJ posl.tl.ve 'J' . II<M.-G.)NII < 2-k-j , k s J, 1.nteJ J kl gers k, j, we have (7.1)

Then AM= (W-lLj:l 2-jMj) [

(~k:l Gk) +Pker A]= w-lLj:l 2-jGj,

which is compact because 2-j + 0 (j + "') and G.G. = 0 if i -F j. J ]. Similarly, NB is compact. Observe that (l-M} = [
( 7. 2}

Now (Mj-Gj} (Nk-Fk} = (Mj-Gj}MjNk(Nk-Fk} is compact because MjNk E K(H). It follows from (7.1} that (1-M} (1-N} is the norm limit of the aompaat partial sums of (7.2). Hence, (1-M) (1-N} € K (H). D PROOF OF THEOREM 7.2. By Lemma 7.1, we can restrict ourselves to 165

the case when Tk = 0 for some k ;,; 1. We proceed by induction over k. I f

-1 = 0, then T is compact. Suppose that Tk

T

E:

K (H) for some k ;,; 2 and

the result is true for k-1.

By Corollary 7.5, there exists a projection P such that T (1-P)T

K(H);

E:

k-1

P,

then

T =

[

A

X

)ran P

K21 K22 ker p

'

where K 21 and K 22 are compact operators and A= Tran p· Observe that

( k-1 Hence, A

T -

[0

0

) )k-1

K21 K22

[Ak-l p =

0

*)

0 p

K(H).

E

K (p H). By our inductive hypothesis, there exists a com-

£

pact operators K 11 Define

E

k-1

L (ran P) such that (A-K 11 >

= 0.

then

CORO~LARY

7.6. Let Q be an essentially nilpotent operator of es-

sential order k; then there exists a sequenae {Kn}n:l of aompaat oper~ tors suah that

11Kn11

(i)

(ii)

+

0

(n

+ oo);

o(Q-Kn) ={O}u[o(Q)\D(O,l/n)}, n=l,2, ••• ;

(iii) nilpotent.

Q-Kn- Rn&Fn• where Fn

Furthermor~,

i f Q is.quasiniZpotent,

so that rank R J ~ rank QJ ~

.

that QJ

n

E

= OIH([o 0 (Q)\D(O,l/n)];Q)

oo-

t~en

and Rn is

the Kn's aan be ahosen

and rank R J is finite for aZZ j suah

n

K(H).

PROOF. Let on=o 0 (Q)\D(O,l/n)

(n=l,2, ••• ). Clearly, on isaclopen

subset of o(Q) and

where Mn=H(on;Q), Fn=OIMn and An=QNn' so that o(An) =o(Q)\On· By Proposition 3.45 (and its proof) operator N

E

there

e~ists

a normal compact

K (Nn) such that II N II= sp (A ) < 1/n and B =A -N is quasi

n n n n nilpotent. By hypothesis, Qk = 0; i t readily follows that = 0. By Theorem 7. 2, Bn = Ln+Cn, where Ln 166

k

= 0 and Cn

E:

-

n~

(A ) n

K (Nnl.

=

.

(B ) k n

s~nce

o(Bn) ={0}, it follows from the upper semicontinuity of the spectrum (Corollary 1.6) and Proposition 3.45 that there exists a fi~ite rank operator Gn such that llcn-Gnll < 1/n and cr(Ln+Gn) ={0}. Define R = L +G and n n n Kn=Q-

(~n :n)n = (~ Nn+(~-G)) n n

Then Kn € K(H), 11Kn11 < 2/n + 0 (n + oo), i.e., {Kn}n:l satisfies (i) • On the other hand, (ii) and (iii) are immediate consequences of the above constructions: Observe that R k = (L +G ) k is a finite rank n n n quasinilpotent; therefore, R k is a nilpotent. A fortiori, so is R • n n In the case when a <9> = 0 , ~e Rn' s can be cho~en so as to sa~ fy the condition"rank R J ~ rank QJ ~ oo- and rank R J is finite for . n n all j such that QJ E K(H) ",by an "ad hoc" modification of theprev:icus argument. 0 Corollary 7.5 yields the following m

COROLLARY 7.7. Let {Aj}j=l be a finite family of operators aating on Hand assume that the produat A1A2 ... Am is aompaat. Then there e~­ ist aompaat operators K1 , K2 , .•• , Km suah that (A 1 -K1 ) (A 2 -K 2 ) .•••• (Am-Km) = 0. PROOF. By Corollary 7.5 and an obvious inductive argument, we can find orthogonal projections P1 , P 2 , ••• , Pm-l such that K1 = A1 P1 , Kj = (1-P. 1 )A.P. (j = 2,3, ... ,m-1) and K = (1-P 1 )A are compact operators. JJ J m mm Then, (Al-Kl) (A2-K2) (A3-K3) .. • (Am-1-Km-1) (Am-Km) [A1 -A1 P1 J[A 2-(l-P 1 )A 2P 2 J[A 3 -(l-P 2 )A3 P 3 J ••• [Am-l-(l-Pm- 2 )Am-lpm~] x [Am-(l-Pm_ 1 )AmJ= CA1 (1-P 1 )

J[A 2 (1-P 2 )+P 1 A 2 P 2 JrA 3 (1-P 3 )+P 2 AfrP~J

•.. [Am-1 (l-Pm-l)+Pm-2Am-lpm-1J[Pm-1Am] = [Al (l-P 1 )A 2 (1-P 2 )J[A 3 (1-P 3 )+P 2A3P 3 J ... [Am-l(l-Pm-l) +Pm-2Am-lpm-l][Pm-1Am] = .•. = [Al(l-Pl)A2(l-P2)A3(l-P3) ··· Ar-1 (l-Pr-1) ][Ar(l-Pr)+Pr-lArPr][Ar+l(l-Pr+l)+PrAr+lpr+l] .•• [Am-l(l-Pm-l)+Pm-2Am-lpm-l][Pm-1Am] = [Al(l-Pl)A2(l-P2) .•• Am-1(1-Pm-l)][Pm-lAm] = 0.

0

7.3 Restrictions of nilpotent operators 167

Recall that qk is the k x k Jordan nilpotent cell (k = 1, 2, •.• ) de fined by (2.1). LEMMA 7.8. If Q is a niLpotent of

o~de~

k > 1 and M is an invaP-

iant subspace of Q such that OIM - qk (a.) foro some caPdinal a.. 1 :Sa.~ ""·· then the!'e e:xists an invariant subspace N fo!'Q such that MnN={O}, M+N is closedandH = M+N. PROOF. Let M.=Mnker Qj, j=l,2, ••. ,k. By induction on j we shall prove that there

~xists

an invariant subspace Nj

c

ker Qj such that

1) N • nM . = { 0}, and

J J . 2) N •+M . = ker QJ • J J For j = 1 we can take N1 = ker QeM 1 . Suppose N. has been construct-1

ed for some j < k. We have

Q

J

(Nj)nM = Mnker Q = M1 . Define

Nj+l = [Q

-1

(Nj)9(Nj+M 1 ) ]EDNj'

It is completely apparent that N.+N 1 is closed and j+l J If x E ker Q , then Qx = y .+z., where y. " M. and (a.) . J J J J - qk J.t follmvs that y. =Qy.+l for some y.+l E 1 -1 J J J E Q (Nj) =Nj+l+M 1 , and therefore x E Mj+l+Nj+l+M 1 Since N. is invariant under Q (j = 1,2 1 ••. 1k) 1 ]

that Nj+lnM = {0}. zj

E

Nj. Since OIM

M'+l' Hence x-y.+l J ] =Mj+l+Nj+l' if we define N

0

we are done. LEMMA 7.9. (i)

Let Q be a niLpotent of o!'deP k and let (5.1) be

the canonical representation of Q; then Q +lQ +l + 2 ••• Q _ 1 maps 1 p r r 1r r 1 r p P -1 ker QPeker Q injectively into ker Q 9ker or-1 fo!' 1 s r < p and p 213,. "lk.

k-1

(ii) Assume that ran Q contains a subspace M of dimension a.. Then there e:xist a subspace R0 c (ker Qk-1 ) .i of dimension a. such that M = Qk-lR and a constant o > 0 such that (7.3) 5
II or ,r+l 0 r+l, r+2 • • • 0 p-1,p ll

PROOF. (i)

For each p=213, ••• ,k-l, we have 0 0

...

Qp-1

*

0 0

168

1 ker Q eker oo ker o 2 eker Ql

0 Ql2Q23''' 0 p-l,p 0

k-1 ker 1leker Q -

(The first p-1 columns of the matrix of QP-1 are O's and the (l,p)-entry is equal to o 12 o 23 •.. Qp-l,p") By definition of the representation (5.1), it follows that o 12o 23 ... Q -l maps ker QPeker Qp-l into ker o 1eker QO and ker(Q 12 o 23 .•..• p ,p p-1 p p-1 Qp- 1 ,-p ) c ker Q n[ker Q eker Q ]= {0}.

Therefore, o12 o23 •.. Qp-l,p is injective. A fortiori, Qr,r-fi.Qrt~r+ 2 ... Q 1 is also injective for 1 ~ r < p. p-. ~p . k-1 k-1 1 . (~~)Thus, J.f M c ran Q =Q 12o23 .•• Qk·-l,k[(ker Q )] ~sa sub space (hence, closed) of dimension a, then there exists a subspace R0 c (Jeer Qk-l) 1 such that o 12 o23 •.. Qk-l,k:P- 0 ~ M is an isomorphism: this sho\o~s, in particular, that dim R0 =a and o 12 o23 ••• Qk-l,kJR 0 is bounded below. Now it is not difficult to show, by induction, that Qr,r+l maps Qr+l,r+ 2 ..• Qk-l,kR 0 isomorphically onto Or,r+lQr+l,r+ 2 ••. Qk-l,kRo and therefore there exists a positive constant o such that (7.3) is satisfied f~r all x E R0 and for 1 ~ r ~ p ~ k-1, whence it readily follows that QJR 0 must be closed (i.e., a subspace) for all j = 0 ,1,2, ••• ,k-1. Let y = Qpx for some p, 0 ~ p ~ k-1 and some x E R0 , and let z = p+l p+2 k-1 Q xp+l+Q xp+ 2+ ..• +Q xk-l' where xp+l' xp+ 2 , •.. , xk-l E R0 : then II y-z II

:2:

(1/ I!Ok-l-p II> llok-l-p ll = <11 I!Ok-l-p II> I!Ok-lx II (l/IIOk-l-pll> 11012023" •• Qk-l,kxll

:2:


(o/IIQk-l-pll>llxJJ

(o/IJOk-l-pll·lloPII> IIYII•

j k-1 p whence it follows that the subspaces V{Q R0 }j=p+l and Q R0 form a po~ itive angle. Therefore the algebraic direct sum R = L~:~ QjR 0 is closed • • 2 • • k-1 J and direct, i.e., R = R0 +QR 0 +Q R0 + ... +Q R0 is a subspace of H con. . M "" Qk-lR . taJ.nJ.ng . 0 It is completely apparent that R is invariant under Q and that the action of QJR can be described by: Q:~jR ~ Qj+lR isomorphically for j=O,l,2, ••• ,k-l, and Q:Qk-lR 0 ~ {0}, whe~ce it i~ediately follows that o 1 = OJR - qk (a). (iii) Let

Q

and assume that M

(

Q * )R 01 02 Rl

ran Qk-l(c R ) • Observe that k-1 Q

(Qlk-l 0

*

Q k-1

)

2

and Qk-lH = M. It follows that ran o/-l = {O}, i.e., Q k-1 2

= 0.

0

Corollary 7. 6 implies, in particular, that i f Qk = 0; cr·(Q) = { 0} 169

and rank Qj =

co-

(

see Section 1.3)

i.e. , Qj is a compact opera tor, not of finite rank; k, then Q is the limit of a sequence

~

for all j

{Rn}n:l of nilpotent operators such that rank Rnj < rank Qj for all n and j

(n,j=l,2, ... ) and Q-R

rollary 7. 7

e K(Hl for all n=l,2, . . • • By using Co-

k

n

\'le

can obtain an analogous result for the case when Q = 0.

PROPOSITION 7.10. Let Q be a niLpotent of o~de~ k and assume that k-1 k-2 r Q • Q •...• Q a~e non-zero finite ~ank ope~ato~s • and rank Qj = ..,_ fo~ j = s,s+l, ... ,r-1. but Qs-l /. K(H)(l,;; s,;; r,;; k). Then the~e exists a sequence {R } : 1 in Nk(H) such that Q- Rn E K(H) fo~ aH n, j . n nrank Rn =rank QJ fo~ j r,r+l, ••. ,k-1, {rank RnJ}n:l is st~ictZy increasing fo~ eaah j, s ,;; j ,;; r-1. and ]IQ- Rnll ... 0 (n ... co).

PROOF. By Lemmas 7.9 and 7.8 and an obvious inductive argument, we have Q

(: ;) = W(F$T)W-l,

where W is invertible, F- $. k+l q. (aj), the u. 1 s are non-negative ig k-1 J=r J · J tegers, uk =rank Q f< 0, Tr=o and rank TJ=co- for j=s,s+l, ... , r-1, but Ts-1 is not compact. If s = r, there is nothing to prove. Assume that 1 ,;; s < r ,;; k and let

Tl,r-1 Tlr

0

• T2,r-l T2r • T3,r-l T3r T 0

0 Tz-l,r Mr-1 0 Mr be the canonical matrix ofT r

€

·

L(M) with respect to the decomposition · 1

M = $j=l Mj' where Mj=ker TJeker TJ-, j=l,2, ••• ,r. Observe that T 12 T 23 ••. Tr-l,r is a compact operator with rank co-. Thus, by Corollary 7.7 there exists compact operators K12 , K23 , ••.• , Kr-l,r such that (Tl2-Kl2) (T23··K23) .•. (Tr-l,r-Kr-l,r) = 0. ( j = 1, 2, •.• , Given n ~ 1, we can find compact operators C. 1 . -1 J- ,J,n r-1) such that IIC. 1 . II < (rn) and J- , J ,n n < rank(Tl2-cl2,nl (T23-c23,n) · · • • (Tr-l,r -cr-l,r,nl = dn < "'-· Let Tn' be the operator obtained from T by replacing T. 1 . by )I J € K(H),

T. 1 .-c. 1 . , j=2,3, •.. ,r. It is easily seen that T-T' J- ,J J- ,],n n 170

IIT-T~II

< (rn)-l,

(T~)r=O, rank(T~)j :s

is not compact and rank(T')r-l=d

n

co- for j=s,s+l, ••• ,r-l,(T;}s-l

is finite.

n

Now we can repeat the same argument with

T~

in order

t~

obtain

T"n EN r (M) such that T-Tn" E K(M), IIT-T"II
n

n

j = r-l,r-2. After r-s steps, we obtain an .operator Tn

E

Nr(M) such

E K (M), II T- T II < 1/n, rank T J is finite and larger than n n s-l n is not compact. n for j = s, s+l, ... ,r-1 and Tn

that T- T

Passing, if necessary, to a subsequence we can assume that

rank

tTnjb:l is strictly increasing for each j, s :s j s r-1. Finally, define R = W(FGlT )W- 1 . It is completely apparent that co

n

n

the sequence {Rn}n=l satisfies all our requirements.

0

7.4 Operators similar to Jordan operators A Jordan operator J in L(H) is an operator unitarily equivalent to

m

mj

Glj=l[).j+Glk=l qk

(akj) ],

(7.4)

where 1 :s m .m are distinct complex numbers, 0 :s akj :s co and mj is finite for all j = 1,2, ••• ,m. THEOREM 7.11. If Q E N(H), then Q is similar to a Jordan operator if and only if ran Qj is closed for all j = 1,2, . . • • PROOF. If J is a Jordan nilpotent operator, then it is completely apparent from the definition (7.4) that ran Tj is closed for all j = 1, 2, • . . • It is also clear that if Q = WJW-l for some invertible W, then ran Qj = W(ran Jj) is closed for all j = 1, 2, • . . . Assume that Q E Nk (H) and ran Qj is closed for all j = 1, 2, .•• ,k-L Since ran Qk-l is closed, it follows from Lemmas 7.9 and 7.8 that Q has an invariant subspace Mk with invariant complement Rk such that OIMk ~ qk (ak) (0 :s ak = (1/k)dim Mk :s co) and 01 Rk E Nk_ 1 (Rk). Now we consider OIRk. By induction, we conclude that Q is similar to an oper~ k

tor of the form Glj=l qj

(c:x • )

0

J •

It is easily seen that . 1

.

a.=a.(Q) =dim (ker Qnran QJ- )S(ker Q(lran QJ), j=l,2, ••. ,k, J J . and tha~ this set of cardinals is a similarity invariant. Theorem 7.11 says that this is, in fact, the only similarity invariant of Q. More precisely, we have the following.

171

COROLLARY 7.12. {i)

Tf

o1 ,

02

E

Nk(H) and ran

o1 j

and ran

o2 j

are alosed for aZZ j = 1,2, •• ,k-1, then 0 1 - 0 2 if and onZ.y if aj (0 1 ) aj(Q 2 J for all j=1,2, ••• ,k-l. (ii) If Q ~ Nk(H) and ran Qj is dosed for all j=l,2, ••• ,k-1, then Q- Q* and Q- XQ for aZZ X e: ~\{0}. By Lemma 7.1, we have

COROLLARY 7.13. If T e: Alg(H), then Tis similar to a Jordan ope~ ator if and onZ.y if ran q(T) is alosed for all polynomials qjp, where p is the minimal. (monic) polynomial. of T. Operators similar to Jordan operators are actually dense in Alg

(H). We shall need an auxiliary result. L~ 7.14. Let Q e: Nk(H) with aanonical representation (5.1). Then QJ has closed range for aZZ j (j = 1,2, ••• ,k-1) if and only if Qj,j+l has closed range for aZZ j (j=l,2, ••• ,k-1J.

PROOF. Observe that (j) 0 0 1, j+l ( j) 0 Q2,j+2

0 0

*

0

0 • 0

Q ( j) k-j,k 0

0

.0

0

0 0

0

0

where Q(j)+. m,m J =Qm,m +lQm+ 1 ,m+ 2 ••• Qm+'J- 1 ,m +'· J If Q.1,1'+l has closed range (for all i=l,2, ••• ,k-l), then o. '+lis bounded below by o > 0 forall ( .)

1, 1

.

i=l,2, ••• ,k-l, and 0 J +'is bounded below by oJ, m=l,2, ••• ,k .. j. Ask m,m J . . . 1 sume that xn=L·-'+l xt:l e: (ker OJ).L, x~ e: ker Q 1 9ker Q1 - , i=j+l,j+2, 1-J

••. ,k, !lxnJI

=

1

1

.

1 for all n=l,2, ••• , and !IOJxnll _,. 0 (n _,.

oo).

It is not

difficult to see that

ojllxj\~1

s

llo{~~+i

xj+i" ... 0 (n ...

00 ) ,

i=l,2, ••• ,k-j,

and, a fortiori, that II xnll s L~=Jl. II x~+·ll ... 0 (n ... co), a contradiction. . . ·1J 1 Hence, ran OJ= QJ[ (ker QJ) ].L is closed, j = 1,2, ••• ,k-1. Conversely, if Oj has closed range, then Qjl (ker Qj)~ is bounded below by some positive constant n. Then oi~~+l = 0 12 0 23 ••• Oj, j+l is also bounded below by n and therefore it is left invertible (j = 1,2, ••• ,k-:0.

172

This clearly implies that ran Q.] , ]"+l is closed for j = 1,2, .•. ,k-1.

0 THEOREM 7.15. Let Q E Nk(H). Then there exis~s a sequena~ {Qn}n:l in Nk(H) suah that JJQ-Qnll ~ 0 (n + ~; and (ran QJ)- =ran Q J for alZ n j = 1, 2, .•. ,k-1 and for aZZ n = 1,2, . . . . PROOF. Assume that Q has the canonical representation (5.1) and let Q. 1 . =V.H. be the polar decomposition of Q. 1 ., j = 2,3, ... ,k. J- , ] ] ] ]- , ] Since Q. 1 . is injective, ker H.= {0}. Define Q by (5.1) with Q. 1 . J- ,J J n J- ,J replaced by Q. 1 . =V.(H.+l/n), j=2,3, ••• ,k, n=l,2, •..• J- , J, n J J . Then Qn E Nk(H) and (by Lemma 7.14) ran QnJ is closed for all j = 1,2, •.• ,k-l (n= 1,2, ... ) . On the other hand, it readily follows from the definition that ran Qj c: ran Q j c: (ran Qj)- for all j=l,2, ••• , n k-1 and all n = 1,2,... • 0 By using the arguments of Lemma 7.1, we obtain COROLLARY 7.16. Let T be an algebraia operator with minimal monia polynomial p. Then there exists a sequenae {Tn}n:l in Aig(H) suah that p(Tn) =0 for aZZ n=l,2, .•. , JJT-Tnll ~ 0 (n ~~;and (ran q(T)]-=ran q(Tn)for aZl qJp and aZ"l n=l,2, • . . .

7.5 A similarity invariant for polynomially conpact operators PROPOSITION 7.17. If p(T)

E

K(H) for some (monia) polynomial p,

then the following are equivalent:

T - J+K, where J is a Jordan operator and K is aompaat. (ii) T - J for some Jordan operator J.

UJ

ran q(T) is the aZgebraia sum of a subspaae Hq and the range R of a aompaat operator R, for eaah polynomial qJp. q q (ivJ 0 is an isolated point of cre[q(T)*q(T)] for eaah polynomial (iii)

(v)

p(T)

is similar to a Jordan operator for every unital ~-re­

presentation p of C*(T). (vi)

p(T)

is similar to a Jordan operator for some faithful unit

al *-representation p of C*(T). (vii)

p(T)

is similar to a Jordan operator for every unital *-re

presentation p of A(H).

The equivalence between (iv) and any of the stateMents (v}, (vi} or (vii} follows easily from Corollary 7.13 and the well-known 173

fact that ran A is closed if and only if 0 is an isolated point of a(A*A). The equivalence between (iv) and (iii) follows from the analysis of the possible structures of an operator range (or Julia manifol~ see [71], [96]) and (i) => (ii) => (iv) are trivial implications. In order to complete the proof (By showing that (vi) => (i)), we shall need the following auxiliary result. LEMMA 7.18. If T" L(H), o(T) =o 1 uo 2 , where o 1 and o 2 are two di!!._ joint compact nonempty subsets, H1 is the Riesz spectral subspace corresponding to o 1 , H2 = HeH 1 , T =

C) H1 (A 0 B H ' 2

and Pj is the orthogonal projection of H onto Hj, j = 1,2, then P 1 and P 2 belong to C*(T). Moreover, there also exists a unique operator X:H 2 + H1 such that

W=

(~ ~) (~ ~) [~ ~)-l = [~ ~) =A$B, (~ ~), W-l = (~ ~rl = (~ -~), A$B

" C* (T)

and C*(T) =C*({W,A$B}).

PROOF. It is well-known that a C*--algebra with identity is inverse closed. If the idempotent E =

(~ ~):~

is the projection of H onto H1 =H(o 1 ;T) along H(o 2 ;T), then E is the limit of rational functions in T and therefore 1 1) Hl (l+Y*Y 0) E, E* = ( Y* l H2 ' EE* = O O and Pl =(norm) lim(n

+ oo)

pn(EE*)

belonq to C*(Tl, where {pn}n:l is a sequence of polynomials converging uniformly to 0 on the real seqment [-1/3,1/3] and converging uniform!¥ to 1 on the real segment [2/3,1~1Yirl.It is clear that P2l~P1 , A$0 = P1 TP 1 , 0$3=P 2TP 2 , A$B=A$0+0$B and A$8-T also belong to C*(T). Since o(A) =o 1 is disjoint from o(B) =o 2 , it follows from Corollary 3.20 that T is invertible and AB -1 -1 -1 -1 X:TAB (C)= 27Tifr(J.-A) C(J.-B) dA, where r is the (suitably oriented) boundary of a Cauchy domain n containing o (A) su~h that o (B) nil-= fl. 174

Given ,\

E

r,

belongs to C*(T) and (by taking limits of suitable Riemann sums) we conclude that w=

(~

:) =1+ 2!i

Ir

X(A.l dA. "'C*(T).

A fortiori, w, w- 1 and AIBB belong to C*(T). Since T=W- 1 (AIBB)W C*({W,AIBB}), it immediately follows that C*(T) =C*({W,AIBB}).

0

Now we are in a position to complete the proof of Proposition 7. 17. Assume that there exists a faithful unital *-representation p of C* (T) on H

such that A= p (T) is similar to a Jordan operator with

minimal poiynomial p, p(A) = ni:l (A.-A.i)ki. Since p is faithful, it fol lows that p(T) = 0. (Indeed, p is also the minimal polynomial of If.) Hence, by Theorem 7.2, T (or some compact perturbation of T) is an

a~

gebraic operator with minimal polynomial p. Let

*

T

( 7. 5)

0

Tm be the matrix of T with respect to the decomposition

n

H = llli:l Ri'

where the Ri's are inductively defined so that llli=l Ri =H({A. 1 ,A. 2 , •.• , A.n} ;T) (n = 1, 2, .•. ,m) . Applying Lemma 7.18 (and an obvious inductive argument) toT, we m

can find a (unique!) invertible operator W in C* (T) such that T = W(llli=l T.)W-l and C*(T) =C*({W,I!l.ml T.}). A fortiori, ~

~=

A

~

*

p (T)

0

m

~

with respect to a decomposition H =Ill. 1ran p(R.), where R. denotes the p .1= 1 1 m orthogonal projection of H onto Ri (1=1,2, ••• ,m), V=p(V/), llli=l Ai = m ~ p(lll. 1 T.), and A. is similar to a Jordan operator with minimal polyl.= 1 k· 1 nomial (A.-A.i) 1 175

Clearly, this reduces our problem to the case when cr(T) is a si~ gleton: moreover, replacing (if necessary) T by T-A, we can (and sha~ -k -k-1 directly assume that T = 0, but T 'I 0 for some k <: 1. Then Ak = 0, Ak-l 'I 0 and 0

Hp,l

Al2 0

f-1 p,2 Hp,3

A23 *

0

(7.6)

A 0

0 ~-l,k f-lp,k-1 0

f-1

p,k where Hp,J. = ker Ajeker Aj-l, ] · = 1 ,2, ... ,k. Since A is similar to a Jordan operator, it follows from Lemma 7. 14 that the operators Aj,j+l:Hp,j+l ~ Hp 4 j .are bounded below and 0 is an isolated point of the spectrum of A*JAJ for all j=l,2, •.• ,k-l: k-1 .. therefore there exists n > 0 such that (O,nJn[u ._1 cr(A*JAJ)] = (O,nJn k-1 - j-j ][uj=l cr(T* T )] = 9. Thus, if Ej(.) denotes the spectral measure of T*jTj, then rank Ej ([n/2,n]) = Bj is finite f~r .all j = 1,2, .•. ,k-1. I t follows that 0 is an isolated point of a (T*JTJ) (j=l,2, ... ,k-l) and [n/2,nJ~u. ~ cr(T*jTj)J is finite. Hence,e[n/2,nJ contains a closed sub interval[~7aJ disjoint from uj~O cr(T*jTj) and therefore (by the Stone= Weierstrass theorem) there exists a sequence {rn}n:l of polynomials converging uniformly to 1 on [O,a]and converging uniformly to 0 on k . . . . [S,~ln[uj=O cr(T*JTJ)J. It readily follows that {rn(T*JTJ)} converges in the norm to the orthogonal projection L.=E.([O,a]) "C*(T) (j=O, J J . . . -1 1, 2, ••• ,k), and p (L.- t. 1 > = P. = (norm) lim(n ~ ~l {r (A*JAJ) -r (A*J .] _1 J JJ n n )} E C*(A) .is the orthogonal projection of Hp onto Hp, J., for all A j=l,2, ••• ,k. Since p is isometric, there e~ist compact perturbations Rj " C*(T)+K(H) of Lj-Lj-l' j=l,2, •.• ,k, such that R1 , R2 , ... , Rk are pa~ wise orthogonal projections, L]-.L .~, R.J = 1. and p l = "'j, tor all j =; 1,2, •.. ,k. LetT= (Tijli,j~l be the matrix ofT with respect to the decompo·· . . k h . . (- --;<. ) =P. s~t~on H=ID. 1 H., were H. =ran R., J=l,2, ••. ,k. s~nce p R.T.tc J= J J J ~ J ~ AP. and p is isometric, it follows that T .. must be compact for all J ~] (i,j) such that 1 ~ j ~ i ~ k. Thus, up to a compact perturbation, we can directly assume that T has an upper triangular matrix of the form ( 7. 6) (with the A .. 1 s replaced by T .. 1 s) with respect to the above de·· ~J ~J composition. Now observe that

rnj

176

E&j~2

p(lED

=

Tj,j+lTj,j+l) =p(Rl+

P 1 +}:j~ 2 (PjAPj+l) *(PjAPj+l)

Lj~2(RjTRj+l)*(RjTRj+l) =lED

E&j~ 2

Aj,j+lAj,j+l

is invertible in C*(A), whence we i~~ediately obtain that Tj,j+ 1 :Hj+ 1 '+l (c H.; j = 1,2, •.• ,k-1) is a Fredholm operator with 0 :!>

~ranT.

J, J ind T j , j +1 <

J

oo.

If k = 1, then T = 0 is a lifting of T, and we are done. Assume that k > 1; then, by using the above construction and an inductive ark-1 gument, we can assume that (1-~) (T+Kk-l) Jker ~ (ker ~ =E&j=l Hj) is equal to 0

*

0

0 T'

k-2,k-l 0 d

with respect to a decomposition ker Rk= (ED~:i Hj)E&a: k-l (0:; dj <"" for all j), where Kk_ 1 is a finite rank operator, Hj is a subspace of finite codimension of H. and T~ '+l:H~+ 1 ~ H~ is bounded below for all J ],] J J j=1,2, ••• ,k-l, and Od denotes the zero operator acting on the fik-1 dk-1 nite dimensional space a: Let Rk-l be the orthogonal projection of Hk-l onto Hk_ 1 • Since Tk- 1 ,k:Hk ~ ran Tk- 1 ,k is a Fredholm operator with negative index, Rk-lTk-l,k:Hk ~ ran Rk-lTk- 1 ,k is also a Fredholm operator with ind Rk-1Tk-l,k = dk 2: 0 and Tk-1,k = Rk-1,kTk-l,kJ [ker Rk-1Tk-l,k].L·is an is~ morphism o~ Hil~ert spaces between Hk = [ k~r Rk- 1 Tk-l, k] .Land ran Tk-l, k. Thus, i f a: k =a: k-lEDker Rk-1 Tk-1,k (dim a: k = dk = dk-1 +dk: < co) I then there exists a finite rank operator Kk such that if Kk = Kk-l +Kk, then T+Kk is unitarily equivalent to the orthogonal direct sum of an oper~ tor matrix of the form (7.6), with the A .. 's replaced by the T~ .'s, ~] d~J and the operator Od acting on the finite dimensional space a: k. k

Since T~ .+ 1 :H~+l ~ ranT~ '+l is an isomorphism of Hilbert ],] J ],] spaces for all j = 1, 2, .•• , k-1, it follows from Lemma 7 .14 and Theorem 7.11 that T+Kk is similar to a Jordan operator, and this completesthe proof of Proposition 7.17. 0 REMARK 7.19. It is convenient to observe that in the above proof (i.e., (vi) => (i) in Proposition 7.17) we did not use the hypothesis 177

•p(T) is compact". Indeed, this hypothesis is an easy consequence of any of the hypotheses (i), (ii), (v), (vi) or (vii), but false, in ge!!_ eral, if we merely assume (iii) or (iv) (see [30,p.868]).

7.6 Nice Jordan operators A Jordan operator J, given by (7.4) will be called niae if, for each j (j = 1,2, .•• ,m), akj is an infinite cardinal for at most oneva1 ue of k, k=1 12, ..• ,mj, and very niae if, for each j, akj is an infinite cardinal for exactly one value kj of k and akj = 0 for k f kj. Na·· mely 1 Q is a nice Jordan nilpotent if and only if (ao) k (a.) I; k a.
q 6 (ao) for some s.

z

LEMMA 7.20. Let C be a C*-algebra with identity 1 and let t ~ Nk (C) (k ~ 2). Then the folLowing are equivalent (i) tk·-j+t*j is invertib'le for all j = 1121 ••• ~k-1. (ii) tk-j+t*j is invertible for some j, 1 s j s k-1. (iii) There exists a faithful unital *-homomorphism T:L(~k) + C and an invertible element v € C suah that VT (qk) v-l = t. (iv) There exists a faithful *-representation p:C + L(H ) and j, p 1 s j s k-1, suah that the sequenae H p(tj) H p(tk-j) H p( ....tj) H p .... p .... p p k

.

.

.

k

.

is exaat (i.e~, ker p(t -J) ran p(t~ and ker p(tJ) ran p(t -J)J. (v) For every faithful unital *-representation p:C + L(Hpl• the sequenCJes H p (tj) p (tk-j) p (tj) 1121•••1k-l, P .... HP .... HP .... HP s j are exaat. (vi) p(t) +

~

qk91 for every faithful unital *-representation p:C

L(H ).

PROOF. (i) => (ii) and (iii) => (vi) => (v) => (iv) are trivial implications. (v) => (i) and (iv) => (ii). Since ran p(tj) and ran p(tk-j) are . . k . k . closed, so are ran p(tJ)*=ran p(t*J) and ran p(t -J)*=ran p(t* -J). Observe that and ran 178

k . ker p (t -J)

[ran p(tk-j )*J J. 1

so that ran p(tk·-j+t*j) =ran p(tk-j) +ran p(tj)* and

H

p

k . . k . j ran p(t -J+t*Jl * = ran p(t -J) *+ran p(t ) = H • p

Hence p(tk-j+t*j) is invertible in unital, it readily follows that tk-j+t*j (ii) => (iii)

L(H } • Since p is faithful and mu~t be invertible in C.

Assume that tk-h+t*h is invertible (for some h, 1 s

h s k-1) and let p:C*(t)

L(H } be a faithful unital

+

an~

If p(t) =T, then T e: Nk(HP}

0

*-representatio~

Tk-h+T*h is invertible. Let

Tl2 0

T23 0

*

T

0

j j-1 (where H.= ker T eker T , j = 1, 2, ••• ,k-1) be the canonical represenJ tation of T. Now

[

0 T(k,h)) T(k*,h*) 0 '

where

(h) Tl,k-h+l

(h) T2,k-h+2

*

T(k,h) 0 T(h)J h,k (T (h)

k-h,k

)

*

(h) (Tk-h-l,k-1)*

0

T(k*,h*)

*

(T(h) )* l,h+l

and T (h) - T T T Since Tk-htT*h is invertible, m,m+h- m,m+l m+l,m+2" · • m+h-l,m+h • it is not difficult to conclude that the operators

H , m= 1,2, ••• ,k-h, m '+lis injective for all j=l,2, ••• ,k·-l, an

T(h)+h:H +h m,m m

+

are invertible. Since T. J,J elementary inductive argument shows that T. '+l:H.+l +H. is invertible J,J J ' J

179

for all j = 1,2, ..• ,k-1. In particular, this means that all the subspaces H . ( J' = 1, 2 , ..• , k) have the sane dim ens ion. J Thus, up to a suitable identification of Hj+l with Hj (via a fixed unitary mapping form Hj+l onto Hj) we can directly assume that T. '+lis a positive invertible operator for each j=l,2, ••• ,k-l. ),) (') k-j . Clearly, T J +. (m = 1, 2, .•. , k-j) and T +T*J are also invertible m,m J operators (on their respective spaces) for all j "" 1, 2, ••• , k-l,. If A. "" k-. k-. . . J T JT* J and B . = T*JTJ, then J

= (Tk-j+T*j) (T*k-j+Tj) J J · is invertible and A.B. = Tk-jT*kTj = 0. Since A. and B. are positive, R. . an h errol..tJ.an . J l.'d empotent; J . easJ..J A. (A. +B . ) -l (.J "" 0]1]2 , , , ••. , k) J.S l.. t J.S J J J ly seen that P. = R. - R. 1 is the orthogonal projection of H onto HJ., JJ J j=l,2, .•. ,k. A .+B. "" (Tk-j+T* j ) . (Tk-j+T* j ) *

Let lj denote the identity on Hj' j = 1,2, ... ,k. Ilow it is easily seen that for each i, 1 s i < k, C*(T) contains the operator P l..TP.l. +l[l 1 el 2e ..• $1.J.- 1 e(T~l.,l.. +lT J.,l. . . +') fBl.+lfB ... elk]-~ ~ l. PiTPi+l[l 1 el 2 e ••• eli_ 1 eTi,i+leli+le ... elkJ 0

-1

H1

H2

0

0 1 0

0 Hk (the empty entries are 0 's) , where 1 = 1.J.,J.. +l denotes the identification between H.+l and H.l. (i.e., the operator U.l.,l.'+l). l. Since this is true for all i, 1 s: ~ < k, it is not difficult to conclude that all k x k operator matrices (with respect to the decomposition H =e.k 1 H.) with "constant entries" belong to C*(T). Hence, J=

J

0 1

Hl

0 l 0

Q=

0

H2 H3

"' qk 0 0 1 Hk-1 0 Hk

180

(oo)

E:

C*(T).

(k-1) (k-2) (2) If W= Tl,k EDT 2 ,k Gl ••• GlTk- 2 ,kftk-l,kltlk' then 0

1 Bll

.Bl,k-lBlk Hl

0

.B2,k~B2k H2 • 8 3,k-18 3k H3

1 0

0

0

1

0

0 Hk-2 1 Hk-1 0 Hk

Similarly, i f

• 8 i,k~

Bh. .Bj,k~ Bjk

Hl

.B4_k~ B.h

H3

H2 Slk-lltlk'

0 1 8 k-2,k Hk-3 1 Hk-2

then

181

0

1

B)s Bj 6

0

1

B,4 6

0

1 0

B" 3k B4k B" Sk B" 6k

0

0

0

0

0 Hl 0 H2 0 H3

0

0 H4

1

0 Hk-2

0

1 Hk-1 0 Hk

0

By an obvious inductive argument, after k-1 steps we obtain Q

v

-1

-1

, where v=ww 1w2 ..• wk-l E C*(T). Since p is faithful and unital, we deduce that there exist

TV, T=VQV

(unique) v, q

E

C* (t)

c

C such that t = vqv -1 , p (v) = V and p (q) = Q. If

~1=

(A..) . . k 1 is a matrix with constant entries in C*(T), then there ~] ~.]= exists a unique m E C* (t) such that p (m) = M.

Define T:L(~k) + C*(t) c C as follows: If {ej}j~l is the canonical miB of ~k and A= (A. •• ) • • kl is the matrix of A E L (~k) with re~J

~.]=

spect to this basis, then T {A) = m. The properties of p make it clear that T is a faithful unital

D

*-representation. Since T (qk) = q, we are done. . Observe that qk

k-.

.

k .

J+qkJ is a unitary operator. Hence, so is T(qk-J

+qkJ), j = 1,2, •.• ,k-1. Let n = [k/2] (= the integral part of k/2) and for a € C assume that 6 = lla-T (qklll
k-n

n k-n n k-n k-n n n +a*) -T(qk +qk >II s lla -T(qk) ll+lla* -T(qk)* II

s [(1+6)k-n_l]+[{l+6)n-l] s 2(1+6)k-n_2 < 1 and therefore ak-n+a*n is invertible in C. By Lemma 7.20, we conclude that i f a

Nk{C) and lla-T(qklll
-1

E

and lla-bll < ak/(llvl!.llv

-1

II>,

then

llv- 1 bv-T(qklll = ljv- 1 (b-a)vll
therefore b - T(qk). Hence, we have COROLLARY 7.21. Let C be a C*-algebra with identity 1. The set

C:

a- T(qk)} (where Tis some faithful unital *-homomorphism of L(~~into C) is open in Nk{C) (k=l,2, ..• ).

{a

E

Thus, the subset 182

N~(A(Hll = {T e Nk(A(H)):

Tk-l+T* is invertible}

(7.8)

is open in Nk(A(H)) and

N~(H) =71-1:N~(A(II))] = {T intersects Nk(H)

E

-k -k-1 . . . +T* ~s ~nvert~b1e} ( 7. 9·) T = 0, T

L(H):

in an open subset of Nk(H)(k=1,2, ... ).

PROPOSITION 7.22. If T e L(H) and

~ is aompaat, but

Tk- 1 ~

0 (fop some k ~ 2), then the fo~Zowing aPe equivaZent (i) T - J+K, whePe J is a niae JoPdan niZpotent and K is compact. (ii) T = WJV+K, whePe W, V aPe FPedhoZm opePatoPs such that ~ = VW 1. J is a niae JoPdan niZpotent and K is compact. (iii) T = WJV+K, whePe W, V aPe FPedhoZm opePatoPs suah that ~ · VW = l, J is a VePy nice Jopdan niZpotent and K is compact. (iv) Tk-j+T*j is invePtible foP aZZ j = 1,2, ••• ,k-1. (v) Tk-j+T*j is invePtible foP some j, 1 $ j $ k··l.

(vi) ThePe exists a faithfuZ unital *-homomoPphism <:L(~k) + C*(T) and an invePtible element V" C*(T) such that v,(qk)V- 1 ='1'. (vii) ThePe exists a faithfuZ *-PepPesentation p:C*(T) + L(H p ) and j, 1 $ j s k-1, such that the sequence p(Tj) HP

+

p(Tk-j> Hp

+

is exact. (viii) FoP evePy faithfuZ unital *-PepPesentation p:A(H) the sequences Hp

p(Tk-jl

p(Tjl +

HP

+

Hp

p(Tj)

...

HP

+

L(H pl

(7.10)

are exact foP aZl j = 1,2, ••• ,k-1. (ix) p(T) - qk91 foP evePy faithfuZ unital *-PepPesentation p: C*

(T)

+

L (H

p

l•

We shall need an auxiliary result. LEtlllA 7.23. Suppose H = H1efl 2 with H1 "' H and 0 < dim H2 < "' and LetT= T1 eT 2 be in L(H); T." L(H.), j = 1,2. Then, undeP the identi] ] fiaation of H with H1 , T is unitaPily equivalent to T1 in A(H).

-

PROOF. Let U:H 1 + H1 eH 2 be a unitary mapping from H1 onto H1 eH 2 and let S=UT 1 u• in L(H); it suffices to prove that 5"' T. Define V:i/ 1 H1eH 2 by Vlt = xeo • Now UV*TVU* = S, VU* e L (H) is an isometry and corank VU*=nul:VU*)*=nul V*=dim H 2 is finite; thus '\YO* is unitary in A(H) and it follows that T "' 5. 0

+

PROOF OF PROPOSITION 7.22. (i) => (ii) and (iii) => (ii) are triv 183

ial implications, (ii) => (iii) follows immediately from Lemma 7.23, and the equivalence between any two of the conditions (iv) - (ix) follows from Lemma 7. 20 and its proof. (Take t = T, and C = C* (T) and C = A(H) .)

If T has the form of (iii), then J z qk(oo). Since U(qk (oo)) is closed (Proposition 4.22), it easily follows from Proposition 4.21 that p(J) " qk (co) for any faithful unital *-representation p of A(H) on a Hilbert space H of dimension a. By Lemma 7.20, (iii) => (ix). p Assume that C*(T) admits a faithful unital *-representa~ion p0 such that p 0 (T) "qk(oo). By Proposition 7.17 and its proof, T=W(J+K) . k-1 w-1 , where \J E G (H) , K c K (H) and J " E9. k=l q . (a·) J w1. th ak = "". IE L.=l (oo) (oo) J J k-1 J o:j = oo, then J" qk eqs E9R, where 1 ~ s < k and R' =0. Let p:A(H) + L(H ) be a faithful *-representation. Since U(J) is p closed (Proposition 4.22), it is easily seen from Proposition 4.2lthat p(J) " qk (o:)eqs (B)eL, a f 0, B f 0, Lk-l = 0 and therefore the sequence H

p

it readily follows that the sequence (7. is not exact. Since T = 10) cannot be exact for j = s, contradicting the equivalence between (ix) and (viii). ltle conclude that I~:i o:j-l < oo, i.e., J " qk (oo)EDF, where F acts on a finite dimensional space. Since J is a nice Jordan operator, we see that (h) => (i). The proof of Proposition 7.22 is complete now. D

COROLLARY 7.24. Assume that Q E L(H) is a quasinilpotent oper•ator> sueh that Qk = 0, Qk-l f 0 and Qk-l+Q* is invertible (for> some k ~ 2). Then Q is similar to a niee nilpotent operatoP if and only if rank Qk is finite.

PROOF. The necessity of the condition "rank Qk is finite'' is clear, since :ran Qk must be closed if Q is sir.tilar to a n±ce i'lilpotent. Assume that rank Qk is finite. Then rank Qj is finite for all j ~ '+1 . .
(a·)

is a nilpotent and, by Lemmas. 7. 9 and 7. 8, Q - RE9F, where F = Glj=kqj J , Lj:k jaj • • 'bl e, so 1.s . ...,.k-l S1.nce R +R" 1.s 1.nvert1. v +'J* ( ProposJ.' t '1.on 7 . -?2) , but 184

this is impossible unless J is nice. Hence, Q

~

J$F and J$F is a nice Jordan nilpotent.

COROLLARY 7.25. If Q

k-1 Nk{H), Q

E

~

0 and.the canonical decompo-

sition of Q is given by (5.1) with ker QJeker QJ-l = Hj, then the

fol~owing

(i~

D

j = 1,2, ••• ,)<.,

are equivalent:

If Qj,j+l = VjHj (polar decomposition), where Hj = (Qj,j+l*

'+l) ·2 :tl '+l + tl.+l and V. :II '+l + II. is a (necessarily injective) par J,] J J J J J tial isometPy, then nul V~ < co for> all j = 1,2, •.• ,k-1.

Q.

(ii) If Qk-j+Q*j = ~.R. (polar decomposition), then nul J J foP aZZ j=l,2, ••• ,k-l. (iii) nul W~ < co for some j, 1 s j s k-1. J ( iv) nul W. < oo foP aZZ j=l,2, ••. ,k-l. J (v) nul W. < oo for some j, 1 ~ j s k-1.

W~ J

<

co

J

PROOF. Given £ > 0, let Q

£

be the operator obtained from .Q by re-

placing Q. '+l by V.(H.+£). By Theorem 7.15 and its proof, Q is simiJ,J J J £ lar to a Jordan nilpotent of order k. (i) => (ii), (iv). If Q satisfies (i), then Q

£

Jordan nilpotent.

is similar to a nice

(Use Theorem 7.11 and its proof.) By Proposition 7.

22, Q k-j+Q*j is a Fredholm operator. On the other hand, it is easily £ £ k . . k . . seen from our construction and (5.1) that ker(Q ··J+Q*J: =ker(Q -J+Q*1 k-.

and [ran(Q

.

J+Q*J) J

-

= ran(Q

k-.

£

.

£

£

J+Q*J) for all e: > 0 and all j = 1,2, •.• , £

k-1, \IThence the results follo\11. (iii:=> (i)

Conversely, if nul

V~=oo

for some j, j=l,2, •.• ,k-l,

then M= V

ran(Q k-j+Q*j) has infiniEe codimension in tl. Since (ran £> 0 £ £ = ran W. c. M, it follows that nul W~ = oo, a contradiction. J J (iv) => (i) Similarly, if nul V~=oo for some j, l s j s k-l,then ker (Qk-j+Q*j) = V 0 ker (Q k-j+Q*j) is {nfinite dimensional and therefore e:> £ £ nul W. = oo, a contradiction. J Since (ii) => (iii) and (iv) => (v) are trivial implications, we

W. R.) J J

are done.

D

COROLLARY 7.26. Assume that Q has the form of CoPollary 7.25 and nul Vi! is finite for al.Z j = 1,2, .• ,k-1; then the foZZowing are equivaJ Zent: (i)

(ii) E)) =

~k-1

Q

~

+Q* is not invertible.

If Hj =

oo (j'or a"ll

£

J

A dEj (spectral decomposition), then rank Ej([O,

> 0) for some j, l

s j s k-1.

If R. = J A dF. (spectral decomposition), then rank F.([O, J J J E]) = oo (for all c > 0) for some j, 1 s j s k-1. (iv) rank FjUO,e:]) =oo (for aZZ £ > 0) for aZZ j=l,2, ••• ,k-l. (iii)

185

PROOF. Since nul V'i! is finite for all j=l,2, ••• ,k-l, it is not J difficult to conclude from (5.1), Corollary 7.25 and Proposition 7.22 that the following four conditions are equivalent (a) Qk-l+Q* is invertible1 (b)

Qj, j+l is a Fredholm operator for all j = 1, 2, •.• ,k-11

(c)

Q

J+Q*J is a Fredholm operator for all j = 1, 2, •.. ,k-11

(d)

Q

J+Q*J is a Fredholm operator for some j, 1 ~ j ~ k-1.

k-.

.

k-.

.

0

The contrapositive of this equivalence yields the result.

COROLLARY 7.27. Assume that Q E: Nk(H), rank Qj is finite for j=r, j -s-1 r+l, .•• ,k-1, rank Q =oo- for j=s,s+l, •.. ,r-1, Q FO and either s=l -s-1 or s ~ 2 and Q +Q* is not invertibZe. Then there exists a sequenae {Q } : 1 in Nk(H) suah that 110- Q II -+ 0 (n -+ oo) and n n-. k (a·dn (?..) on -ne[61. +lq. J J; ~n J=s J (ii) t To a
for j

r+l,r+2, •.• ,k; j

(iv)

00

{rank Qn }n=l is striatZy inareasing for j = s+l, ... ,r-1; (V) Mn is a Jordan operator of order s; and (vi) Mn s-l F 0 and either s = 1 or s ~ 2 and M s-l+M* is not inn

n

vertibZe.

PROOF. By Proposition 7.10 there exists a sequence {R } 00 in Nk n n=l E K {HI for all n, rank R j =rank Qj for j = r, r+ 1 , .

(H) such that Q- R

n

n

oo

{rank R J} is strictly increasing for each j such that s n n=;t. ~ j ~ r-1, and IIO-Rnll-+ 0 (n-+ oo). By Lemmas 7.9 and 7.8, R!l- LnlllFn' whereL s=OandF =lll.ks+lq~ajnl actsonafinite dimensional space. (It n n 1= J is easily seen that a. is independent of n for j = r+l ,r+2, ••• ,k.) Jn Since Q- R is compact, it follows from Lemma 7.23 that L s-l F 0 .. ,k-1 ,

n

n

and either s = 1 or s ~ 2 and L s-l+L* is not invertible. Thus, in order

n

n

to complete the proof, it suffices to show that if T c

Ns (HI, Ts-l

F0

and Ts-l+T* is not invertible, then T can be approximated arbitrari:!.y by operators similar to Jordan models with the same properties. Let

£

> 0 and let

0 T12 Tl3 0 T23 0

.Tl,s-iTls Ml .T 2,s·-a?2 s M2 .T3,s-i T3s M3

T

(7 .11)

0 0 Ts-i.s Ms-1 0 Ms 186

be the canonical decomposition (given by (5.1}} ofT. Since 'lls-lf:O,it is not difficult to see that M. = ker Tjeker Tj-l is infinite dimension al for all j=l,2, ••• ,s. SinceJTs-l+T* is not invertible, it follows from Corollary 7.26 that either (ranT. '+l} has infinite codimension J' J or ran Tj,j+l is not closed, for some j, 1 ~ j ~ k-1. Le: Tj, j+l = V jHj be the polar decomposition of T j, j+l. If (ran Th,h+l} =ran Vh has infinite codimension for some h, 1 ~ h ~ k-1, then we choose T

£

as the operator defined by (7.11} with T.

'+l

J,J

re~

placed by V.(H.+e:/2}, j=l,2, ... ,s-l. It is easily seen that liT-Til e:/2 < e:,

J

J

£

(7.11) (with the new (j,j+l)-entries, j = 1,2, ..• ,s-1) is the

EN IH) and (by Lemma 7.14) ranT j is £ s £ closed for all j = 1, 2, .•• , s-1. By Theorem 7.11, TE: is similar to a Jar dan operator. It is completely apparent that T S-l t= 0 and, since the

canonical decomposition ofT

£

(h,h+l) -entry of T£ is vh (Hh +e:/2) and ran Vh (Hh +e:/2) =ran vh has infinite codimension, it follows from Corollary 7.25 that T s-l+T* cannot £

£

be invertible. Assume that s

2 and (ran Tj,j+l)

2

has finite codimension forall

j=l,2, ..• ,s-l, and let h be the last index such that ran Tj closed. Since Ts-l

t= 0,

'+l isnot

Th,h+~ J cannot that if Hh = f A

it is not difficult to see that

be compact. Hence, there exists 11, 0 < 11 < e:/2, such

dEh (spectral decomposition) then Eh( [ 0 •11)} =Eh( (0 •11}) and Eh ( [11 ,ao}) are both of infinite rank. In this case, we define T' as the operator de£

fined by (7.10) with Tj,j+l replaced by Tj,j+l =Vj(Hj+e:/4) for j=l,2, •.• ,h-1 and j = h+l,h+2, .•• ,s-1 and Th,h+l replaced by Th,h+l = Vh (/ [11 ,ao) A dEh}. It is easily :;ecn thatiiT··T'II=max{e:/4,11} < e:/2, T's=o and

T£•s-1 t=

£

0. Furthermore,

£

(use ( 7.10) :)

_o- o-.-.-.-o~T.i2:ir ~ ·:~:1!.1Ml

*

T,s-l+T'*

I

u

I I

• •

I

0

M2 • •

M s

Since Tj, j+l is a Fredholm operator for all j I= h and nul Th,h+l = oo, it is not difficult to check that nul Ti 2 Ti 3 • •• T~-l,s =oo. Thus, if T~s-l+T'* =\<'1 1 R1 (polar decomposition), then nul w1 =ao. Let 0

A 12 A13

0

A23 0

ker *

ker ker

T~ 1 eker T~ 0 T~ 2eker T~ 1 T~ 3 eker T~ 2

T'£ 0

187

be the canonical decomposition of T~. Since nul w1 = ""• it follows from Corollary 7.25 that (ran Aj,j+l) has infinite codimension for some i, 1 ~ i ~ s-1. Thus, we can proceed exactly as in the Iirst part of the proof in order to find an operator T£ similar to a Jordan nilpotent of order s such that II T' - T II < £/2, T ~:~-l "I' 0 and T s-l+T* is not invert £ £ £ £ £ ible. It is completely apparent that II T- T II < £. Since £ > 0 can be £ chosen arbitrarily small, we are done. 0 RE!~P~ 7.28. It is convenient to observe that, by using translations, direct sums and similarities, all the results of this section can be rephrased as results about compact perturbations of similarities of nice (not necessarily nilpotent!) Jordan operators, etc.

7.7 Notes and remarks The theorem about the structure of a polynomially compact operator (Theorem 7.2) is due to C. L. Olsen [159]. The auxiliary result Lemma 7.1 is essentially contained in [139] (see also [43]). Corollary 7.6 is a trivial consequence of C. L. Olsen's theorem and Corollary 7. 7 is a remark of D. A. Herrero. Lemma 7.8 is an unpublished result of c. AposLol and D. Voiculescu [34]. (A particular case of this result, for a = 1, can be found in B. Charles' article [65,Theorem 7].) Lemma 7.9 and Proposition 7.10 are unpublished observations of J. Barria and D. A. Herrero [43] and D. A. Herrero, respectively. Theorem 7.11 has been simultaneously and independently discovered by C. Apostol and J. G. Stampfli [30], L. J. Gray [109] and L. R. Hilliams [l97J. Theorem 7.15 is due to L. J. Gray [110], but the proo: given here is based on c. Apostol's Lemma 3.2 of [12] (Lemma 7.14, above). The results on sintilarity invariants of Section 7.5 have been taken from L. A. Fialkow and D. A. Herrero's article [93] and the no·tion of "nice Jordan operator" is also due to these authors [94]. The J~ey result given by Lemma 7.20 is an unpublished result o:: C. Apostol and D. Voiculescu [34]. The statement given here is a mild improvement due to L. A. Fialkow and D. A. Herrero and Proposition 7.22 and Corollaries 7.25, 7.26 and 7.27 also correspond to these authors (unpublisheu results). On the other hand, Corollaries 7.21 and 7.24 are contained in [34].

188

8 Closures of similarity orbits of nilpotent operators

With the help of Lemma 7.1, the problem of characterizing S(T)for T a polynomially compact operator can be easily reduced to thecase when T is nilpotent. This chapter is an account of what is known and what is unknown about this problem. The results include some information about quasi~ potent operators, the complete solution of the corresponding problem in the Calkin algebra and several conjectures and examples that illustrate the difficulties of the still open part of the problem.

~Universal

operators

Given a class W of operators acting on H, we shall say that R € W c S(R)-. Namely, qk is a universal operator for Nk(a:k) (k=l,2, ..• ), but N(H) (Han infinite dimensional space) and {A: A c a:} c L(H) do not contain any universal operator. is a universa~ operator for W if W

8.1.1 Universal quasinilpotent operators k k-1 Let Q € L (fl) be a quasinilpotent operator. I f Q = 0 and ran 0 contains a subspace of dimension n (1 sa s oo), then Q- qk (a)eR (Lemmas 7.9 and 7.8), where R c Nk(H 8 ) for some space Ha of dimension a (0 sa s oo). Dy Rota's corollary (Corollary 3.36(ii)), R ~ q 1 (a) (=the Sl.m 0 operator acting on H8 ). A fortiori, (by a new application of Rota's corollary) we have ~ q (B)eq (a) ~ q (oo)EDq (a) 7 q (oo)eqk. (8.1) s1.m 1 k s1.m 1 k s1.m 1 Assume that Qk t- 0 for all k = 1,2, ... , and let Rk (Q) = {x € H: Qkx 0}. It is easily seen that Rk(Q) is a aZosed nowhere dense subset of H. Hence, by Baire's category theorem, H\uk:l Rk(Q) t- ~, i.e. there exists a vector y c H such that Q1 ~y t- 0 for all k = 1, 2, • . . • Let II = Q

V k?O Qky and let

189

*)M

M Q = ( 0 L MJ."

Clearly, a (Q) = { 0} =a (M) ua (L) • Thus, by Lemma 3. 36 and Corollary 3.35(ii), Q s!m !1&L s!m M&O, where M=QIM is a cyclic quasinilpotent operator such that 14k ~ 0 for all k = 1, 2,... •

• • • .J

LEMMA 8.1. If Q E: L (H) is quasinilpotent and Qk ~ 0 foP all k = 1,2, then S(Q) aontains aZZ aompaat quasiniZpotents .

PROOF. By Lemma 2.5 and Proposition 2.33, it suffices to show that Q s!m qkeo for all k large enough and, by our previous observations, we can restrict ourselves to the case when Q has a cyclic vector y. Let {en} be the Gram-Schmidt orthonormalization of the sen-quence {Qn-ly}n: 1 • Since y is cyclic, it is straightforward to check that the matrix of Q with respect to this ONB has the form

:t

.

qll ql2 ql3 q21 q22 q23 q32 q33 q43

qlk q2k q3k q4k

. .

ql,k+l" q2,k+l" q3,k+l" q4,k+l"

Q

qkl-l,k qk+J.,kil. qk+2,kil.

0

1/k Moreover, qk,k-l~O for all k 2: 2 and lq 21 q 32 ••• qk,k-lls IIOklll/k-+ sp(Q) =0 (k-+ oo), so that inf{iqk,k-ll: k 2: 2}=0. Fix k 2: 2 and let W be the diagonal normal operator diag({1,~ 1 , j

w2,. • .,wk-2'wk-l'wk-l'wk-l'wk-l' ••• }), where wj = (lli=l qi+l,i) 2, ••• ,k-1. If N = V{e 1 ,e 2 , ••• ,ek}, then we have WQW-l = where

190

(~ :) ~J.

,

-1

.

, J = 1,

1

w1 w1 q23w q24w 2 3 w2 q33 q34w 3

q22 1

1

q44

A=

0

qk-1,k-1

q1,k+l q2,k+lw1 B

ql,k+2 q2,k+2wl

ql,k+3 q2,k+3wl

1 wk-1 qk-l,k+lwk-2 qk-l,k+2'"'k-2 qk-1,k+3wk-2 qk,k+lwk-1 qk,k+2wk-l qk,k+3wk-l 0

0

0 qk+1,k 0

c

0

and

D

0

It follows that

k-1

IIA- Lj=l

k ej+leej II ~ Lj=l

(qijli,j:k+l.

.

II Oil), II Bll ~

llcll ~ 11011 and IIDII "'lloll. . "" -- qk* -- qk, we cone 1 u d e t h at S 1nce l.~;k-1 j=l ej+l .,ej

k Lj=l

.

II ole,

dist[qk&lO ,S (Q) J s

11wo1·r 1 -q~eo11"' 4Lj~ 1 11ollj· By Rota's corollary (Corollary 3.35(ii)), there exists a sequence -1 {V} n n-_ 1 such thatiiV _1 n OVn II< 1/n (n=l,2, •.• ). It is completely appa!, ent that VnOVn is also a cyclic quasinilpotent. Thus, we can apply 00

191

the above argument to VnQvn-1 in order to obtain a sequence {Wn}n: 1 in G(H)

such that IIW (V QV - 1 ) W - 1 - qk*6l0 II = II<W V ) Q(W V ) - 1 - qk*6l0 II < 4k/n + 0 n n n n nn nn

(as n

+

0

co). Hence, Q sim qk6l0 "' qk6l0 for all k = 2,3,... •

From Lemma 2.5, we obtain the following auxiliary results: qk(k-l) slm qk-ik) for aZZ k > 1.

COROLLARY 8.2. (i)

q 6lq (k- 2 ) ~ q (k-l) for aZZ k > 2 • 1 k sJ.m k-1 "" (oo) + (oo).., (co) + (oo) f ( •••) vvv qjwqk sim ql wqk sim qk-1 or a ZZ J·

(iiJ

~

1

an d a ZZ k

> 1.

.

s-1

(oo)

k (a·) k Ell[6lj=s+l qj J ], Lj=l,j~s aj < oo, (oo) k (T ·) \' k j T = qs 6l[6lj=s+l qj J ], Lj=s+l T j < oo and rank A ~ rank Tj for aZZ j = s, s+l, ••• ,k-1, then there e:x:iata a non-negative integer a. aueh that T ~ q (a.)I&A.

(-z.v)

sJ.m(v7

If A= [6lj=l qj

If

L~=i

'j =

oo,

(a·)

J JCDqs

then

[6l~:i

qj (Tj) ]6lqk(oo) i q 1 (oo)Glqk (oo)

aZZ k > 1. (vi) If k > 1, then S(q 1 (oo)6lqk(oo))-=Nk(H)\N~(fl)

:>

N~(H) ia defined by (7.9)). PROOF.

(i)

for

(where

Nk_ 1 (11)

Since k(k-1) = (k-l)k, both operators act on the same

finite dimensional space. On the other hand, rank [q

(k)Jj=k maa{k-1-j,O} ~ (k-l)max{k-j,O}=rank [qk(k-l)]j k-1

(j = 1,2, ••• ,k-1). Now the result follows from Lemma 2.5. (ii) (iii)

This can be proved by using the same arguments as above. By (i),

(ii), Lemma 2.5 and an obvious inductive argument,

we have q.Glq (co) =q.Gl[q (k-l)](oo)Glq (oo) "!" q (j)Gl[q (k)](co)$q (oo) J k J k k sJ.m 1 k-1 k cr {j)6lq (co)6lq (co)= q (j)(ll[q (k-2)] (oo)(ll'l (co) "!" q (j)Q ~1 k-1 k 1 k-1 k SJ.m 1 k-2 6lq (co) "!" + q ( j) 6lq {00) 6lq ( 00) = [ q 6lq ( k- 2) ] (00) -t ( k -1) (co) k sJ.m sim 1 1 k 1 k SJ.m qk-1 (co) qk-1 (iv) I f rank Aj ~rank Tj for all j=s,s+l, ... ,k-1, it is notdif

.

.

ficult to find non-negatJ.ve J.ntegers a, + \' k Ljcs+l jTj· By Lemma 2.5,

T

\' k

.

such that a.+L. 1 '.J. Ja.J. = s-r J= ,JrS

q.<'j)J "!" (a) k q.(aj)]. q (-r)..,["'.k s w wJ=s+l J sJ.m ql 6l[6lj=l,jf"s J T ... (a)..,,. . . A f or t J.OrJ., sim q 1 ~· (v) By {iii), q 1 (co)Glqk(oo) slm

192

[6lj~l qj (Tk)]Glqk(oo) and, by Lemma 2.

->. (j-r]·) 5 , q.J (<]·) s:un ql ql (oo) Ellqk (oo)

(. -1 2 k- ) th k-1 (-r]·) (oo) .,. J- , , •.• , 1 , so at [Eilj=l qj ]Eilqk sim

Hence, [Eil~:i qj

(T

jl ]Eilqk (oo) # ql (oo) Ellqk (oo)

(vi) By Corollary 7.21, S(q 1 (oo)Eilqk(oo))- c Nk(H)\N~(H). Let A E + Nk(H)\Nk(H) and lets (1 ~ s ~ k) be the largest exponent such that .As-1-lo. I f s=k, then it follows from Corollary 7.27 that!IA-Tnll-> 0

(n for a sequence {Tn}n:l in Nk(H) such that k-1 (T' ·) (oo) T - [Ell. q. n] Jeq n J=l J k with I~:i T n. = oo. Since (by (v)) Tn # q 1 (oo) Ellqk (oo) , it readily follows that {~n}n:l]c S(q 1 (oo)Eilqk(oo))-. A fortiori, A E S(q 1 (oo)Eilqk(oo))-. .,. oo)

If s < k, then it follows from Theorem 7.15 and Corollary 7.12 that IIA- Snll.,. 0 (n.,. oo) for a sequence {Sn}n:l in Nk(H) such that Sn · f.~n~•te. By () ... ) , - "'.k "']=1 q.< J 0 nJ·>, where ons=oo and onk ~s v and ( ~~~ q (oo)Eilq (co)# rEil.k q. (Onj) ]Glq (oo) ; [Ol.k q. (Onjl]Eilq (co) =Gl.k q~Onjl 1 k J=l J k s~m J=l J s J=l J ' which is similar to sn. Proceeding exactly as in the first case, we conclude that q (co)Eilq (co) 7 A. 1 k s~m Since the inclusion Nk_ 1 (H) c Nk(H)\N~(H) is trivial, we are don~ 0 PROPOSITION 8.3. The quasinilpotent operator Q € L(H) is a univer sal quasinilpotent (and therefore S(Q)

=N(tr)

if and only if Uoreovei•, Q (HI= (defJ ll

={A< (DQT); o(A) and oe(A) are aonneated and aontain the origin})

5

is not a nilpotent.

{Q

E

={Q ( L[H): is a G6 in

( 8. 2)

L(H):

Q is a universal quasinHpotent}

co

Q is quasinilpotent}\uk=l

-1 TI

[Nk(AlHJ)]

(8. 3)

L(HI, dense in the set Q(H) of alt quasinilpotent operator& -1

PROOF. It is completely apparent that if Q E TI [Nk(A[HI I] for -1 some k ~ 1, then S(Q) is also contained in the (closed) set TI [Nk(A [H) J , so that the condition is necessary. Assume that Q is quasinilpotent and Qk -;. 0 for all k = 1, 2, • • • • By Voiculescu' s theorem (Proposition 4. 21 (iii) ) , if R = p (Q) for some fafrh ful unital *-representation p of C*(Q), then QEilR(oo) E U(Q)-. A fortio~ ri, Q : QEilR (oo) and it is completely apparent that o (R) = {0} and Rk I' s~m 0 for all k = l, 2, . • . • By Corollary 3.35(ii) and Lemma 8.1,

193

QtllR(ao)

~

Otll(q tllO) (ao) =q

(oo)tllq

(ao)

sJ.m ·k -1 ··k for each k, k = 2,3, •••• Hence, by Corollary 8.2 (vi)

Since Q " N (HI (Theorem 5 .1) , we see that S (Q) = N (HI , whence we obtain formulas (8.2) and (8.3). It is completely apparent that Q (H) is dense in Q(H). Namely, u i f Vis the Volterra operator, defined by Vf(x) =fox f(t)dt.inL 2 ([0,1], dx), then Vis a compact ~uasinilpotent and vk ~ 0 for all k=l,2, •••• It readily follows that S(V(ao)) c Q (H) and S(V("")) is dense in Q(H). u By Corollary 1.3(ii), Q(H) is a G6 in L(H). Since n- 1 [Nk(A(H))] is a closed subset of L(H) for each k=l,2, ••• , it follows that Q (H) is u also a G6 • 0

8.1.2 Universal compact quasinilpotent operators Let K " K(H) be a compact quasinilpotent operator such that Kk ~ 0 for all k = 1, 2, • • • • By Lemma 8. 1, K is a universal operator for the class of all compact quasinilpotents. By using the same arguments as in the last part of the previous proof, we obtain the following PROPOSITION 8.4. The compact quasiniLpotent operator K a universaL compact quasiniLpotent (and therefore S (K)

= N (H)- nK (H) = {C " K IHI:

C

E:

L(H) is

is quasiniZpotent} J

if and onLy if K is not a nilpotent. The set of aLL universaL compact quasiniLpotents is a G6 in L(H), dense in the set of aLL compact quasiniLpotent operators.

8.2 Compact perturbations of not nice nilpotents Assume that T" LIHI is an essential nilpotent of order k (i.e., fk = 0, but i'k-l ~ 0) such that fk-l+i'* is not invertible. If IIA -l'JnTWn -lll + 0 (n + co), then II .A -WnTWn -lll + 0 (n + ao) and it follows from Corollaries 1.8 and 7.21 that A is an essential nilpotent of order at most k such that xk-l+A* is not invertible; therefore (8. 4)

This 194

observa~ion

reduces the problem of characterizing S(T)

to

that of characterizing those operators in the second member of (8.4) that can be uniformly approximated by similarities ofT. Since ae(A) {O} for all A in S(T)-, it is not difficult to see (use Corollary 1.6) that a(A) =a(T) and dim HP.1~l =dim H(A1T) for all A E a(T) 1 furthermore, TJH(A1T) s~ 7 AJH(A1A) for each A in a(T) (use Corollaries 1.8 and 2.8). By using these results and Corollary 3.22, the problem of characterizing S(T) can be reduced to consider two cases: a) the case when T is a quasinil9otent, and b) the case when a(T) is the union of {O} and a sequence of normal eigenvalues converging to 0. This will be done in two propositions. PROPOSITION 8.5. Let Q E l(H) and assume that Q is quasinilpotent, ok = 0 and Qk-l 'f o. but Qk-l+Q* is not invertible; then S (Q)

={A

€

Q.(H):

~k

~k-1

A = 0, A

~

+A* is not invertible,

rank Aj s rank Qj s oo-, for alZ j

2

k}.

PROOF. Let R= (0) for some faithful unital *-representation p of C*(Q)1 then we have

F$q (oo)eq (oo) 7 F$q (oo) 7 F$q (oo)$q (oo) 7 F$ (oo) + ·1 k s~m ·k-1 s~m 1 k-1 s~m qk-2 sim + F$q (oo) 7 0 sim ·1 s1m ' where F is any Jordan nilpotent acting on a finite dimensional space such that rank Fj s rank Qj, for all j 2 k. (To see this, apply: 1) Proposition 4.2l(ii)1 2) Lemmas 7.9 and 7.81 3) Lemmas 7.9, 7.8 and 0.11 4) Corollary 8.2(ii)1 5) Corollary 8.2(i), etc.) Assume that A is a quasinilpotent such that Ak=O, Ak-l+A* is not invertible and rank Aj s rank Qj s oo-, for all j 2 k, and let t (1 s t ~t ~t-1 s k) be the exponent such that A = 0, but A 'f 0. If either t > 1 and At-l+A* is not invertible or t =1 (i.e., A is compact), then it follows from Corollaries 7.6 and 7.27 that A can be uniformly approximated by a sequence {An}n:l c N(H) such that An~ Fneq1 (oo)eqt(oo) (for suitably chosen ~perators ~n acting on finite dimensional spaces, such that rank FnJ s rank gJ, for all j 2 k1 n=l,2, ••• ). As in the proof of Proposition 8.3, we conclude that 0 7 A. ~t-1 ~ s~m If t > 1 and A +A* is invertible, then t < k, A is similar to a compact perturbation of ~l (h)eqt(oo) (for some h, 0 s h < t1 use Proposition 7.22). By Corollary 7.6, A can be uniformly approximated by a oo + (oo} · sequence {Bn}n=l c Nt(H)nN(H) such that Bn ~ Gneqt (for suitably chosen operators Gn acting on finite dimensional spaces, such that

195

rank Gn j that

$

rank Qj, for all j

?:

k; n = 1, 2, .•. ) . l•Je conclude as above

Q sim A. Conversely, if Q s!m A, then it follows from our previous obser-

vations (at the beginning of this section) and Corollary 1.8 that A is quasinilpotent, Ak =0 and Ak-l+A* cannot be invertible.

D

PROPOSITION 8.6. LetT" L(H) and assume that Tk=O, Tk-l '/' 0, Tk-l+T* is not invertible and o(T) ={O}u{An}n:l is an infinite set

(where An~ 0 for all n, An~ Am if n ~mandAn+ O, as n + ~J; then S(T)- is the set of all those A" L(H) suah that Ak =0, Ak-l+A* is not invertible, o(A) =o(T), dim H(An;A) =dim H(An;T) and TIHCAn;T) s!m AIH(An;A) for all n =1,2, •••. PROOF. Let p be a faithful unital *-representation of C*(T) and

T$R(~) slm Teg 1 <~l eqk <~>.Let Am ={A 1 ,A 2 , •.• ,Am}. By Corollaries 3.22 and 3.35(ii), Tlet R =p(T). By Propositions 4.2l(ii) and 8.5, T sim

T $[$ ml R], where R =A +lA IJ and J is the jordan form of the nilm n= n n n n n n potent operator TIH(An;T) (n =1,2, .•• ),a(Tm) =a(T)\Am and 11Tm11 < 2 max fiAnl: n > m}. By taking suitable limits, we conclude that Teq <~>eq <~> -t s =q ("'leq (co)$[$ co R ] . 1 k sJ.m 1 k n=l n + (oo) (co) m 00 Clearly, S sim Sm -q 1 egk $[$n=l Rn]$[$n=m+l AnlM ] , where T

-t

sJ.m

n

lMn denotes the identity on a space Mn of dimension equal to dim H (An;T) (n =m+l,m+2, •.• ). Let em+j be a unit vector in Mm+j' j =1,2, ... ,p; then

0

0

A

m+p-1

1

A m+p

where F = e. P V{ }• 1 A .1M m J=' m+J m+j 9 em+j Since An + 0 and [I Rn II $ 21 An I + 0 (n + co) , it is easily seen that there exists a sequence {S~}m:l of operators such that S~ - Sm reor all m=l,2, ••• ) and llq es-s'll+ 0 (m + 00 ) .

P

Hence, S sim qpes

m

~ qpeg 1 ("'les for all p =1,2, . . • . By Lemma 2.5

and Proposition 2.33, S sim Kues, where Ku is a universal compact 196

quasinilpotent. Assume that T sim A. I t follows from our observations at the beginning of this section that:

ae

(1)

(A) =cr (T) ={0}, o(A) =o(T) and

e

dimfi(A ;A)=dimfi(A ;T) foralln=l,2, ••• ; (2) Tlfi(A ;T) ~Aifi(A ;A) · n n ~k -k-l _ n SJ.m n for each n, n = 1, 2, ••• ; (3) A = 0 and A +A* is not invertible. Suppose that A

€

L(fl)

satisfies (1),

(2) and (3) and let B~p 1 (A)

for some faithful unital *-representation p 1 of e*(A). Then Bk~o, Bk-l+B* is not invertible and (by Proposition 8.5) ql (~)Glqk(~) ~ B(=) (=) s1m On the other hand, by Proposition 4.2l(ii), AGlB ~a A. Thus, given e: > 0 there exists W invertible such that llwcq{"">e qk

c~>,w- 1 -B(oo) II< e:/2. By continuity, llt7(ql (=>eqk (oo)+e)w- 1 -a<=>ll < e:,

provided llell <

e:C21~7II-IIw-lll)- 1 •

Let Ia Nl I 1 IB2 N2 I I

A

I•

0

'B N m 1 m

Amm

I • I

.:. _I.:. tAw N~

0

where Nl = fl (A l ; A) and Nm =fi({A 1 , ••• ,A m};A)9H({A1 , ••• ,A m-1};A), and let I Bl Nl All A12 Aln Alm Al,m+l" I I B2 N2 A2n A22 A2m A2,m+l" I I I I B Nm A Amm Am,m+l" mn I m ,am+ Nrn+-1 Am+l,rn+-1- Am+l" Arn+-l,n I em I 0 I I B N A -A nn n I n n

.

.

-- -- - -- --

I

0

--- - - --

I

- -,- -

I A~ N00

I t is easily seen that A -em is a normal compact operator, IIA-emll=max{IAnl:

n > m}-+ 0 (m-+ =)and

0

197

. quas1n1lpotent, . . - k-1 +Om -. 1s . not 1nvert1 . . b le. By where Om 1s 0- k =0 and Om (co) m Proposition 8.5, Ku$qk sim om. A fortiori (use properties (1) and ( 2)) ,

[$ m R ]$K $q (co) ~ Cm. n=l n u k s1m

tr

Choose m large enough to guarantee that II A - em II < e:( 21 IWII ·II ~I) -l: then II A -em II < e:/2 and there exist V1 and V 2 invertible such that V2 W and m (co) (co) co (co) -1 II (Vl$V2)({[$n=l Rn]$Ku$qk }${ql $[$n=m+l Rn]Sqk }) (Vl$V2) - A$B(co)ll = maxfiiv 1 f[en:l Rn)$Ku$qk(co)lv 1 - 1 -AII, IIV2{ql (co)$[$n==+l Rn]$qk (co) }V2-l-B(co)ll}

s maxfllv 1 { [en:l Rn] $Ku$Qk (co) lv 1 -l- emil +II em- All, [IW(ql (co)$qk (co) )lol-l -B(co) II+ e:/2} <

E.

Since {[en:l Rn]$Ku$qk(co)}${q 1 (co)•l•n==+l Rn]eqk (co)} ~ Ku$S and e: can be chosen arbitrarily small, we conclude that S

T sim ' s!m

K $S 7 A$B(co) ~ A. u s1m s1m

0

REMARK 8.7. By Lemma 2.5 and Remark 2.6, . the condition ."TIH(A n :T) sim AIHCAn:A)" can be replaced by "nul (An-A)J ~nul (An-T)J for all j =1,2, ••• ,dim H(An:T)-1".

8.3 Quasinilpotents in the Calkin algebra

8.3.1 General quasinilpotents Now assume that Q E L(H) is an essential nilpotent operator such -k = 0, Q -k-1 '/' 0 and Q -k-1 +Q* - is invertible. Proceeding as in the that Q first part of the proof of Proposition 8.5, we can find R - qk (co) such that Q$R "'a Q • Hence, Q "a QeR sim geq 1 (co) , so that S (0)- ::> S (t;2$ql (co))-. Combining this fact (7.9) and Propositions 7.22 and 8.5, we conclude that if 0 is similar to a compact perturbation of q 1 (h)eqk (co), where k ~ 2 and 0 s h < k, then S(Q)- is the disjoint union of S(Qeqico))-and

+ S + (Q) =S(Q) - nNk(H), 198

(8.5)

which is open and dense in S(Q)-. As a first step toward a eescription of the structure of S+(Q), we shall consider similarity orbits in the Calkin algebra. Let us remark from the beginning that we have b1o kinds of similarities, namely those corresponding to invertible elements in n[G(H) J= {i E G(A(H)): ind W= 0} and those of arbitrary index. Accordingly, we shall write T s-sim ~ A for T, A E A(H) if there exist invertible elements Wn in 1 A(H) such that ind Wn=O for all n=l,2, .•• , and IIA-~nNn-11 ~ 0 (n ~ ~> and, respectively, T ~. ~ if there exist invertible elements w-sun wn € G(A(H)) such that IIA- wnwn-~1 -+ 0 (n ~co}. LEMMA B.S. LetT and Tl be two faithful unital *-homomoPphisms of

L(~k) into A(H); then (i) Tl

(X)

U

Thepe exists a unitaPy element

fop a Zl X

(ii)

{ind W: (iii)

{ind

W:

£

L(~k) .

U€

ThePe exists an integeP h, 0 s h s

A(H) suah that OT(X) = k-~

suah that

WT(qk) =T 1 (qk)~, WE G(A(H))}=h+k:
~h(X) =T 1 (X)W, foP aZl

X

E

L(~k), ~

E

G(A(II))}=h+k~.

PROOF. (i) Let W be a partial isometry (i.e. ~·~ is a projectio~ such that W*W=T(e 1 ee 1 ), WW*=T 1 (e 1 ee 1 ). Then O=}:j~l T 1 (ej®e 1 )~T(e 1 ® ej) is a unitary element of A(H) and th (X)= Tl (X)O for all X in L(~k). (ii) It is clearly sufficient to prove that {ind

V:

V

E

G (A (H)),

V commutes with T (qk)}

=

k?l.

Furthermore, it follows from (i) that T can be replaced by any other faithful unital *-homomorphism of L(~k) into A(Hl, because of the invariance of the index under conjugation by unitary elements of A(H). Thus, we can assume that there exists a unitary operator U:~k®H ~ H such that T (X) = n (U (X®l) U*) . If V commutes with '(qk) , it is easily seen from the matrix form of T(qk) that ~ k-1 j v = n[U{}:j=O qk ®Aj)U*] (Aj E L I H) l k-1 and, since the diagonal part of the triangular operator matrix Lj=O qkj®Aj is equal to l~k®A0 , it follows that l~k®A0 is Fredholm and ind V = ind (l~k®A0 ) = k ( ind AJ :: 0 (mod k) . On the other hand, given m E 1l and A E LIHl a Fredholm operator of index m, the element n[U(l~k®A)U*J of G(A(H)) commutes with ,(qk) and ind U( l~k®A) U* = k ( ind A) km. (iii) Use (i) once again (as in the proof of (ii)). i'Te have to show that

=

19.9

{ind V:

V E: G(AIHI I, V commutes with t{X)

for all X

in case t {X) = rr {U{XelH) U*). Now it is easily seen that

E

L{O:k)} =k?l

v commutes

with

t[L{O:k)] if and only i f V=t{U{la:k&A)U*) and hence, Vis invertible i f

0

and only i f A is Fredholm and ind V = k {ind A) • h k h . .h Let H = H0 EB fBj=l Hj , where h = dim H~ s k-1 {k for j = 1, 2, ••• , k. Let

0 1 0 1

~

2) and

H~J "' H

0

0 .

0

(8.6)

L (H):

£

h 0 1 Hk-1 0 Hh k

then 1HhEBHh

o sh 0 1

g

1

H2

0

Hh 3

0 q {h)fBq (oo)

1

k 0

h

0 1 Hk-l

0 H~ where ~

T2

Sh:H~

H~EBH~

+

is an isometry of corank h. By Lemma 7.23,

T0 ~ T1

~ ••• ~ Tk-l in A(HI; indeed, straightforward computations show

that {ShfBlfBlEB ••• EJl) T0 (ShfBlEBlfB ••• fJl) * = Th' rr {Sh6llEBlEB ••• EBl) is unitary in A(HI and ind{ShfBlEBlEB ••• EJl) =-h, h=O,l,2, ••• ,k-l. It is easily seen that 'h {qk) = Th uniquely determines a faithful unital *-homomorphism from L(a:k) into A(H) for each h, h = 0,1,2, ••• ,k-L However, by Lemma 8.8, there is noVE: G(A(H) I, ind V=n such that Vth {qk) = tr{qk)V, unless h- r- n :: 0 {mod k). Furthermore: ~

~-1

'I 0 {mod k), then tr{qk) is"far"from Vth{qk)V Given k

~

!

If h- r- n

(Corollary 8.11 below)

2, define

+ Nk,h(HI = {T E: L(H):

T is similar to a compact perturbation of Th}

{h=O,l,2, •.• ,k-l) and ~+

Nk,h (A (H))

+ =rr[Nk,h (H) J.

By {7.9), {7.8), Proposition 7.22 and Lemma 8.6, N;(HI

200

OJ~(A(HIIl

is the disjoint union of the sets Nth (H)

(fJth (A (H) ) , resp) , h = 0,1,

2, ••• ,k-l. LEMMA 8. 9. Let C be a C*-aZgebra with identity 1 and Zet ~

C be a faithful unital *-homomorphism. Assume that x k

s = Lj=l T(ejee 1 )x

Then T(qk)s = xs and

II s - 1 II

s kll X -

1: ( qk) II

1::

L {a:k)

Nk(C) and Zet

€

j-1

{ 1-+j I X Ill

k-1

•

PROOF. Indeed, k-1 k j-1 -r(qk)s-[Li=l -r(eieei+l)][Lj=l T(ej9e 1 )x J -

k-1 k j-1- k-1 j Li=lLj=l T ((eieei+ll (ejee 1 llx - Li=l -r (eiee 1 )x = sx,

and, proceeding as in the proof of Proposition 1.10, we have k j-1 k lls-111= IILj=l dejee 1 lx -Lj=l T{ejeejlll t k j-1 k j-1 = llt.j=l -r (ejeel) x - Lj=l -r (ejeel) T (qkl II

k s Lj=2 II

T

(ejeel) [x

j-1

- -r (qk)

k-1 j-1 i sllx-T(qkliiLj=lLi=O llxll LEI4MA 8 .10. Let k ?. 2 and j

(i)

k-1

j

1 II s Lj=l II x -

s Jt!lx--r(qklll (1-+Jixlll

1:

{qk)

k-1

j

II

0

•

then

?l,

If j _ h (mod k), then

{VT 0 ~-l: (ii)

E

j-1

~

E

G(A(H)) and ind V=-j}

=1.1~,h(A(H)).

If j t h (mod k), then

{V'.i' 0 v- 1 :

V

E

G(A(H)) and ind

V=-j}n~~,h(A(H)) =~.

(iii) The subsets {N~,h(A(H)) }~=~are open in N~(A(H)). Indeed, these subsets are the aomponents of ~~(A(H)). PROOF. By our previous observations, i t only remains to show that

-+

Nk,h(A(H))

-+

is connected and open in Nk(A(H)). The connectedness is an

immediate consequence of (i),

(ii) and the fact that {V

€

G(A(H)): ind

V=-h} is connected for each h, h=O,l,2, ••• ,k-l. ""91 N-1 W W Let -Xn E -+ Nk(A(H)) and -X=v·.~: 0 v , ind V=-h, be such that ll.r·- ..... nll ... 0 (n ... oo). Then we have lim(n ... oo} IITo-v-lxnVII=O and therefore, by 1 x ~s Lemma 8. 9 I there exist elements s E G (A (,.,) ) such that To = n n n n (for all n = 1,2, ••• ) and IISn- 111-+ 0 (n ... oo). Hence, for n large

s

v-

enough, sn is invertible, ind sn = 0, and xn = (VSn) 'o € ~:. h(A(H) l. n ., + Therefore, Nk,h(A(H))

so that x

(~n) -1 +

I

is open in ~k(A(H)).

0 201

From Lemma 8.10, Corollary 7.21 and comments preceeding it, we mediately obtain the following

i~­

"-1 • ind V = -h (0 ~ h S k-1), Y E COROLLARY 8.ll. {i) If -X = ;-.~;: v·.L· 0 v -+ Nk(A(Hll and jjX-Yjj < ~(IIVII·IIV-ljj)-1, then Y e Nk, h ( A ( Hl l . riiJ If i = VT 0v-l € 'N~,h lA IHJ 1. Y = \iT 0w-l E Nk'+- ,r (A ( H l J and h- r ~ 0 (mod It) , then

IIX-YII

~ ~[(IIVII·IIV- 1 1ll- 1 +(11WII·IIW- 1 1lll·

PROPOSITION 8.12. Let Q be a quasini~potent element of A(HJ and € A(H); then (i) If Q E Nk(A(H))\N~(A(H)), and Qk-l I 0, then

let A

Q· s-sl.m +. A <= > 'Q w-sl.m +. A <= > A € Nk (A (HJ J\ Nk+ (A ( HJ I • (ii)

If

Q E N~,h(A(Hl

l for some k ~ 2, and h, 0 s h s k-1, then k-1 -+ Q s-;im A<=> A E Nk(A(H))\uj=O,jlh Nk,j(A(H) J

and

- w-;im -A <=> A Q

(iii)

If

Q;

E

Nk (A (H) J.

uk:l Nk(A(H) ), then

Q s-sl.m +. A<=> 'Q w-s1.m .... A<=> A PROOF. (i) We have Q s-sim A=> Q w-sim A=> A

E

€

I~(A(HJ J-.

-+ Nk(A(H))\Nk(A(H)J

(The last implication follows from the fact that the non-invertibility of ~-l+X* for X E Nk (A (H) J) is a similarity invariant: see Lemma 7.20.) -+ On the other hand, if -A E Nk(A(H))\Nk(A(H)), then (Theorem 7.2) there exists Q0 , A0 € Q.(HJ such that Q0k=A 0k=O, 1f(Q 0 ) =Q, 1dA0 l =Ji.. By Proposition 8.5, Q0 simA0 .+A fortiori, Q s-sim A. (ii) Assume that Q e ~k,h(A(H) J, 0 s h s k-1. By Lemma 8.10,

fWQW- 1 :

w"

G(A(HJJ, ind W=O} =N~,h(A(HJJ

and G(A(HJJ} =S(Q) =N~IAIHJJ. On the other hand, if -A" -+ Nk r(A(H)), 0 s r s k-1, and r I h,then it readily follows from Lemma 8.101 (or Corollary 8,U(ii)) that ~~~~- 1 AII is bounded away from zero for all We G(I.(HJ) such thilt ind W = 0. Hence, 'Q s-tlJ. ...4-.m A. It is clear that

{VQV- 1 : v

0

€

s--;im A => Q w-Sim A=> A

E

Nk(A(H) l

and, by our previous observations and Theorem 7.2, it readily follows 202

that Q s-s~ ~.m A for each At: NkiAIH))\Nk+IAIHl). (iii) It is completely apparent (from Theorem 5.34) that

Q

+.

s-s~m

A=>

Q

~.

w-s~m

A=> At: NIAIHl)-.

Conversely, if A e N(A(H)) then (by Proposition 3.45) there exists Q0 , A0 e LIHJ such that rr(Q 0 ) =Q, rr(A 0 ) =A, o0 is a quasinilpo- • It is easily seen that -Q k ~ 0 for all k tent and o (A0 ) = o e (A 0 ) = o (A) 0 = 1,2, .•• and (by Theorem 5.1) that A 0 e NIH)-. By Proposition 8.3,

o0

sim A 0 • A fortiori, Q s-~im A.

0

COROLLARY 8.13. (i) For eaah k ~ 2, Nk(A(H)) is the disjoint un-+ -+ -+ -+ ion of the sets Nk(A(H) )\Nk(A(H) ), Nk,O(A(H)), Nk,l(A(H) ), ... , Nk,k-l (A(H)). (ii) N~,h(A(H)) = {WThW- 1 : We G(A(H)), ind W = 0}, for eaah h 0,1,2, .•• ,k-l. (iii) N~(A(H)) =S(Th) for aZZ h=O,l,2, ••. ,k-l. -+ - -+ -+ - -+ (ivJ Nk(A(H)) \Nk(A(H)) = Nk h(A(H)) \Nk h(A(H)) Nk(A(H)) \ + , , Nk(A(H)) for eaah h, h=O,l,2, •.• ,k-l. As in Chapter II, let [T]s = (R. e A(H): T s-~im Rand R s-sim T} and [T]w={R e A(H): S(R)-,.,S(T)-}; [A]s < [T]s means that T s-~im A and [A] w < [T] w means that T w-s1m ~. A, respectively. These two relations are partial orders in the corresponding quotient sets obtained from A(H) by using the equivalence relation #s and #w' respectively. If Q(A(H)) denotes the set of all quasinilpotent elements of A(H) ,then (Q(A(H))/#s,<) is a lattice and (Q(A(H))/#w,<) is a chain. Indeed, it is not difficult to see (by using Proposition 8.12) that their structures are the ones depicted in Diagram (8.7) (below), where Qu denotes an arbitrary universal quasinilpotent operator.

203

[ ij 1

w

OIAGRAM ( 8 • 7)

8.3.2 Nice elements of the Calkin algebra In the case when T is similar to a compact perturbation of a nice Jordan operator with more than one point in the essential spectrum, the analysis of S(T) present some subtlety that cannot be detected with the sole help of Lemma 7.1. The phenomenon will be better explained with an analog of Lemma 8.8 for this case. The proofs of Lemmas 8.14, 204

8.15 and 8.16 below follow the same lines as those of Lemmas 8.8, 8.9 and 8.10, respectively, and will be omitted. LEl~ 8.14. Let t and r 1 be two faithful unital *-homomorphisms of L(~kl)$L(~k2)$ ••• $L(~km) into AIH); then (i) There exists a unitary element DE A(H) such that 6r(X) = m k· tl (X)fi fo1' al.Z X< $j=l L(~ J). (ii) There exists an integer h, 0 s h s k 0 -l, whe1'e

k 0 = GCD{k 1 ,k 2 , ••• ,km}, such that if Al' .h 2 , ••. , Am a1'e m distinct complex numbe1's, then

{ind

~'J:

(iii)

{ind W:

m m +"~ 1 wr(Ol._ 1 [A.+qk ]} =r 1 ($._ 1 [A.+qk J}w, W, G A(H))}=h+k 0 2Z.

..,.

]-

]

j

]-

'j

J

The1'e exists an intege1' h, 0 s h s k 0 -l, such that

Wr(X) =r 1 (X)W, fo1' aZZ X< $j:lL(~kj), W
LEMMA 8.15. Let C be a C*-algeb1'a with identity 1 and let r:$j:l

L(~kj) ~ C be a faithful unital *-homomo1'phism. Assume that Al' A2 , .•• aPe m distinct complex numbe1's and let~:" C with minimal polyp, p(A) =rrj:l
nomia~

J

where Cp is a constant depending only on p.

LEMMA a.16. Let A1 ,x 2 , ••• ,Am bg m distinet eomplex numbersandlet Th=ql

(h)

m

(co}

$[$j=l(.>.j+qk.

)] "L(H);

J

where 0 s h s k -1, k 0 =GCD{k 1 ,k 2 , •.• ,km}· 0 Then S(T 0 ) has exactly k 0 components:

S(Th)

={WT 0

w- 1 :

W<

G(A(H)), indW_h (modk)},

h = 0,1,2, .•• ,k-1.

8.4 Compact perturbations of nice Jordan operators From our observations (at the beginning of Section 8.3.1) and Proposition 8.12(ii), we obtain the following COROLLARY 8.17. (i) If Q E L(H) is simila1' to a compact perturbation of q 1 (h)$qk(oo)' where k 2 2 and 0 s h s k-1, then S(~)- is the disjoint union of S(q 1 (oo)$Q)- and S+(Q) (defined by (8.5)). 205

s+(Q) is an open dense subset of SCQ). (iii) s+(Q) c N~,h(H) and S(Q)-nN~,j(H) =~foP aZZ j ~ h, 0 s j (iiJ

s k-1.

Let Q be as above. If Q is quasinilpotent, then S(q 1 (=)~Q)- is completely described by Proposition 8.5. If cr(Q) = {O}u{A 1 ,A 2 , .•. ,Am} is a finite set, then Q - Q0 $F, where Q0 is a quasinilpotent and F acts on a finite dimensional space; in this case S(q 1 (=)$Q)- can be easily described by using Lemma 7.1, Proposition 8.5 and Theorem 2.1. If cr(Q) ={O}u{A } = 1 , where A F 0 for all n, A ~A if n ~ m, and A n n= n n m n + 0 (n + =), then S(q 1 (=)$Q)- also admits a simple description, given by Proposition 8.6. On the other hand, it follows from Corollary 8.17 that S+(Q) = + + + S(Q) nN 1{.(H) =S(Q) nNk,h(H). Since every element of Nk,h(H) is similaP to some operator of the form -

q 1 (h)$qk(=)+C, C

€

(8.8)

K(H),

we can directly assume that Q=q 1 (h)$qk(=)+K, k ~ 2, 0 :s h s k-1, K

€

K(H).

(8.9)

Uow it is clear that the structure of S(Q) will be completely determined if we can specify which operators of the form (8.8) belong to S(Q)-. Furthermore, by our previous observations, we only have to analyze the case when Q is quasinilpotent and the case when 0 is not isolated in cr(Q). There are only partial answers to these problems. However, these partial results are sufficient to exhibit the complexity of the general problem. The case when 0 is not an isolated point of cr(Q) will be considered in Section 8.45. Sections 8.41 through 8.44 will be devoted to the case when Q is a quasinilpotent. A conjecture will be presented concerning the classification of the closures of sim ilarity orbits of these operators.

8.4.1 Nice Jordan nilpotents LEfil-11\ 8.18. Let {Qn}n:l be a sequence of nice nilpotent opePatoPs, Q - qk(=)$F and let. Q- qk(=)$F, whePe F c l(~rn), F E L(~r) aPe n n n

JoPdan nilpotents. Then the following aPe equivalent. (i) There exists operatoPs Wn in G(H) such that

IIWnQnWn-l_ 206

oil+

0 (n

+ =);

(ii) n

~

There exists a natural number p such that Qn s!m Q for all

p;

(iii)

an and Bn (n

There exist a natural number p and non-negative integers ~ p)

such that

(a ) (Bn) kB 0 +r = ka 0 +rn and qk n $F n sim qk $F. PROOF. (iii) => (ii) => (i) are trivial implications. (i) => (iii) If Q ~ q <~)$F let H (n) and H (n) denote the n k n' 1 2 subspaces corresponding to qk(~) and F, respectively, and let P. (n) denote the orthogonal projection of H ~nto H • ( n) , j = 1, 2. Simila~ly, let H = H1 $H 2 , where H1 is the space of qk(~? and H2 is the space of~ and let P. be the orthogonal projection of H onto H . , j = 1, 2. Let Q~ w Q w -l,Jx = H1 eoH 1 and X = {x £X: \'1 P 2 (n)w -lx,;"JO}. nnn n n n k-1 k-1 Define Yn = Xn VQXn V •.. VQ Xn and Zn = Xn VQ~Xn V ••. V Q~ Xn

.

Since the spaces X , QX , ... , Qk-lX n

n

n

are pairwise orthogonal and

QJ IX is isometric (for j = 0,1,2, .•• ,k-1), the algebraic sum X +Q'X n k-l n n n + ... +Q' Xn will be closed and direct for all n large enough; moreo-n ver,

liP y

- P Z II -> () ( n -> ~) . n n (n) -1 since W0 P 2 wn (Xn) =

{

1 0}, it follows that wn- (Xn)

c

( ) H1 n ; so

that Q'kX = {0} and therefore z is invariant under Q' and, for n n n n n large enough, Q~IZn- qk (~). Furthermore, since Yn has finite codimension, z also has finite codimcnsion. Since Q' and Q are similar via ( ) n 1 n n W and Wn- (Xn) c H1 n , we can use Lemmas 7.9 and 7.8 in order to f~nd an invariant subspace Rn c H (n) of Qn such that H (n) = R +w -l 1

1

n

n

(Xn)

(dim Rn < oo) . Since qk(oo)$(Qn1Rn)- qk(oo), it is easily seen that there exists a non-negative integer an such that QniRn- qk(an). On the other hand, Yn reduces Q and therefore there exists a non-negative integer Bn such that (Bn) Q 1 HeY n = P HeY Q 1 HeY n ~ qk $F. n

Since IIPz -Py 11 -> 0 (n

n

+

n

~) ,

for all n large enough we can find

unitary operators Un " L(H) such that I!Un -111-> 0 and U~PZ u = PY • n n n Hence, we have: (8.10) But

PHez n

Q~IHezn-

(a )

0 n1Rn$H2 (n) -

qk

n $Fn

Hence, (8.10) implies the existence (for n large enough) of invcrtiblc operators na +r ne +r S =~ n n _. ~ n n 207

such that

-1

(an)

lim(n ..,.

oo)

liS n (qk

EDF r- ) Sn

(a )

- qk

n EDFII = 0.

Now it is easy to derive from this relation that, given j then

(a )

rank(qk for all n

~

~

1,

.

n EDF)J

(8.11)

n0 (j).

Since (qk 16 nlfDFn)m= 0 for some m ~ k, it follows that {8 .11) hcil.ds for all j = 1, 2, ••• , and for all n 2: p. If n ~ p, it follows from Lemma 2.5 that PROPOSITION 8.19. Let T

€

N~(H) be simiZar to a niae Jordan nil-

potent: -

(oo)

1

(T .)

m

J J}W-, m ~ k, Lj:l,jfk jTj=dT < oo. + Then S+(T) is the set of aZZ A E Nk(H) suah that (aJ.) -1 m (oo) m ]} V , L . _1 . ~k ja . = d < oo ( 8. 12) A - V{qk fD[fDj=l,jfk q J· ]- , ] r J A T - W{qk

ED[EDj=l,jfk qj

and there exist non-negative integers Tk' ak suah that

l

min{Tk,ak} = 0, kTk+dT=kak+dA, and m

(a . ) r

rank[EDj=l qj

(T . )

s rank[ej:l qj

J J

r

(8

.13)

J J ,

fol' aZl. r=1,2, ..• ,m-l. Furthermore, there exists a aonstant ek S (T)

>

,m

0 suah that i f

B

£

and (oo)

liB- qk (oo)

m

then B- qk $[$._ 1 ·~k q. ]- ,]r J dense subset of S(T)-.

m

ED[$._1 ·~k q.J ]- , ]r

(T ·)

J ] -

(T

j)

Ill

< Bk ,m•

T. In partiauZar, S(T) is an open

PROOF. It readily follows from Lemma 2.5 that if A satisfies (8. 12) and there exist Tk' ak ~ 0 such that (8.13) hold, then m

ej=l qj

(T.)

J

...

m

· sim ej=l qj

(aj)

(8.14)

and, a fortiori, T s1.m ~ A. _ 1 Conversely, if IIWnTWn ·-All-+ 0 (n-+ oo) for a suitable sequence 00 {Wn} n=l in G (1-1) , then i t follows from Lemma 8 .15 (\..ri th Qn :: T and Q = A) that there exist non-negative integers Tk' ak such that kTk +dT = kak +dA and (8.14) holds. Since, by Lemma 2.5, this is equivalent to the validity of the inequalities m m (a.) r (T .) r rankrej=l qj J J s rank[EDj=l qj J J, r=l,2, •.• ,m-l, it readily follows that Tk' ak can be chosen so that min{Tk'ak} = 0. 208

Assume that {Bn}n':l c S(T)- and IIT-Bnll-+ 0 (n-+ co). Since s+(T) is open and dense in S(T)- (Corollary 8.17(ii)), B must be similar to n a compact perturbation of T for all n large enough. Furthermore, since k k rank Bn s rank T for all n and rank Tk is finite, it follows from Corollary 7.24 that Bn is similar to a nice Jordan nilpotent: _

B -V {qk n

n

(oo)

•

m J- , Jr

ID[ID._l '4k q.

J

((3

-1

.)

n] ]}V

n

By Lemma 8.18, there exist a tegers Bnk' tnk (n

~

p) such that (Bnk)

rank{qk

m ID[IDj=l,j"lk

for all r = 1,2, .•• ,m-1, and for all n ~ p. But, on the other hand, the first part of the proof indicates that the reverse inequalities also hold in (8.15), i.e., these inequa! ities are actually equaZities. It readily follows that if n ~ p, tl1en 13 .=t. for all j=l,2, ••• ,k-l,k+l,k+2, ••• ,m. I.e., B - T for alln~p. nJ J n From this result and the proof of Lemma 8.18 we deduce the existence of constants Bk,m > 0, depending only on k and m (m ~ k), such that the intersection of the open ball (in L(H)) of radius Bk about q 1_ ("")e[e.:1 . 4 k q. (Tj) J with S(T)- only contains points of S(TJ. Now ' ] 'Jr J (oo) m (T •) -1 it is easily seen that i f R=S{qk ID[e._ 1 . 4 k q. J ]}S - T, then ]- r]r ] 1 -1 the intersection of the open unit ball of radius Bk ,m about R with S(T) is also included in S(T). Hence, S(T) is an open (and obviously dense) subset of S(T)-. D From Proposition 8.19 and Theorem 2.7, we obtain the following COROLLARY 8.20. FoP each k ({T

E

N~,h(H):

~

2 and h, 0 s h s k-1.

T is a nice JoPdan niZpotent}/#,<)

is a Zattice.

PROOF. I f q.

(a.)

J.

J J

gat~ve

+

Nk h!Hl are nice Jordan nilpotents, then there exist non-ne·' htha t 'l.j=l m .JTj = , m .]aj = d • ~ntegers tk' ak sue l.j=l E

m (8·) I f B =e. 1 q. J , J= J

c =e. m1 q. (Y·) J

d E N (a: ) are the Jordan nilpoJ= J tents determined by Theorem 2.7 such that m (t .) m· (a·) m (t ·) m (aJ·) [B] = [IDj=l qj J Jvrej=l qj J ],[C)= [IDj=l qj J ]A[IDj=l qj ],

then it readily follows from Proposition 8.17 that [T]v[A] = [qk (oo)IDBl and rTJA[A] = [qk (oo)IDC].

D

209

We shall see later that this lattice is not closed under "sup" or "inf" (for denumerable subsets). From this result, Propositions 8.3, 8.4 and 8.5, and Corollary 8.1~ we obtain COROLLARY 8 • 21 • ~ + = ([Q.(H) \uk= 2 Nk (H)Ju(uk= 2 {T

€

+ Nk (HI:

T is a niae Jordan nUpotent}/i,<)

is a Zattiae.

This lattice contains the infimum of each of its nonempty subsets, but it is not closed under "sup•. CONJECTURE 8.22. (Q.(H)/i,<) is a lattice. \'1e shall say that a similarity orbit S(T) is ZoaaZZy alosed i f (1) there exists £ > 0 such that the intersection of the alosed ball B(T,£) (in L(H)) of radius£ about T with S(T) is closed in L(H). This notion admits several equivalent formulations; namely: (2) Given A= ~JTW-l E S (T) there exists £(A) > 0 such that 8 (A,£ (A)) nS (T) is closed in L(H) (Take £(A) =£<11WII·IIW- 1 11J- 1 .); (3) There exists£> 0 such that B(T,£)nS(T) c S(T); (4) Given A.: S(T) theL·e exists £(A) >0 such th_at B (A,£ (A)) nS (T) c S (T) ; ( !:i) S (T) is an open dense subset of its closure. (Take 0 < £ < ek to show that (5) => (1). The converse ,m implication is obvious.)

COROLLARY 8.23. (i)

If Tis a niae Jordan operator, then S(T) is

ZoaaZZy closed. (ii)

If Tis a niae Jordan operator, then S(') is ZoaaZZy closed

in A (H).

PROOF. (i)

By using the same arguments as in the proof of Lemma

7.1, we can reduce the problem to the case when o (T) = { 0} • If rank T

is finite (T acting on a finite or an infinite dimensional space),then the result follows from Lemma 2.5, Corollary 2.8 and Proposition 1.12 ( i) • If T is a nice Jordan nilpotent and rank T = =, then it follows from Proposition 8.19. ( ii) This follows from Corollaries 8.11 and 8 .13. 0

L(H):

PROPOSITION 8.24. (i) If S (T) is ZocaZZy closed, then [T] ={A AiT} coincides with S(T). (ii) If S (T) is Z.oaaZZy aZ.osed, then [T J = S (T).

PROOF. (i) all T € L (H). 210

It is completely apparent that S(T)

c

[TJ

c

€

S(T)- for

If A# T, then there exists a sequence {An} n":.l inS (A) such that IIAn- Til+ 0 (n + "") and An € S (T)- for all n= 1,2, •••• If S (T) is locally closed, it readily follows that An - T for all n large enough. A fortiori, A- An-T (for all n large enough). Therefore, A € S(T). The second statement follows by the same argument. 0

8.4.2 Nilpotents of order 2

If T € N 2 (H), then either Tis nice and similar to q 1 (j)eq 2 ("") for some j, 0 s j < oo, or T # q 1 ( "") 6lq 2 ("") , or T is compact. The structure of the poset (N 2 (H)/i,<) is the one given by the following diagram: [q

(oo)]

2~ [q

<

1

>eq <'"'>J [q' (4)6lq (co)]

1

~

[q

(6)6lg (oo)

1~

J

(oo)t (3)]

[q

I2

1

[q ("") Qlq (2)]

I

1

2

[q (oo)elq ]

1

I

2

[0 J

DIAGRAM (8.16)

211

8.4.3 Quasinilpotent perturbations The next lemma provides a device to construct quasinilpotent ope~ ators T such that Tk = 0 and Tk-l+T* is invertible (for some k l!: 2). LE!mA 8.25. Let Tn

operators such that T n +l

+ Nk(H) (k

E

T

7

s~m

n

Wn E G(H) and a quasinitpotent T (n->- oo) and {Tn}n:l c S(T)-.

:2:

2; n=l,2, ... ; be quasinilpotent

for aZZ n= 1,2, •.•. Then there exists + _1 E Nk(H) such that IIWnTnWn -Til->- 0

PROOF. Let E > 0 be such that Ak-l+A* is invertible for any A € (n) n-1 L(H) with II T 1 -A II :s; E. We shall recurrently construct {Sn,{Rj }j=l' e: } 00 1 , where S , R. (n) are invertible operators and e; > 0, such that n n= n J n s 1 = 1, e: 1 = e:/2 and for all n > 1, we have: l) e:n < e:n-1 12 ; 2) liSTS - 1 -s 1 T 1 s - 11 11 < £ 1 /2; n n n n- n- nn3) IIX-S T S -lll < E => IIR.(n)XR.(n)- 1 -T.II < 1/n (j=l,2, .• ,n-l). nnn n J J J This will be done as follows. Since Tn s!m Tn-l' it is easily seen that we can find S € G (H) satisfying 2) (n = 2, 3, .•. ) . Next we choose R. (n) such that n J -1 IIR.(n)s T S -lR.(n) -T.II < 1/n J nnn J J ' j = 1, 2, ••. ,n-1. Clearly (by continuity), we can find e:n such that 1) and 3) are satisfied. -1 00 + Then {S T S } 1 is a Cauchy sequence in Nk(H); let T denote n n n n= its limit. Since II T- T1 11 < Ln:l £n :s; e:, i t follows that Tk-l+'l'* is in-

vertible. It is also clear that Tk=o and (by corollary 1.2(ii)) o(T) -1

= {0}; moreover, IIT-S T S II< L· e;./2 < £ so that (by using 3) n n n J=n J n for j < n) , we have IIR.(n)TR.(n)-l- T.ll < 1/n. J J J Hence T s!m Tj for all j = 1,2, •.. 0 00

Some extra notation will be needed. If {Tn}n:l is a sequence of quasinilpotent operators in N;.,h(H)

(k ~ 2, 0 s h s k-1) such that

Tn+l s!m Tn and T sim Tn for all n = 1, 2, .•• , and there exists {Wn} n:l c G(H) such that IIW T W -1_ Til-+ 0 (n-+ oo), we shall write: n n n

T

t

nsim

T

The following result summarizes the most immediate 9roperties of II

t

II

sim PROPOSITION 8.26. (i)

212

Given a quasini Zpoten t

R

exist a sequence of nice JoPdan niLpotents {Tn}n:l in N~,h(H) and a se quence {Wn}n:l in G(/1) sut:!h that IIWnTnWn-l_ Rll-.. 0 (n-> co) and Tn+l s !m Tn for a L L n = 1 , 2 , • • • • (ii) For every sequence

sim Tn+l sim Tn sim

sim

T2

s!m Tl

of quasinilpotents in N~,h(H), thePe exists a quasiniLpotent (H) such that T t T. nsim (iii) If T t T, then nsim S(T)- = [u"" S(Tn)J-. (8.17) n=l (iv) If T t T for a sequence {T } : 1 of quasiniZpotents in Nk+ h ns~m n n, (H) and Tn s~m f. Tn+l foP infinitely many n's, then S(T) is not LocaLLy elosed. This hoLds, in paPticulaP, if Tn is simiLar to a nice Jordan nilpotent foP alt n, but T is not similar to a nice Jordan nilpotent. t T and A sim T foP all n = 1,2, .• • , then A sim T. nsim n PROOF. (ii) is an immediate consequence of Lemma 8.25 and (iii), (v) and (vi) follow trivially from the definition of " t ". sim (i) From Corollary 7.6, Proposition 7.22 and Corollary 7.24, there exists a sequence {Rn}n:l c N~,h(H) of operators similar to n~e Jordan nilpotents such that II R- Rn II -> 0 (n -> co) • Define T1 = R1 . By Corollary 8. 20, there exists a nice Jordan nil·· potent Tn such that [TnJ=[R 1 JvLR 2 lv ... v[Rnl (n=2,3, ... ). It is easi-

(v)

If T

ly seen that Tn+l si:m Tn si:m Rn (n=l,2, •.. ), whence the result follows. (iv) Observe that if Tn sim Tn+l for infinitely many n's, then T cannot be similar to any of the Tn's. ~his is true, in particular, if Tn is similar to a nice Jordan nilpotent, but T does not have that form. Clearly, S(T) cannot be locally closed.

0

REMARK 8.27. Observe that Proposition 8.26(i) and (ii) guarantee that, if R ~ N~,hiHI is a quasinilpotent, then there exists a sequence {T } col of nice Jordan operators in Nk+,hiHI satisfying T t T for some nsim n n= quasinilpotent T in N~,h(H) such that T sim R. However, this proposition does not guarantee that the Tn 's can be chosen so that T lt R, i.e., T t R. This is, precisely, the main gap to be filled in order to comnsim pletely determine the closures of these similarity orbits. (

( (l • )

)

co ~r~ m q J J € N~,h(H) is similar to a nice - qk "'"'j=l,j'/:k j Jordan nilpotent, then F = 61. m _ 1 . ~k qj (aj) acts on a space of finite di

If J

)-

r)r

213

mension equal to L.~1 ·~k ja. = rk+h for some r ~ 0. Define the number rn ]- r ] r J Stk'(T), the m-th stable kernel ofT, by Stkm(T) =nul ~-nul(qk(r)E&q 1 (h))m (m=l,2, .•• ).(8.18) 8.1

The following assertions are easy consequences of Lemmas 2.5 and and Corollary 7.24:

a) Stkm(T) does not depend on the particular operator qk(®)E&F similar to T. Hence, it is a well-defined similarity invariant; b) If T1 , T2 € N~,h(/1) are similar to nice Jordan operators,then T1 s!m T2 i f and only i f Stkm(T 1 ) s Stkm(T 2 ) for all m = 1,2, .•• ; c) If T, Tn € N~,h(/1) are similar to nice Jordan nilpotents and IIT-Tnll + 0 (n +®),then lim sup(n + ®) Stkm(T) s Stkm(T), for all m=l,2, ••. ; n

Now we shall extend the definition of m-th stable kernel for an arbitrary quasinilpotent T in N~,h(/1). Stkm(T) = inf (sup{Stkm(T'): £

T'

E

N~,h(H) is similar to a

> 0

nice Jordan nilpotent, II T - T 'II

( s .19)

< £}) •

By c), it is easily seen that if Tis actually similar to a nice Jordan nilpotent, then the new definition (8.19) coincides with the previous one (8.13). It is also clear that this extended m-th stable kernel is a similarity invariant. + PROPOSITION 8.28. Let T, R, Tn• Rn € Nk,h(H) be quasinilpotent operators and assume that Tn• Rn are similar to nice Jordan niZpotents for all n=l,2, ••• ; then (i) If IIT-Tnll + 0 (n + ®), then

lim sup(n + ®) Stkm(Tn) s Stkm(T), for aZZ m= 1,2, .•.• (ii) If T sTm R, then Stkm(T) s Stkm(R) for all m=l,2, . . . • (iii) If T t T, then Stkm(T) = inf Stkm(T ) , for aZZ m = 1, 2, ..• nsim n n (iv) If T t T and R + R, then T 7 R if and only if Stkm(T) s nsim nsim s~m Stkm(R) for a'll m = 1, 2, . • . • PROOF. (i) follows immediately from c) and (ii) is a consequence of ( i) • (iii) Since T sim Tn for all n, it follows that Stkm(T) s Stkm (Tn) (n = 1,2, ... ; use (ii)). Since T is the limit of a sequence of similarities of the Tn's, we have lim sup(n + ®) Stkm(Tn) s Stkm(T) (m= 1,2, ••• ) and Stkm(Tnl is non-increasing because Tn+l s!m Tn for each n = 1, 2, •••. 214

Hence, Stkm(T) = inf Stkm(Tn) n

(iv)

(m= 1,2, .•. ) •

If T sim R, it follows from {ii) tha~ Stkm(T) ~ S~km(R) {m

1, 2, ••. ) .

Conversely, if Stkm{T) s Stkm(R) for all m=l,2, ••• , and R

n

is a

nil~otent of order pn' then we can use (iii) to see that Stkm(R) s

~::n:R~) 5 ~::(~::,;~~) 1a:dmt: ::~ws:::: ::e:: :x~:~:a:~n:u:: ::::rs::: this guarantees that Tsn s!m Rn and therefore T

s!~:~

••. ). Since R t R, it readily follows that T s!m R. nsim

Rn {for all n = 1, 2,

0

It follows from Proposition 8.28 that the conditions Stkm(T) s Stkm {R) {m = 1, 2, .•• ) are necessary to have T -t R and, in the case s1m ~ when T t T and Rn : R for suitable sequences {Tn}n:l and {Rn}n=l of nsim s1m nice Jordan nilpotents, these conditions are also sufficient. So it is natural to raise the following + CONJECTURE 8.29. If T, R e Nk,h(H) are quasinilpotent operators and Stkm (T) s Stkm {R) for all m = 1, 2, .•• , then T 7 R. Moreover, given "' + s1m T as above there exists a sequence {Tn}n=l c Nk,h(lf) of nice Jordan operators such that T t T. nsim

An affirmative answer {together with the previous results) could be regarded as a complete solution to the problem of characterizing the closures of the similarity orbits of quasinilpotent operators. By Proposition 8.26, an affirmative answer to this conjecture would also provide an affirmative answer to Conjecture 8.22; Indeed, if {A } ~l m m= is a denumerable family of quasinilpotents inN~ h(H) and T t A , m= , mnsim m 1,2, •.• , then we can define Tn be [Tn] = V[Tij 1l:si,jsn 1 then Tn+l s!m T for all n = 1, 2, •.• , and T t T e Nk+ , h ( Hl , where [ T J = Vm[Am] • n nsim Another consequence would be that S{T) {T as above) cannot be locally closed unless T is similar to a nice Jordan nilpotent.

+ 8.4.4 Universal operators in Nk,h(H)

LetT e N:,h(H) be a quasinilpotent.operator. If T is not a nilpo t tent, then it is easily seen that rank TJ = .,._ for all j~k.IfT=O, Tt-l;60, rank Tj=co- for j=k,k+l, ••• ,r-1 and rank Tj is finite for r t :s j :s t-1, then it follows from Lemmas 7.9 and 7.8 that T - Re[ej=r+l 215

q.<<jl], where d=}:.t +1 jT. < ao, Rr=O and rank Rj=co- for j=k,k+1, J J=r J k+2, •.. ,r-1; moreover, by (2.2), . 1 . '+1 r.=rank TJ- -2rank TJ+rank TJ :2:0, for j=r+l,r+2, ••• ,t. J

(A finite sequence of finite ranks of operators will be called compat£ b~e if the above conditions are fulfilled.) It is also clear that R € Nk+ , h'( fl 0 l , where fl 0 is a subspace of codimension d and h', 0 s h' s k-1, satisfies d+h' h (mod k). The following results provide some insight about the structure of

=

+ the poset (Q(fl)nNk,h(fl)/#,<) • PROPOSITION 8.30. (i)

Given k :2: 2, h (0 s h s k-lJ, r

t-1

:2:

a compatible finite sequence {mj}j=r of ~anks, the~e exists Tu (fl), Tt = 0 such that

k+l and + E Nk,h

u

k-1 (8.20) r-1 u for s j s t-1 J + ~1 and Tu : s a universal o~e~~ior for the class Nk,h(fl;r;{mj}j=r) ={A E Nt(H) nNk,h(H): {rank AJ} . . satisfies (8.20) }. J=.L + t-1 (ii) Besides the maximal element [Tu], (Nk,h(H;r;{mj}j=r)/#,<) ao,

rank Tj

contains a chain equa~

{[Tn,l]~€~

l

ao-

m.:

for 1 s j for k

$

$

j

$

r

o~de~-isomo~phic

to~.

to [Tu].

whose

sup~emum

is

+ t-1 k+2, then (Nk,h(H;r;{mj}j=r)/#,<) contains at least two "parallel" chains {[Tn,lJ}n€~ and {[Tn, 2 J}ndZ o~dero­ isomo~phic to ~.The supremum of each chain is [Tu], Tm,l s1m Tn, 2 fo~ any pair (m,n) and T 2 ~ T 1 if and only if m :2: n. m, Sl.m n, + t-1 (This partial structure of (Nk,h(H;r;{mj}j=r)/#,<), for the case when k > 2 and r :2: k+2 is depicted in Diagram (8.21) .) (iii)

If k > 2 and r

:2:

DIAGRAM {8.21) We shall need the following auxiliary result. 216

LEl4MA 8.31. (i)

If K is aompaat, k

~

2, 1 s p s k, and

0 1

Hl

0 0 1 0 0 0

Qk (K;p)

H2

.

H3

0

0 0 0

0 1

Hp-1

K 0 0

0 0 1

uP

0 0 0 1

Hp+l

0 0 0

.

0 0 0 0 0 0

.0 .0

0 0 0 0

..0 ..0

(8.22)

1 Hk-1 0 Hk

... c. then Qk (K; p) sim Qk (C;p) for eaah c suah that K sim (ii) If p < k, 0 s j s k-p-1 and Qk(K;p,j) is the operator defined by (8. 22) with K in the (p,l)-entry reptaaed by 0 and the 0 in the (p+j,j+l)-entry repZaaed by K, and K is quasinitpotent, then Qk(K; p,j) # Qk(K;p) •

(iii) If Ks is a aompaat niZpotent of order s ~ 2 suah that K j=co- for j=l,2, •.. ,s-l, (namely, K =$col~), then s s n= n s (co) (s-1) (n) qk fB[qk+p(s-l)fBqk-p ] ! Qk(Ks;p) sl.m

rank

and

o. for m ~ k+p(s-1) {-co, for k-p < m < k+p(s-1) o. for 1 s m s k-p.

(8.23)

m

(In partiauZar, Stk (Qk (Ks;k)) =-co for 1 s m < ks.) (iv) If K is a universaZ aompaat quasiniZpotent (nameZy, K

co 1 u fBn=l ~n)• then

u

and m

Stk (Qk(Ku;p)) =

{-co, for m > k-p

0, for

(8.24)

1 s m s k-p.

(In partiauZar, Stkm(Qk (Ku;k)) = •. co for aZZ m ~ 1.) PROOF.

(i)

If W

£

G(H 1 ), then W(k)Qk(K;p)W(k)-l =Qk(WKW-l;p) .Now

the result follows by takinq limits as W runs over suitable sequences in G(H 1 ). (ii)

Similarly, K

7

s~

C implies Qk{K;p,j)

K is a compact quasinilpotent, it follows from

+

s~

Qk(C;p,j). Since

Propositi~n

8.5 that 217

S(K)

is uniquely determined by the finite rank nilpotents that it

co~

tains. ( .) al. m If F = OCB[f&i= 2 qi ] is a ::::inite rank nilpotent, then Qk (F:p, j) m . (O.·l ~ $i= 2 Qk(Of&qi:p,J) l. and a straightforward analysis of these operators shm-1s that

Hence, Qk(F:p,j) - Qk(F:p) for 0 :-;; j :-; k-p-1, whence the result follO\ITS. ( il.'l.')

The formula qk

(oo)

(s-1) (n) CB[qk+p(s-l)CBqk-p J

t Qk(Ks:p) is an sim obvious consequence of (i) and (8.25). Now (8.23) follows from Propos~

tion 8.28(iii) and some straightfon'/ard computations. (iv)

0

This follows by the same arguments as in (iii).

PROOF OF PROPOSITION 8.30. (i) By our previous observations, all . + t-1 the operators 1.n Nk,h (H:r:{mj}j=r) have the form T - RIBF, where F = t (Tj) . • . Cllj=r+l qj , Tj -mj-l 2mj+mj+l (J -r+l,r+2, ••• ,t, mt l.S definedequal to 0) and R € N~,h'(~:r:~). By Lemma 2.5 (see also the proof of Corollary 8.2(i), there exists a sequence {Cn}n:l in L(a:kr) such that en- qr(k) and IICn-qk(r)ll +

0 (n + oo) • 00 (h' ) Define 'l'u = (CBn=l en) EBq 1 QlF. It is completely apparent from this definition (and Lemma 2.5)

that qkt<•>sq (kn)EBq (h') ..CBF t '1' and therefore, by Proposition 8.26\ili), r 1 sim u S(T ) [ oo S(r. (oo)f&q (kn)f&q (h')eF) J-. (8.26) u = un=l ~k r 1 + h ( H: r: { m .. }t-1) . . Nk Let A b e an ar b l.. trary opera tor l.n . _ • By Proposl., J J-r tion 8.26(i) there exists a sequence {Ai}i:l of operators similar to nice Jordan nilpoten~s such th~t IIP.-Aill + 0 (i + oo), Ai+l s!m Ai (i= 1,2, ••• ) and rank AiJ s rank AJ for all j=l,2, ••• ,t-l, and for all i 1, 2,... •

(oo) t (aij) t , _ Clearly, Ai- qk f&[CBj=l qj J, where Lj=lJaij = h (mod k), a, . . =-r. for j=r+l,r+2, ••• ,t (for all i > i ) and a·k=O for all i=l, l.J J (k) - 0 (r) l. 2, •••• Fix i :<: i 0 • Since rank[qr JP-rank[qk ]P :<: 1 for all p=l, 2, ••• ,r-l, it is not difficult to see that there exist ni

~

1 and ai

~

0 such that

L

L

. .. -k . .+h' k ra.+ . t 1 Ja - rn.+ . t +l JT l. J= l.J l. J=r J

and rank(q(rai>e[CB.t q.J)P :-; rank(q (kni>eq (h'le[CB.t q~-rj)J)P k J=l J r 1 J=r+l J for all p = 1, 2, ••• , t-1. 218

(Observe that ni- a i = (kr) -l

L~:baij +t j~r+l

j ((X ij -T j) - hI ] o)

By Lemma 2.5, it follotfs that q (oo)IBq (kni)IBq (h 1 )tBF -t A.• k r 1 s1m 1 By (8.26), we conclude that Tu s!m Ai (for all i = 1,2, .•• ) and, a fortiori, that Tu s!m A. + h(H;r;{m.}._). t-1 Hence Tu is universal for Nk , J J-r (ii) Define TO, 1 = ql

(hI)

IBFIBQk (Kr-k+l; l) '

where Qk (Kr-k+l; 1) is the operator given by (8. 22) with Kr-k+l as in (iii) of Lemma 8.31 and

T 1 = n,

"' (nk) ' {TO,lwqr T0 , 1 tBq 1

(-nk)

.f

1

n

>

0 '

, if n < 0.

By Lemma 8.3l(iii), T. =q (oo)IBq (i)IBq (i(r-k))IBq (h 1 )IBF t TO,l' 1 k r k-1 1 sim T.ftq (nk) t T (n > 0) and T.~tq 1 (-nk) t T 1 (n < 0). By (i) and 1 r sim n, 1 1 sim n, Lemma 2. 5 Tu sim Tn+l, 1 s!mTn, 1 for all n 10 :IZ. It only remains to show that Tn,l sfm Tn+l,l for any n. Formula (8.23) implies that k-l Stk

{n(r-k) (k-1), if n (Tn, l) = ck-l (F) -

2

0,

( 8. 27)

nk, if n < 0

(where ck_ 1 (F) is a constant that only depends on F). Now the result follows from Proposition 8.28(iv). (iii) Assume that k > 2 and r - k 2 2. Define T n,l as in ( ii) and By Lemma 8. 3l(iii), (nk) (i) TifBqr IB(qk+2 19qk-2) t Tn,2 (n sim

2

O)

and (-nk)

{i)

t Tn 2 (n < O) • s1m • Since qk+21Bqk-2 s1m qk+l$qk-l ~!m qk (2) (Lemma 2.5), we can see that Tu s!m Tn+l, 2 s!m Tn, 2 s1m Tn,l for each n e :IZ. Ti$ql

${qk+2 19qk-2)

Since (use (8.23)) Stkk-l(T 2 ) = - for all n e :IZ, it follows n, from (8.27) and Proposition 8.28(ii) that Tm, 1 s1m f. Tn,2 for any pair of integers m, n. By (8.23) Stk

k- 2

(T

n,

l) = Stk

k- 2

(T

2 > = ck_ 2 (F) n,

{n(r-k) (k-2), if n . nk, 1f n < 0

2

0,

(where ck_ 2 (F) is a constant that only depends on F). By Pr?position 8. 2 8(ii), 219

Tn-1,2 stm Tn,l

for any n

€

~.

On the other hand, since Stkk- 2 (T ) - Stkk- 2 (T ) > 0 for all n-1,2 n,2 n1 Proposition 8.28(ii) also implies that T 1 2 n- 1

f.

s~m

T

n1

2

for any n

€

~

0

•

REMARKS 8.32. (i) Let k 1 h 1 r and {m.}7-l be as in Proposition J J=r 8. 30. The same argument of the above proof can be used to sho\07 that (Nk+ h(H;r;{m.}7::: 1 )/i,<) actually contains s "parallel chains" {[T 1 ]}, 1 J J-r n1 {[T 2 J}, .•• I {[T ]} (n € :
(2)

EDQ5(K2;1)]

are two incompaPabZ.e elements of The same phenomena appear in the case when we consider quasinilpotent non-nilpotent operators. The proof of the follmling proposition follows exactly the same lines as the one given for Proposition 8.30 and will be omitted. PROPOSITION 8. 33. Given k ~ 2 and h, 0 s l1 s k-1, the opera to!' T = Q (K ·k) EDq (h) is universal. for the cZ.ass u k u' 1

l.t~,h(H;oo)

={A

€

Q.(1f)nN~,h(H):

rank Aj =co- for aZ.Z. j

and satisfies

Stkm(T

and T ,

n ,p

220

,

7

s~m

r

?: k-p ifm ) = -m,(n+l), if 1 :<;; m < k-p and n ?: 0 n,p -n(k-m), if 1 s m < k-p ancl n < 0

Tn,p if and onZ.y if n'

~

n and p'

~

p.

~ k}.

+ . (This partial structure of (Nk ,J gram 8.28.)

(H:~)/#,<)

is depicted in Dia-

DIAGRAM ( 8. 28)

8. 4. 5 A general criterion for universality All universal elements ~or the above studied quasinilpotent classes) satisfy Stkm(Tu) = -oo, for 1 s m s k, and this seems to be the "abstract" characterization of universality. It will be sho~m that, if T E N~,h(H), then Tis a universal operator for its own class if andonly if T s!m T$qk+l (k). In order to make it rnore precise, we shall establish the result (and proof) as three se~arate propositions. In the case when T is a quasinilpotent, this result provides some support to the above conjecture about universality. We shall need two auxiliary results.

221

LEMMA 8.34. Let T=g 1 (h) OJ. then there e:z:ist WE" G(H) and m=m(E) suah that llw TW- 1 -q 1 (h+km(E)sqt")ll < 2e; rriol'eover. lltiEllllwe-~1 on~y E

1

£

depends on ~(r) =max{IIO.-T>-11:

l:\1 =r} (r > £).

PROOF. Let Pm denote the orthogonal projection of

H onto the fi-

nite dimensional subspace corresponding to q 1 (h) and the first m copies of qk; then A (m) [ ll

A

(m)

12 A (m) q (co)+A (m) 21 -k 22

T=

)

H

m

HeH ' m

where Hm =ran Pm. Since Pm + 1 (strongly, as m +co), it readily follows that

and (by the upper semicontinuity of the spectral radius; Corollary 1.2 (ii)) sp(A11 (m)) < (3/2)£ for all m > m0 (E). Furthermore, it is also clear that II<:\ -All (m)) -~~ < 2~ (r) for all r > (3/2) E and all m >

m1 (E) > m0

(E).

By Rota's construction (see (3.5) in Remark 3.34) there exist invertible operators V depends on ~((3/2)e)

m

" G(Hm) such that llvmll·llv

-~~

m

lfvmAll (m)vm -~~ < (3/2) e. Define wm

€

G rH) by wm I Hm = v m, wm I H~ = 1; then

l'l TW -1 =

m

m

< M, where M only 1 II < M1/2 \and m m '

(Clearly, ~1e can assumethatllv 11,11v-

v A (mlv -l m [ m ll A (rn)V -1 21 m

v A gk

m 12 (co)+A

(m) 22

(m)

]

"'q (h+km)tlq (co)+E(m;e) 1 k

1

where E(m;e) is a compact operator such that IIE(m;e)jj < (3/2)E+r.i1 1 2nm (m > m1 (e)). Since nrn + 0 (m + oo) that M1 12 n

m

< E/2 for all m

1

we can find m(E) > m1 (E) such

~ m(e).

Define We =Wm(E). It is completely apparent that NE and m =m(e) satisfy all our requirements.

0

LEMMA 8.35. Let L =q 1 (h)
ni ~po!ent acting on a spaae of (finite J dimension dn (n = 1 1 2 1 • • • ) and {)._n}n=l is a sequence of distinct non-zel'o comp~e:z: numbel's oonvel'ging to 0. It Tu is a universaZ opel'atol' for the cZass N~ 0 (H;k+l;~). then I + L
LQ)Tu

+ LQ)q (.. >eq sim ·k km for all m=l,2, ••. Clearly, for all practical purposes we can (and shall!) assume that h=O, i.e., that the direct summand q 1 (h) is absent. Define

0

0

(k x k OP.erator matrix; ln denotes the identity on a space of dimension d) and let D. ={n: d = j(mod k)}, j=O,l, •.• ,k-1. Clearly, itis n J n possible to write ED col R =R ED[ED col S ], where R =ED{R: n € D. for n= n o n= n o n J some j such that Dj is a finite set} acts on a finite dimensional space, Sn acts on a space of finite dimension krn' and o(R0 )no(Sn) = a (Sn 1 ) ncr (Sn") = IJ for all n, n 1 , n", n 1 .;. n" (each Sn is the sum of k Rm 1 s such that the corresponding dimensions dm are all congruent modulus k). It is completely apparent that L can be written as L = EDL , oo" ,... ,.,. 00 where L0 •n=l Sn and R0 and Sn are defined in the same way as the

R

operator matrix Rn. This reduces our task to showing that CD

A

(EDn=l Sn)EDTu =L0 EDTU slm L0 EDqk

(co)

EDqkm (m=l,2, ••• ) •

Let en be a unit vector in the space of Sn such that Snen=~nen, where ~n is the eigenvalue of Sn of largest modulus (by definition of Sn' ~riO for all r=l,2, ••• , and ~r 1 i~r" if r 1 ir").By Lenuna 3.36,

r~

A

s

n

n

0

* ) V{en} s~

{en}~

sim ~n•s~

and it is straightforward to check that there exists s~ ~ qk_ 1eqk (en =k 2rn-k) such that lis~ -s~ll s 2l~n1·

Fix m and let p > 1; then (by Proposition 4.21) L 0 ~a qk(oo)EDL0 and

q (co)EDL =q (co)$($ 00 S ) =a (oo)ED(ED p k o k n=l n ·k n=l Sn) s!m qk (co)e(en~l "'(ED w

oo

n=p+krn+l

S) n

Sn )8($n=p+l p+km S )8(8 n n=p+km+l 00

Sn)ED[(diag{~p+l'~p+2'···•~p+km})e(en~;~~ S~)]

~[(ED p+km n=p+l

S')EDq (co)]ED((eP n k n=l

2~

p+km .

S )"'q (k Ln=p+l rn) n

w

k

223

•<~n:p+km+l Sn))$(diag{~p+l'~p+2'···•~p+km}) -

<•~!~ 1 {s~- s~} )eo]e I <•n~l

-

[($~~!1 S~)eqk(~)

sn>• ($~!~~\ {Sn- Rneoeoe •.• eo})

Sn)]$(diag{~p+l'~p+2'"""'~p+km})

s(en:p+km+l

[($ p+km S")$q (oo)- ($ p+km{S"-S'})$0]$[($ ""S) -($ p 0) n=p+l n k n=p+l n n n=l n n=l p+km oo •<•n=p+l Rneoeoe ..• eo)e(•n=p+km+l 0)1 -

~p+l

1 1

0

~p+3

0

"qk-1

(km)

$qk

where E(p;m) "

(co)

$Lo$gkm-E(p;m),

[(en~+~~{~~ -s~})eO]e[Oe(en~~~~ Rneoeoe ..• eO)eO]e(diag

{~p+l'~p+ 2 , .•. ,~p+km)) is a finite rank operator such liE (p;mlll ,.;; 2 max{sp(S n ) : n > p} -+- 0 (p -+- oo)

that

Hence, -+[q $q ("")]$L$q (km) qk ("")eL sim km k -k-1

for all m=l,2, •••. Since Tu is a universal ogerator for the class N~, 0 (H;k+l;~), it readily follows that L$Tu"

a

L$q (co)$T ~ [q eq ("")]$Lea (km)$qk+l(km)$qk("") k u s1m km k -k-1

-+[q $q (oo))$L8lq( 2 )eq ("")" [q $q ("")]$L sim km k -k k km k The proof is complete now.

PROPOSITION 8.36. Let T

£

(m=l,2, ... ). 0

+ Nk,h(H) and assume that cr(T) is the

union of {0} and a sequenae {Xn}n:l of infinitely many distinat nonzero aomplez numbers aonverging to 0. Then the following are equivalent (i) S+(T) ={A € N~,h(H): a(A) =a(T) and dim H(Xn;A) =dim H(Xn~T) and TjH(Xn;T) sim AjH(Xn;A) for alZ n=l,2, ... } (i.e., Tis universal for its aZass). (ii) T jt T$2u• where
+ Nk,O(H;k+l;,0). (iv) T sim T81Qu as in (ii)). (v) T sim T$Tu (Tu as in (iii)).

(Ou

(

.)

T

v~

(k)"'

-+-

sim qk+l wT. PROOF. (i) => (ii) <=> (iv) => (iii) <=> (v) => (vi) are trivial implications. on the other hand, it is clear that if T ~ TQiq (k) then (use Proposition 4.2l(ii)) we also have T 7 TQiq ?~f$q ktk) 7 s~m k k+l s~m T$q (co) Qlq ( 2 k) 7 7 T$q (oo) Qlq (km) (m = 1, 2, ••• ). By using k k+l s~m s~m k k+l Lemma 8.31 and Proposition 8.30, we see that T s1m T$Tu. Since (by Lemma 8.35) TQITu s!m T$Qu' we conclude that (ii) -(vi) are equivalent statements. Thus, in order to complete the proof it only remains to show that T$Qu is a universal operator for the class of T. This will be done in two steps. 1) Clearly, there exists a unique (up to unitary equivalence) o:s>erator L =q 1 {h)$Q(en:l Rn:l) E N:,h(H) such that o(L) =o(T) and L\HO.n;L) ~ T\H(A.n;T) for all n=l,2, •••• It will be shown that T$Qu sim L810u· Observe that T81Qu s!m T$qk(oo)$Tu$Qu and (with the notation of Lemma 8.35) TQIT Qlq (co) u k e(q (co)eT ) ~ q (h)eR 81[T eq (do+k-h)Je[q (co)eT 1 k u 1 0 0 1 k u ' where T =T\H(o(T)\o(R );T). 0 (d +k-h) + It is completely apparent that A = T0 Q~Q 1 E Nk, 0 (H) , o (A) o(T)\o(R0 ), A\H(A.n:A) ~ T[H(A.n:T) for all A.n E o{T)\o(R0 ) and that our problem has been reduced to showing that

°

°

A$T Qlq (co) 7 L = $ co S u k s~m o n=l n m m For each m (m=l,2, ••• ), A~ (Qin=l Sn)$Am,whereo(l\n)=o(A)\o(Qil'Flsrn>· Since Sn acts on a space of dimension krn' it is not difficult to see that Am is similar to qk(co)+Km (for some compact Km) and sp(Am) -+- 0 (m-+- co). Proceeding as above (see also the proof of Lemma 8.35), we see that (kL ~ r ) co A$T Qlq (oo) ~ ($ m S )Qiq n-1 n $A $T Qlq ( ) u k n=l n k m u ·k

s

(81 m )e[q (kLn~lrnleA ]QIT Qlq (co). n=l n 1 m u k By Lemma 8.34, we can find a subspace Hm of the space of T

s~m

m

(kLn=lrn) q1 whose dimension is a (finite) multiple of k and Wm invertible such that

•Am

225

w

m

[q

(kL

1

and II
~

n-1

2 ._ 1

r

11 s

~.]-

) n •A ]W 1 m m

=

fA11 (m)

A12 {m) ]Hm

A (m) l 21

q

2 sp(A. ) •

-rn

Let kern =dim Hm; then Tu s!m Tu•qk+l

(m) HJ. m

(co)+A

k

22

(k(k-l)c ) m and, proceeding as

in the ~roof of Lemma 8.34, we conclude that ( ) (kL ml rn) (k(k-l)cm) dist [qk "" ,S (q1 n= tBAmtBqk+l

1 = nm

+

0 (m + "") •

A fortiori, dist[(en:l Sn)$qk(oo)•Tu,S(A$TutBqk(oo))] =nm ... 0 (m ... oo). m Sn)$qk " (oo) - L II Since II <•n=l 0

s 2 sp(Am)

+

0 (m

+ oo),

i t readily

follows from the universality of Tu and Proposition 4.21 that A$T tBq (oo) 7 L $T • u ·k s~m o u Hence, TtBIJu s!m LtBOu =gl (h)tBQ(en:l Rn;l)tBQu =ql (h)•<•n:l Rnl•Ou· 2) Let B =q 1 (hleqk (oo)+C, C E K(H), be any operator such that a(B) =a(T), dim H().n;B) =dim H{).n;T) for all ).n E a{T) and TjH().n~T) 7

BIH(). ;B) for each n, n =1,2, •••• Bv Proposition 4.21, B"' B$ n ·· a qk "").Given E > 0, choose m large enough to guarantee that dn I u{Dj: s~~

D. is a finite set} and J B =

lA

n

I < E/4 for all n > m. Then

rAE CE]H({).n}n:l;B)

0

BE H({).n}n:l:B)J.

and sp(BE) < E/4. As in Proposition 3.45, we can find a compact normal operator K , II K II < £/4, such that 14 = B - K is quasinilpotent. By £ £ E E E Corollary 3.22,

[:• ::J -

I

A,OM,, and

::JII

B- [:•

<E/4,

so that A£ CE]$q (co)

fO

M

k

-

A tBq (oo)$M •

k

E

E

£

be the dimension of H({). } m1 ;B). Then 3k-h +h > k, ("") n n= ("") (2 )o 3 k+h0-h > k and AE$qk $q3k-h +h$q3k+h -h·ME s!m AE$qk $q3k $ME Let d

=kr +h

£

£

o

eqk(oo)~M

~emma

7m A 8qk("")•qk( 6 )eM =A (use 2.5); the above construe£ £ £ E tion and the universality of Uu indicate that s~

Lt90u s!m A£$($n=:+l Rn)$q3k-h +h8 q3k+h -h•M£ 0

...

""

"

sim A£e(en=m+l Rn)eqk Since 226

II qk ("")

- •n=:+l Rn

II

(6)

0

er-iE.

s 2 max {I~ I:

n > m} < £/2, we see that

Therefore, dist[B,S+(L$0 )] < E. u

Since E can be chosen arbitrary small, it readily follows that LSOu sim B. The proof is now complete.

0

CONJECTURE 8.37. If T is similar to a compact 9erturbation of q 1 (h)eqk(m) (for some k ~ 2 and 0 s h s k-1) and o(T) contains infinitely many points, then T i LSQ, where L is the operator defined in the proof of Lemma 8.35 and perturbation of qk (oo) PROPOSITION 8.38. If T

Q is a suitable quasinilpotent compact

E

+ t-1 Nk,h(H:r;{mj}j=r)• then the following

are equivaZ.ent (i) Tis a universal operator for the alass Nk+ h(H;r;{m.}~= 1 ). , J J-r (ii) TIt TST • where T is a universal operator for the class + u u Nk,O(H;k+l;~).

(iii) . ) ( ~v

T sim TSTu• where Tu has the form of (ii).

T

+

sim

T•

(k)

wqk+l

PROOF. (i) => (ii) and the equivalence between (ii),

(iii) and

(iv) follow exactly as in the first part of the proof of the previous proposition. (iii) => (i) Combining Lemmas 7.8 and 7.9 we see that, for each (km) t (T·). m (m=l,2, •.. ), T ~Feq ST, where F=e._ +l q. J ~sa ni.lpotent r rn J-r J . acting on a finite dimensional space such that rank pJ =rank TJ for all j ;;;, r and Tm € N~,h' (N;r;~) with 0 s h' s k-1, h' :: h- Lj=;+l jTj

(mod k). Now we can proceed as in the proof of Lemma 8.34 and the first part of the proof of Proposition 8.36 in order to show that T s"tm TST • u

7

s~m

FSq (km)ST ST ·r m u

7

s~m

A

m

=FSq (km)eq (m)(m=l 2 ••• ). r k ' '

Since, by Proposition 8.30 and Lemma 8.31, A : R , where R is + m s~m u u a universal operator for the class Nk h(H;r;{m.}~-1), we are done.O , J J=r The argument for the proof corresponding to the third case is closely related to the one used in the proof of Lemma 8.1. PROPOSITION 8.39. If T

E

+ Nk,h(H;oo), then the following are equiv227

alent

+ Tis a universal operator for the eZass Nk,h(H~~). (ii) T # T~, ~»here 0u is a universal operator for the elass

(i)

+

Nk,O(H~~).

+

(iii)

T # TeT , u

Nk,O(H~k+l:,IJ).

~»here

T

u

is a universal operator for the class

au

(ivJ T s!m TS~, !»here has the form of (ii). (v J T s!m TSTu, ~»here Tu has the form of (iii). (viJ T sim Tl!lqk+l (k) PROOF. Once again, (i) => (ii) and the equivalence between any two of the statements (ii) - (vi) follow exactly in the same way as in the proof of Proposition 8.36. (iv) => (i) By hypothesis, we can directly assume that T is a quasinilpotent com~act perturbation of q 1 (h)eq~~) such that Tj ~0 for all j = 1, 2,... • Let m . Let {en}n:l be the Gram-Schmidt orthonormalization of the se"" quence {Tn-1 e 1 }n=l" It is not difficult to check that T admits a matrix representation of the form tll tl2 tl3 t21 t22 t23 t32 t33

* T

(8. 29)

0

e

p

0

0

I I

I

*

N

where p=km+h and N is the orthogonal complement of M = V{ej}j~l· Since M is finite dimensional, it is not difficult to check that IIPM
orthogonal to ker Ps for all s large enough. Fix r > m so that £r < 1~ then there exists an invertible operator W=l+G (for some finite rank operator G) such that WM ~ ker Pr· Thus, for practical purposes (replacing, if necessary, T by wTW-1) we can directly assume that M ~ ker Ps for all s > r. Given £ > 0, we can proceed as in Lemma 8.34 and Step 1) of the proof of Proposition 8.36 in order to show that

~ ]~an

B E

H

qk(co)+J ker

(for somes(£) > r depending on£), where all a12 al3 a21 a22 a23 a32 a33

al,p-1 alp el a2,p-l a2p e2 • a3,p-l a3p e3

A

0

ap,p-l app ep (i.e., the matrix of A looks like the (1,1) block of (8.29)), 0

0

0

0 ap,p+l 0 0 0

D

(i.e., the matrix of D looks like the (2,1) block of (8.29)) and [:

:

:]

<
Now we can proceed exactly as in the proof of Lemma 8.1 in order to show that Twhere

wn

E

IB (k[S(£)-rn])IB (co)+E( • ) qkm+h ql qk m,£ '

E (m~ £) is a compact operator such that !IE(m~ £>11 < E· On the other hand (by Corollary 8.2(ii)), given n > 0 we can find G(~k 2 ) such that

IIWn[qliBqk+l

(k-1)

]Wn

-1

-qk

(k)ll


The definitions of Ou, Tu and the above observation indicate that

229

T s!m T$Qu slm T$Tu slm {[qkm+heql (k[s(c)-m])eqk(m)]+E(m:c)} eqk+l ((k-l)k[s(c)-m])eqk(m) {q $q (m)$[q $q (k-1)) (k[s(E)-m))}+E' (m•E) "' Tm,e: km+h k 1 k+l ' ' where E' (m:c) "' E(m:c)$0. Clearly, we have (l$l$W (k[s(c)-m]))T (l$l$W (k[s(c)-m]))-1 n m,c n q $q (m)$[q (k)](k[s(c)-m]) +E"(m:e:), km+h k k where E"(m:c) is a compact operator such that liE" (m: clll < n + c· n n Since E can be chosen arbitrarily small, we can assume that c([iwnll·llwn-~1> < n, whence it readily follows that . (m) + d1st[qkm+h$qk ,S (T)] < 2n. Since n can be chosen arbitrarily small, we conclude that T slm (m) Tm = qkm+heqk (m=2,3, .•. ). By Proposition 8.30 and Lemma 8.31, Tm t Ou' whence the result sim follows. 0 COROLLARY 8.40. If T £ N~,h(H), then given E > 0 there exists K € K(H), !IKI[ < E, suah that T +K is a universal. operator for the al.ass ofT. PROOF. We can directly assume that T =q 1 (h)eqk (=)+C (k ~ 2, 0 ~ h s k-1, C compact). By Proposition 4.21, there exists K1 " K(H) ,!IK1 11 < c/2, such that T+K 1 "' Teqk (=). Now it is easy to find a second compact operator K2 , II K2 11 < c/2, whose action only modifies the direct summand qk(=), such that T+K "' T$T for some universal operator T for + u u the class Nk,O(H:k+l:,6), where K =K 1 +K 2 . Now the result follows from Propositions 8.36, 8.38 and 8.39.0

8.5 Separation of isolated points of the essential spectrum affiliated with nilpotents LetT" L(H) and assume that~ is an isolated point of ae(T). If r > 0 is small enough to guarantee that D(~~r)-nae(T) ={~} andy= ao(~;r) (positively oriented), then -

E~

1 = 2ui

Iy(A-T)-

-1

dA

is a non-zero idempotent of A(H) and can be lifted to an idempotent 230

~

IE

L(H). Let

w

IE

G(H)

be such that P =WE l.l

tion and let H = H IH .l l.l

seen that

where H = ran P

,

l.l

l.l

WTW

-1

l.l

l.l

w- 1 is an orthogonal projecand

1/.l

l.l

= ker P • It is easily l.l

Hl.l H .L I

=

)J

where B and Care compact operators, oe(A) = {lJ} and l.l .l oe(D). Hence T - (1.1+0 ) 190, where Q =A- 1.1 is an essential quasinilpotent. l.l 1.1 It will be convenient to introduce the following definition: Let

l

T t: L(H)and A t:

~

o,

if A

t

oe(T),

n, if A is an isolated point of oe(T) and ~A (as defined above) is a nilpotent of order n, <8 • 30 )

""• otherwise.

If 2 s k(A;T) <""and QA is similar to w(qn(oo)) we shall say that A is an isolated point of oe(T) affiliated with qn(oo). If either k(A; T) = 1 or 2 ~ k(A;T) <""and QA is not similar to n(qn(oo)), we shall say that A is affiliated with q 1 (oo)Glqn (oo). (It is completely apparent that this definition is independent of the particular invertible

oper~

tor W above chosen.) From Proposition 6.19, we obtain the following COROLLARY 8.41. LetT t: L(H). let A={A 1 ,A 2 , ••• ,Am} be a finite set of isolated points of o (T), let n. be the component of o (T) such e J e that A . t: Q .- m. either coincides ~JJith or is disjoint from Qh' if j # J J J m hJ and assume that ind(A- T) # 0 for aZ.Z. A t: n = uj=l nj. Given e > O, there e:x:ists K t: KIHI • IIKII < £ . such that m

T- K - AID[ Glj=l (A j+Qj)]

bJhere oe(A)nA=!I. min.ind(A -A) =0 for every At: nnA and 0 1 , Q2 , ••• ,Qm are quasinilpotent operators. Furthermore, if 1 s k(A ;T) = n. < ""• then Q. can be chosen ni lpo] n.-1 J tent of order nj• but Qj J cannot be compact for any possible choice ofQ., j=l,2, ••• ,m. J PROOF. Assume that ind(A- Al) > 0 for all A t: n 1 • By Proposition 6. 19 there exists a compact operator Kl' II K1 11 < e/3m, such that T - Kl

=(: l

: l ) ... Al CDR!,

( 8. 31)

where o (R 1 ) = {A 1 }, Al ' 0 e(A 1 ) and nul(A- A1 ) = ind(A- T) and nul(A-At* = 0 for all A t: n 1 • (If ind(A- T) < 0 for all A e: n1, then we can apply 231

Proposition 6.19 to T* in order to obtain the same decomposition with o:R1 l ={A 1 }, A1 i oe{A1 ) and nul{A-A 1 ) =0 and nul{A-A1 )*=-ind(;>..-T) for all A e: n1 .) n By Theorem 7. 2, R1 = Ri +Ki, where (Ri) 1 = 0 and Ki is compact. Now we can repeat the argument of the proof of Proposition 6.19: First replace Ki by a suitable finite rank operator Fi so that IIKi -Fill< E/3m and Ri+Fi =

( F"

/

* )

Ri

(8

I

where Fi acts on a finite dimensional space, cr(Fil

l

c

.32)

n1 and Ri is a

nilpotent of order n 1 • Thus the operator [

Aol 0

*

:

0

Ri

Fl

{8 .33)

differs from T by a compact operator whose norm is less than 2£/3m. By Theorem 3.49, there exists a compact operator K3, 11K311 < E/3m such that A"= (Al * ) K' 1 0 Fl - 3 satisfies the properties: all A e:

(1) Al J oe(Ail, and (2) nul{A-Ail*=O for

n1 •

It follows from Corollary 3.22 that, i f c 1 =K 1 +{0Ql{Ki-Fill 1 + (K 3ED0) 2 {where { ••• ) 1 corresponds to the decomposition {8.31) and { ••• ) 2 corresponds to the decomposition {8.33)), then

c1

e: K(H), IIC 111<

£/m and A" * ) T - Cl = ( / Rl

~

AiEilRl. n -1

It is completely apparent that (R" >" 1 1

cannot be compact. Now the

0

result follows by an obvious inductive argument.

PROPOSITION 8.42. LetT e: L(H), Zet A={A 1 ,A 2 , ••• ,Am} be a finite set of isoZated points of cre{T) such that A c interior o{T) and assume that 1 s k(Aj:T) =nj <"'foP each j=l,2, ••• ,m. Given £ > 0 thePe ex-

ists K e: K(H), llKll <

E,

such that T-K

~

m

T 1 =AED[Eilj=l {Aj+Qj)],

whePe oe{A)nA=,9', min.ind{A-Tl)k=min.ind{A-T)k foP aU A c Ps-F{T) and all k=l,2, ••• , Q. is a nilpotent of oPdeP n. foP each j=l,2, ••• , ]

J

m, and, moreoveP (i)

232

s If Aj I [ps-F{T)l, then Qj is quasiniZpotent;

(ii)

If A. e: [ps F(T) ]-, then J sa(A.+Q.)={A.}ufAe:p5 F(T): IA-A.I s ~dist[A.,a (T)\{A.}]}; J J J sJ J e J (iii) If Aj I [p:_F(T)]- and Aj is affitiated with qn~""), then QJ. is a universat operator for N+. 0 (H.;oo); _ nJ, J ( ) 5 (iv) If Aj e: [ps-F{T)] , Aj is affitiated with qnj"" and co 00 s a(A.+Q.) = {A.}u{A.+~. }n ({A.+~. } c p F{T) J J J J J,n =1 J J,n n= 1 sonty accumulates at A.), then Q. tQn.($ 00 1 [~. +I~· IJ. ];l)t9Qu (de] J J n= ],n ],n J,n fined as in Propositition 8. 36 ( i i ) ) for suitabty Jo'l'dan nitpotents J.J,n , n ~ 1. FuPthermore, there aZso exists R

L(H), IIRII < E, such that

e:

m

T- K- R - T 2 = A$[$j=l (Aj+Rj) ],

where

I

I

(oo)

Rj - {t9n=l [~j ,n+ ~j ,n Jj ,n1 }t9qnj 00

for suitabty chosen Jordan nitpotents J,

J ,n

, n=l,2, ••••

PROOF. As in Corollary 8.41, we can confine ourselves to the case when A = {A 1 } (The general case follows by an inductive argument) • Furthermore, we can also translate the operator and assume that Al = 0 is an isolated point of ae (T) such that 1 s k(O;T) = k < ""· Let

be the triangular representation (3.8) of T. Applying Corollary 8.41 to Tr and Tll., we can find compact operators Kr' Kll., 11Kr11 < E/3, IIK11,II < r:./3, such that T

r

- K

r

= (Ar * ) 0 Br

where Br and Bll. are nilpotent operators, Br and Br have the same order of nilpotency, Bll. and Bll. have the same order of nilpotency, 0 I ae(Ar) ucr (A 0 ) , T - K - A $B and T 0 - K0 - B 0 t9A 0 • Thus, if K1 = K $0$K 0 , then e., r r rr "' .. "'"' r"' K1 is compact, IIK 1 11 < E/3 and (by Corollary 3.22)

233

A r B

*

r T

T- Kl

(8.34)

0

BR.

0

AR. where 0 is an isolated point of the essential spectrum of the second direct summand, not contained in the interior of the spectrum of this operator. (iii) If 0 is an isolated point of the spectrum of the second direct summand of (8.34), then it follows from the Riesz' decomposition theorem that T- K1 - Ar~BT 1 6lQIBAR., where 0 I o(T 1 ) and Q is a qunsi nilpotent such that Qk = 0, Qk-1 "f 0. If 0 is an isolated point of 0 e (A) affiliated with qk (®), then it is easily seen that 0k-l+0* is invertible and Q is similar to a compact perturbation of q 1 (h)&lqk (®)for some h, 0 s h s k-1. Clearly, we can directly assume that Q = q 1 (h)$ qk(®)+C (where Cis compact). Assume that 0 € o(A ) and let P be the orthogonal projection in r n (h) the space of Q onto the span of the subspace corresponding to q 1 and the space corresponding to the first n copies of qk; then II Q

-

( P n QP n)

&lqk (®)I i = IiC

- (PnQP n) II

-+

0 (n

-+

®l

and PnQPn acts on a finite dimensional space. By Theorem 3.49, there exists a conpact operator Kn of arbitrarily small prescribed norm such that Ar~BP nQP n- J{n is smooth. It readily follows that we can find a compact operator K2 , 111< 2 11 < e:/3, such that T- K1 - K2

-

A'IBT &lq (®)IDA

r 1 k .t' where A' is smooth. r By Corollary 8. 40, we can find a compact operator such that

K3,

111< 3 II <

e:/3,

T-K- AIBQu' where K=K 1 +K 2+K 3 e K(H), ~KII < e:/3+e:/3+e:/3=e:, A=A~IBT 1 ~BAR. and Qu is a universal operator for Nk,O(H:®). (If 0 I o(Ar), then 0 e o(AR.) and we can apply the above argument to QIBAR..) (i) and (vi) If 0 is isolated in the spectrum of the second di·· rect summand of (8.34), but it is also an isolated point of oe(T) affiliated with q 1 (®) &lqk (®) , then we can use Theorem 7. 2 in order to ob-· tain the decomposition Q = o1 +c 1 , where Q1 k = 0 and c 1 is compact. By Corollary 7.6 we can find a compact operator K2 , IIK 2 11 < e:/3, suchthat

"'A.t' were h -ok = o , -xk-l~o . T-K1 -K 2 - Ar~BT 1 ~BQ 2 6lF 2 ... ~,o~ 2 r , -xlc-1-x*. \.1 2 +\.! 2 ~s not l._!! 2 vertible, o(F 2 ) = {0} c o(Ar)uo(AR.) and F 2 acts on a finite dimensional

234

space. By Theorem 3.48, there exists a compact operator K3 , IIK 3 11<E/3, such that T-K- AEDQ 2 , where K=K 1 +K 2 +K 3 E K(H), IJKII < E/3+E/3+E/3=E A=A~&lT 1 &lAJI, (where A~ is a smooth compact perturbation of Ar&lF 2 , in the case when 0 € a (Ar)), or A= ArEDT 1 EDAR_ (where AR_ is a smooth com-

and

pact perturbation of F 2 &lAJI, in the case when 0 I a (Ar) and 0 " a (A.~~,) ) . Since 2 k-l+Q2 is not invertible, it follows from Proposition 8.5

o

that there exists Q3 - q 1 (oo) EDqk (oo) such that II 0 2 - 0 3 11 < E, whence we obtain (vi) • (iv)

If 0 is not an isolated point of the spectrum of

let {~n}n:l be an enumeration of the normal eigenvalues of B contained in D(O;r)-, where r=~dist[O,ae(T)\{0}] and let J be the canonical Jordan form of the nilpotent Btheorem, 8 - 8 0 618 1 , where a(8 0 B -~ o n

)

n

~

n I H (~ n ;B) • By Riesz' decomposition =cr(8)nD(O;r)-, cr(8 1 ) =cr(B)\D(O;r) and

IH<~

:B)- J for all n=l,2, . . • • n o _ k n _ k-l Clearly, 8 0 = 0, 8 0 -F 0; moreover, 0 is an isolated point of

a (T) affiliated with qk(oo) if and only if 8 0 can be chosen unitarily e~uivalent to q 1 (h)EDqk(oo)+c, where 0 ~ h ~ k-1 and cis compact. By Proposition 4.21 and Corollary 8.40, there exists a compact operator K2, II K211 < E/3, such that T-K 1 -K2-

ArED(B 0 ED~EDqk(oo))EDB 1 EDAJI,,

where B EDOullq 1 (h)EDQ,'k(ED 00 1 [~ +I~ IJ ];l)EDQ .Itiseasily seen that BED o n= n n n u o 0 EDqk h&lqk(oo) II Qk(ED 00 1 [~ +I~ IJ l;l)EDQ (See thep:toof of Proposition B. u

36

-

n=

n

n

n'

u

and that (by using Lemma 2.5) K2 can be replaced by a compact ope£

ator K2 , IIK 2 11 < c/3, such that T- K1 - K2 - ArED[ (8 0 ED~EDqk-h&lqk

( 00)

) EDqk+h ]&lB 1 EDAJI,.

Now thP. proof follows exactly as in case (iii). (If 0

E

cr(Ar),

qk+h is "absorbed" in Ar via a small compact perturbation; if 0 ; cr(Ar), then 0 .o cr(AJI,) and qk+h is "absorbed" in AR..) (vii) Assume that 0 is not isolated in a(B), but it is an isolated point of ae(T) affiliated with q 1 (oo)EDqk (oo). Then it follows from

Proposition 0.6 that B II [ED "" 1 (~ +I~ IJ ) JEDqk(oo). Hence, there exists o n= n n n 8 ' - [ED 00 1 (~ +I~ IJ }JEDqk(oo) such that liB -8'11 <E. o n= n n n o o It is completely apparent from the above constructions that min. ind (A - T 1 ) k =min. ind (A - T 2 ) k =min. ind (A - T) k for all A € ps-F(T) and for all k = 1,2, •••• The proof of Proposition 8.42 is complete.

D 235

Let T, A,

£

be as in Proposition 8.42 and let m

T-K- Tl =AID[EBj=lp'j+Qj)] be the decomposition provided by that proposition with K e K(H), JJKJJ < £/3, where the Q.'s satisfy the conditions (i)- (iv). If Q. is si~ilar to a compact perturbation of qn~oo) (n. ~ 2), J J J then by a formal re9etition of the arguments of the proof we can find a compact operator C. of arbitrarily small prescribed norm such that Q.- C. - Q!EBF., wher~ Q~ "' qn ~""), F. acts on a space whose dimension J J JJ J J J is a finite multiple of n. and o(F.) c o(A). If QJ. does not have that J J form, then we can find C. as above such that Q.- C. - Q~EBF., where F. J J J J J(oo) (oo)J acts on a finite dimensional space, o(F.) c o(A) and Q! 11 q 1 eqnJ· . J J Hence, there exists K' E K(H), JJK'JJ < £/3, such that m m T-K-K'- [A$($j=l Fj)]6l[$j=l(;\.j+Qjl]. Finally, by Theorem 3.48, there exists K"

E

JJK"JJ < £/3,such

K{HI,

that T-

C -

A'$[$j:l (Aj+Qjl],

where C=K+K' +K" E K(H), JJCJJ < £and A' is a smooth compact perturb~ tion of A. Thus, we have the following COROLLARY 8.43. LetT

e

L{H),

Zet II= 0. 1 ,>. 2 , ••• ,/,ra} be a j'in1'te

set of isolated points of oe(T) such that II that l

c: k(I.;T)=n. <"' J

ists C e K(H), JJCJJ <

£,

for each j=l,2, •••

c

interior a(Tl Jnd assume

,m.

C!-ven

F:

>

0 there e:r-

suCJh that

T-C and

8.6 Notes and remarks

Proposition 8.3 has been independently (and almost sinultaneouslY, proved by C. Apostol [131 and by D. A. Herrero [138]. In fact, the results of [137J and [1381 provide a complete characterization of the set of all universal quasinilpotents for L(H) and for each closed bilateral ideal of L(H) strictly larger than K{H), for the case '"hen H is not necessarily separable (9rovided the ideal admits some universal quasini~potent.) On the other hand, the characterization of the set of all universal aompaat quasinilpotents (Proposition 8.4) was obtained by c. Apostol in [13]. The proofs given here combine c. Apos~ol's La~236

ma 8.1, the technical Corollary 8.2 (essentially contained in [44))and an argument of [138]. The results contained in Sections 8. 2, 8. 3 and 8.4 were not published before. In [44], J. Barr!a and D. A. Herrero extended their pr~ vious results about finite rank operators [43] to arbitrary nilpotents and nilpotent elements of the Calkin algebra. On fortunately, the article contained several errors and the main result did not cover all pos~i~ ities as clained there. Correct proofs of the same results (Indeed, slightly better ones) were independently obtained by C. Apostol and D. Voiculescu in [34], by using a different approach. The proofs of Propositions 8.5 and 8.6 are essentially the same as the ones given in [44], corrected with the help of the structural results (due to L. A. Fialkow and D. A. Herrero) given in Sections 7.5 and 7.6 of Chapter VII. The proofs of Propositions 8.12 and 8.17 given here have been taken from [34]. They are more illuminating and "coordinate free" than the corresponding (correct) proofs given in [44]. C. Apostol and D. Voiculescu's approach is heavily based on their Lemma 7.20 and Lemma 8.8, which can also be derived from [182,Lemma 3.3]. Lemma 8.8 can be regarded as the link between the present problem and Brown-Douglas-Fillmore theory. In order to make it more explicit, it will be necessary to introduce some definitions: Let C be a separable C*-algebra with identity. By an extension of K(H) by C we shall mean a pair (E,cp), where E is a C*-subalqebra of L(fl) containing K(H) and the identity operator and cp:E + C is a *-homomorphism with kernel K[H). Such a pair yields the short exact sequence 0

+

K(H) ~ E

i

C

+

0,

where l is the inclusion map. Two such extensions (E 1 ,cp 1 ), (E 2 ,cp 2 ) are called stPongZy equivaZent if there exists a unitary operator V € L[H) such that the following diagram commutes:

(8.35)

Exts(C) will denote the set of all strongly equivalent classes of extensions of K[H) by C. (E 1 ,cp 1 > and (E 2 ,cp 2l are weakZy equivalent if there exists a FPedhoZm partial isometry V € L[H) such that Diagram (8.35) commutes. Let Ext(C) denote the set of all weakly equivalent classes of extensionsof K[H) by C. 237

If (E,~) is an extension of K(H) by c, then w(E) is *-isomorphic to C and this defines a faithful unital *-homomorphism T:C ~ A(H)· Co~ versely, if T:C ~ A(H) is a faithful unital *-homomorphism, then (w-l[T(C) ],w) defines a extension of c. Thus, we can identify extensions with *-monomorphisms from C into A(H). Two *-monomorphisms Tl and T2 are weakly equivalent (strongly equivalent, resp.) if and only if there exists a unitary element U E A(Hl (a unitary operator U € L(H), resp.) such that T1 =uT 2u*. For two *-monomorphisms Tl and T2 from C into A(H) we define T1 + T2 to be the *-monomorphism from C into A ( H$H l such that (T 1 +T 2 )

(X)= Tl (X)$T 2 (x)

1

X

€

C·

Since H ~ H$H, this defines a structure of abelian semigroup in Ext(C) (Exts(C)) by: [Tl]+ [T2]= [Tl+-r2]

([-rlJs+ [T2]s= [-r1+<2Js• resp.) •

In fact, Ext(C) and Exts(C) are abelian semigroups with identity and, in many cases, they are actually groups. It is also apparent that [u ~ [T) defines an epimorphism of semigroups (of groups, in case s s they are actually groups) from Ext (C) onto Ext(C). The reader is referred to [60], [61], [182] and, especially, to [40] and the references given in the different articles contained there. In particular, i f C = L(a:k) (for some k ~ 2), then Ext[L{a:k) J = {0} and Exts[L (a:k) J = 2Zk = 2Z/k2Z , so that the natural epimorphism Exts[L(O:k)] + Ext[L(O:k)] is a trivial mapping [182,Lemma 3.3]. This interesting connection with the Brown-Douglas-Fillmore theory is not apparent from the proofs of Propositions 8.12 and 8.19 given in [44]. Corollary 8.23 and Proposition 8.24 are due to L. A. Fialkow and D. A. Herrero. These two results are related with the local analysis of the similarity orbits (see [16] and [94]). The results of Section 8.4.3 belong to C. Apostol and D. Voicule~ cu [34] and partially fill the gap in the main theorem of [44]. Sections 8.4.4 and 8.4.5 contain recent results of D. A. Herrero. (Actu~ ly, the peculiar behavior of the operators qkn$Qk(Ku;l), where Ku is a universal compact quasinilpotent has been partially analyzed in [44, Section 7] • ) Finally, corollary 8.41 is Corollary 1 of [42] and Proposition 8. 42 can be regarded as a nontrivial improvement of Theorem 3 of that reference.

238

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197. L. 198. L. 199. F. 200

J.

commutators. Acta Sci. Math. (Szeged) 42 (1980) 195199. R. Williams, On quasisimilarity of operators on Hilbert space. Dissertation. (University of Michigan, 1976). R. Williams, Similarity invariants for a class of nilpotent O£ erators. Acta Sci. Math. (Szeged) 38 (1976) 423-428. Wolf On the essential spectrum of partial differential boundary value problems. Comm. Pure Appl. Math. 12 (1959) 211-228. Zem~nek and B. A~etit, Local behavior of the spectral radius in Banach algebras, J. London Math. Soc. (2) 23 (1981) 171-178.

247

LEMMA 8.34. Let T =ql (h)

Index

A

Algebraic direct sum 5 element of the Calkin algebra 128 operator 114 Analytic Cauchy domain 2 contour 2 Jordan curve 2 Jordan curve, regular 2 Apostol-Morrel simple models 135 Apostol-Salinas approach 120 Apostol's triangular representation 62,63 Approximate defect spectrum 42 point spectrum 42 Approximately equivalent *-representations 97 unitarily equivalent operators 100 Arc-length measure 59 Asymptotically similar operators 12

B

Bad properties 78 BDF theorem 41,84-85,94 theory 237-238 Berger-Shaw trace inequality 86 Biquasitriangular operator 145-148 Block-diagonal operator 146 Bro~rrder spectrum 69 Brown-Douglas-Fillmore theorem 41,84-85,94

Calkin algebra, quotient Cauchy domain Closed linear manifold span Closure (in the metric topology) Commutant of an operator Commutator Compact operator Compression of an operator Cyclic operator

1

2 4 22 2 85 84 1

45 14,35

D

Diagonal normal operator 70·, 76 Double commutant of an operator 85

E

Essential m1n1mum modulus 143 nilpotent 162 reducing operator spectrum 106 spectrum 5 Essentially normal operator 84 Exact sequence 177,183,237 Extensions of C*-algebras 237-238

F

Faithful *-representation Finite rank operator Fredholm domain operator

97 13 9 9

c G

C*-algebra 81-82,84,97 C*-algebras, irreducible 97 *-representations of 98 Calkin algebra 1

Gelfand-Naimark-Segal theorem 97 Gram-Schmidt orthonormalization 67

249

H

L

3,58,109 Hausdorff distance Hilbert-Schmidt operator 76 Hyperinvariant subspace 5 Hyponormal operator 85-86

I

Idempotent, Riesz 2 Idempotent operator 25 Index 9 Infinite-dimensional essential reducing operator spectrum 107 reducing operator spectrum 107 Infinite direct sums of operators 130 Inner derivation 81 76,13 product Integral part 20,35,124 Invariant subspace 5,157 under compact perturbations 11 1,11 under similarities under unitary equivalence 10,11 Irreducible 104-105 operator *-representation 98 12 Isomorphic Hilbert spaces

17,209-210 Lattice Left essential spectrum 9 resolvent 45 resolvent set 9 spectrum 9 Locally closed similarity orbit210 Lower semicontinuity of the rank 8

M

Matrix 59 decomposition 92 models for operators 54 Maximal numerical range ~~ximal numerical range, normalized 57 10 r.Unimal index 144 Minimum modulus Minimum modulus, essential 144 Modulus of quasidiagonality 146 140 of quasitriangularity Hodulus, essential minimum 144 144 minimum 214 m-th stable kernel Multicyclic operator 85 Multiplication by A operator 59,89 85 Multiplicity

N J

Jordan block curve curve, regular analytic form nilpotent operator

15 2 2 14 171 171

K

Kernel

250

4

n-dimensional essential reducing operator spectrum 107 reducing operator spectruml07 Nice element of the Calkin 204-205 algebra Jordan nilpotent 206,209-210 171,205 Jordan operator Nilpotent element 6 6,108 operator 162 Nilpatent, essential 117 NilPotents, sums of two 3 Norm 15 Norm-topology Normal 5 eigenvalue 70,76 operator, diagonal restrictions 74-75,101

Normalized maximal numerical range n-tuples Numerical range Numerical range, maximal normalized maximal

57 100 54 54 57

Operator-valued spectrum Orthogonal direct sum projection onto a subsoace Orthonormal basis

100,106 12-13 45 30

p

0

164,188 Olsen's theorem Operator 1 Operator matrix decomposition 59 Operator, algebraic 114 biquasitriangular 145-148 block-diagonal 146 cornrnutant of an 85 compact 1 cyclic 14,35 diagonal normal 70,76 double cornrnutant of an 85 essentially nilpotent 162 essentially normal 84 finite rank 13 Fredholm 9 Hilbert-Schmidt 76 hyponormal 85-86 irreducible 104-105 Jordan 171 multicyclic 85 nice Jordan 171,205 nilpotent 6,108 polynomially compact 162 pseudonilpotent 124 quasidiagonal 146 quasinilpotent 4,122-123,127 quasitriangular 125,140 rationally multicyclic 84-85 reducible 100 restrictions of a nilpotent 168 5 restrictions of an semi-Fredholm 9 74 smooth subnormal 84-85 trace class 75 triangular 62-63 178 very nice Jordan Operator spectrum, essential reducing 106 infinite-dimensional 107 essential reducing infinite-dimensional 107 reducing n-dimensional essential 107 reducing 107 n-dimensional reducing 106-107 reducing

Partially ordered set 17 Perfect spectrum 111-112 5 Point spectrum Point spectrum, approximate 42 Polynomially compact operator 162 17 Po set Pseudonilpotent element of the Calkin 129 algebra 124 operator

Q

146 Quasidiagonal operator Quasinilpotent 4,122-123,127 operator restriction 103-104 135,140 Quasitriangular operator 1 Quotient Calkin algebra

R

4 Range Rationally multicyclic 85-86 operator 2 Rectifiable contour 100 Reducible operator Regular 2 analytic Jordan curve 41 left representation point of the semi-Fredholm domain 10 right representation 41 *-representation of a C*97 algebra *-representation, faithful 97 98 irreducible 97 unital 59,89 Reproducing kernel space 2 Resolvent 2 Resolvent set Restrictions of nilpotent operators 168

251

Riesz

5

operators

decomposition theorem 153 2 idempotent Riesz-Dunford functional 2 calculus Right essential spectrum 9 44 resolvent resolvent set 9 spectrum 9 Rosenblum 79 corollary 41 operator 53 theorem Rosenblum-Davis-Rosenthal corollary 53 Rota's corollary 61,82 Runge's approximation theorem 86

s Schatten p-classes 75 Self-commutator 84 Semi-Fredholm domain 10 operator 9 Separable C*-algebra 97 Hilbert space 1 Similar, asymptotically 12 Similar operators 12 Similarity-invariant 1,11 Similarity orbit 1,13-14,113,114,189 Simple model 135 Singular point of the semiFredholm domain 10 Slight twist 22 Smooth operator 74 Spectral characterization of quasitriangularity 144 measures 77-109 radius 13 Spectrum 2 Spreading spectral measures 77-109 Stable kernel 214 Staircase representation 147-148 Strongly equivalent extensions237 Subspace 4

T

Trace Trace class operator

252

26,35,75,86 75

Triangular matrix operator representation

62-63 62-63 62

u 97 Unital *-representation Unitary equivalence 10-11 group 84 orbit 11,84,100 Unitarily equivalent operators 12 *-representations 97 Universal compact quasinilpotent operator 194 model 59 operator 189,215 quasinilpotent operator 193 Universality, criterion for 221

v Very nice Jordan operator Voiculescu's theorem

178 40,98

w lieakly equivalent extensions measurable square integrable functions Weyl spectrum Neyl-von Neumann theorem Neyl-von Neumann-Kuroda theorem Nolf spectrum

237 59 69 41 104 9

Symbols and notation

A' A (.) AI ( . ) AII

(.}

Aa (.) AI H) Alg(A(H)) Alg(H) A2 (fl) A(a} Be(X) (BD) BP(X) (BQT) B(X) a: a:d C' ( •} C* (.) C*({.}} cP(II)

eo e 0 (X) C 1 (II)

cz 1HI c"" !HI d(.) D (.,.)

dH( •, •) E(H) Ext (.) Ext 8 (.) E(:>.;T) E(a;a) E(cr;T)

43 85 85 85 85 1 129 111,114 89 13 3 146 109 146 3 2 13 99 97 97 75 83,159 43 75 76 75 64,132 108 3,109 25 237 237 5

2 5

(E,cj>)

(FC+,s) F(H) fk (.} G(A(H)) G(H) H

11Abnor(.} H1 (.}

11 Nor (.) Ho (.)

Hr (.} II(T,A) H(a) 11(/.;T} H (a ;T) H2 (r) H2 (r;H} ind k (., • } ker K(H) La ~ (.)

LB (.} (L(O:d}/#,<) L(H) tP l.u.b.

L(X,Y) t""(X) L2 (r) L2 (Q,dA) m(.)

237 159 13 104 129 1 1 101 62 101 62 62 19 13 5 5 59 60 9 231 4 1 41 51 80 17 1 83,159 18 1 43 59 89 144

MB me (.) min. ind (.) Mp(H) m(T,A) M(f) M_ (f)

!1+(r) N(A) N(A(H)) (N (a:d) /i, <) NF(H) NFk(H) (NF(H)/i,<) (NFk(ll)/i,<} NIH I

Nk(A)

42 144 10 29

15 59 59 59 6 129 17 19 19 20 20

6 6

N~(A(H))

183

+ Nk,h(/1) Nk(H)

200 6

N:!Hl

183

+ Nk,h(H) t-1 + Nk h(H;r;{m.}. ) , J J=r +

200

Nk,h(H;r;~)

218

N~,h(/1;<»)

220

N1 (H) NOIL(H) nul N(Q} N_(Q)

216

N+(Q)

32 85 9 89 89 89

ONB

30 253

PF(H)

140 20 PN 32 !D 32 !Dk q(.) 140 159 qa (.) qd(.) 146 (Q.(A(H))/* 6 ,<) 203-204 CQ.(A(H))/tw,<) 203-204 {QD) 146 Q.( H) 193 13 qk 217 Qk(K;p) 14 qlp (QT) 140 (QT)* 145 193 Q.u(H) 41 Ra 51 RA ( •) 80 ~( .) ran 4 45 R.t(.,.) 45 Rr(.,.) s (.) 1 s+ <. > 198 s135 135 s+ .... 12 sim t

sim s(m) Stkm(.)

..... s-sl.m S(A)

TAbnor

210 21 214 199 115 101

Tr tr (.) Tupper

u(.) U(H) W( •) WN(.)

wo c.> w-sim

X' ~

'\ 1\ 6I

k

6T (.) €

nk n'k n.., n'

"'

)J ( • ) 'II'

p (.)

182 20 25 81 3 20 25 20 25 85 1 2 9

Pre(.> Ps-F(.)

10 10

h

Ps-F (.) n Ps-F(.) r Ps-F ( •) s Ps-F( •)

TN TNor

45 101

Ps-F( •)

62

Ps-F(.)

254

43 30,199

9 10

T1ower

To

11,84 11,84 54 57 54 199

PF ( •) pl.(.) PI.e (.) Pr (.)

62 124

TR.

62 26,75 125

p;_F(.)

+

CD

9

10 10

-

Ps-F(.) I: a (.)

I: (.) (I:,s) aB (.) I:d ( I:d, s) ae (.)

I:e (.) CD

I: (. ) I:: (.) I: (k) aR.(.) a R.e (.) a .tre ( ·)

I:n ( • )

I:~ (.) ao (.)

ap C.) ar (.)

136 136 10

107 107 36 9 9 9 106-107 107 5 5 9

are ( ·)

9

aw<. >

69 42 42

a6 a

'II'

Tab TAB T'-AB $k (€) '¥(A(H))

'¥(H) ljlk ljl(nl

10 10

10 20 2 100 20 69 17,18 18 5 106

.

41

50 BO 28 130 125 30 30

11-11

2 3

<7> a

1 2

+

5

( )

.

..,_ "" t

• <

[. 1

5,14

J.

8

e

12 12 12 12 12 12

8

t

<.,. > s

v (.) *

13 13 13,59-60 13,75 13,76 18 22 42

l-Ip 1-11

I· I 1- 12 [. , • 1

"" a [T1s [T1w

75 75 75 76 84

97,100 203 203

255