Domingo A Herrero Arizona State University Approximation of Hilbert space. operators VOLUME I Pitman Advanced Publishing Program BOSTON· LONDON· MELBOURNE PITMAN BOOKS LIMITED 128 Long Acre, London WC2E 9AN PITMAN PUBUSHING INC 1020 Plain Street, Marshfield, Massacllusetts Associllted Contpflllia Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto © Domingo A Herrero 1982 First published 1982 AMS Subject Classifications: Primary 47 ASS, 41A6S, 47 A60; 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 2 1.2 Lower estimates for the distance to Nk(H) 6 1.3 Lower semicontinuity of the rank 8 1.4 Stability properties of semi-Fredholm operators 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 16 2.1.3 The lattice (N(Ek)/i,<) 17 2.1.4 Closures of similarity orbits of finite rank operators 19 2.2 The distance from the set of all non-zero orthogonal projections to N(H) 20 2.2.1 The limit case 20 2.2.2 On the exact values of 6k and nk 23 2.2.3 A companion problem: the distance from the set of all non-zero ide!_11potents to N{ H) 25 2.3 On the distance to Nk{H) 27 2.3.1 A general upper bound 27 2.3.2 Two illustrative examples 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 35 2.4 On the distance from a compact operator to N(H) 37 2.5 Notes and remarks 38 3. The main tools of approximation 41 3.1 The Rosenblum operator: X + AX -XB 41 3.1.1 Linear operator equations 41 3.1.2 Approximate point spectrum of a sum of commuting operators 42 3.1.3 Local one-side resolvents in L(H) 44 3.1.4 The left and the right spectra of TAB 50 3.1.5 Rosenblum-Davis-Rosenthal corollary 53 3.1.6 The maximal numerical range of an operator 54 3.1.7 The norm of TAB 56 3.2 Generalized Rota's universal model 58 3.3 Apostol triangular representation 62 3.4 Correction by compact perturbations of the singular behavior of operators 69 3.5 Apostol-Foia~-Voiculescu's theorem on normal restrictions of compact perturbations of operators 74 3.5.1 Schatten p-classes 75 3.5.2 Normal restrictions 76 3.5.3 Density of sets of operators with bad properties 78 3.6 Notes and remarks 79 4. Two results borrowed from the theory of C*-algebras 84 4.1 Essentially normal operators 84 4.1.1 Brown-Douglas-Fillmore theorem 84 4.1.2 Berger-Shaw trace inequality 85 4.1.3 Examples of essentially normal operators 89 4.1.4 An application to approximation problems 90 4.2 Matrix models for operators 92 4.3 Spectra of compact perturbations of operators 94 4.4 Voiculescu's theorem 96 4.5 Closures of unitary orbits 99 4.5.1 Operator-valued spectrum and unitary orbits 99 4.5.2 Concrete examples of closures of unitary orbits 100 4.5.3 On normal and quasinilpotent restrictions 101 viii 4.6 Irreducible operators 104 4.7 Notes and remarks 106 5. Limits of nilpotent and algebraic operators 108 5.1 Limits of nilpotent operators 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 113 5.3 Limits of algebraic operators 114 5.4 Normal operators in closures of similarity orbits 115 5.5 Sums of two nilpo~ents 117 5.6 The Apostol-Salinas approach: an estimate for the distance to Nk(H) 120 5.7 Salinas' pseudonilpotents 124 5.8 Limits of nilpotent and algebraic elements of the Calkin algebra 128 5.9 On the spectra of infinite direct sums os operators 130 5.10 Notes and remarks 132 6. Quasitriangularity 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 141 6.2.2 Two lower estimates for the distance to (QT) 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 154 6.6 Notes and remarks 157 7. The structure of a polynomially compact operator 162 7.1 Reduction to the (essentially) nilpotent case 162 7.2 The structure of a polynomially compact operator 164 ix 7.3 Restrictions of nilpotent operators 167 7.4 Operators similar to Jordan operators 171 7.5 A similarity invariant for polynomially compact operators 173 7.6 Nice Jordan operators 177 7.7 Notes and remarks 188 8. Closures of similarity orbits of nilpotent operators 189 8.1 Universal operators 189 8.1.1 Universal quasinilpotent operators 189 8.1.2 Universal compact quasinilpotent operators 194 8.2 Compact perturbations of not nice operators 194 8.3 Quasinilpotents in the Calkin algebra 198 8.3.1 General quasinilpotents 198 8.3.2 Nice elements of the Calkin algebra 204 8.4 Compact perturbations of nice Jordan operators 205 8.4.1 Nice Jordan nilpotents 206 8.4.2 Nilpotents of order 2 211 8.4.3 Quasinilpotent perturbations 212 8.4.4 Universal operators in N~,h(H) 215 8.4.5 A general criterion for universality 222 8.5 Separation of isolated points of the essential spectrum affiliated with nilpotents 230 8.6 Notes and remarks 236 REFERENCES 239 INDEX 249 SYMBOLS AND NOTATION 253 X 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 in variant under similarities. Many interesting subsets of L(HI have this property: nilpotent operators: algebraic operators (satisfying a fixed polynomial): poly nomially compact operators: triangular, quasitriangular and biquasi triangular 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 com pact operators, etc, etc. The following list illustrates the kindsofproblems to be consid ered 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 alge braic 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 sub sets 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