Table Of ContentDomingo A Herrero
Arizona State University
Approximation of
Hilbert space. operators
VOLUME I
Pitman Advanced Publishing Program
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© 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)
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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
