Table Of ContentOTtO:
Operator Theory: Advances and Applications
to
Vol.
Edited by
I. Gohberg
Editorial Board
K. Clancey B. Gramsch M. Livsic
L. Coburn W. Helton E. Meister
R. Douglas D. Herrero B. Mityagin
H.Dym M. Kaashoek 1. Pincus
A. Dynin T. Kailath M. Rosenblum
P. Fillmore H. Kaper 1. Rovnjak
C. Foias S. Kuroda D. Sarason
P. Fuhrman P. Lancaster H. Widon
S. Goldberg L. Lerer D.Xia
Honorary and Advisory
Editorial Board
P. Halmos R. Phillips
T. Kato B. Sz.-Nagy
S. Mikhlin
Editorial Office
Department of Mathematics
Tel-Aviv University
Ramat-Aviv (Israel)
Springer Basel AG
DaoxingXia
Spectral Theory of
Hyponormal Operators
1983
Springer Basel AG
Library of Congress Cataloging in Publication Data
Spectral theory of hyponormal operators.
(Operator theory ; v. 10)
Includes bibliographical references and indexes.
1. Hyponormal operators. 2. Spectral theory
(Mathematics) I. Title. II. Series.
QA329.2.X5 1983 515.7'246 83-17913
CIP-Kurztitelaufnabme der Deutscben Bibliotbek
Hsia, Tao-bsing:
Spectral theory of hyponormal operators / Daoxing
Xia. - Basel ; Boston ; Stuttgart : Birkhluser,
1983.
(Operator theory; Vol. 10)
NE:GT
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No part of this publication may be reproduced, stored in a retrieval system, or
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copying, recording or otherwise, without the prior permission of the copyright
owner.
© 1983 Springer Basel AG
Originally published by Birkhiiuser Verlag Basel in 1983.
Softcover reprint of the hardcover 1st edition 1985
ISBN 978-3-0348-5437-5 ISBN 978-3-0348-5435-1 (eBook)
DOI 10.1007/978-3-0348-5435-1
Dedicated to my wife Chen/an Pan
VII
TABLE OF CONTENTS
PREFACE
CHAPTER 1. Elementary properties ~f hyponormal operators
and semi-hyponormal operators 1
1. Introduction and definitions 1
1.1 Introduction 1
1.2 Definition of hyponormal operators 2
1.3 Definition of semi-hyponormal operators 3
1.4 Semi-hyponormality of hyponormal operators 4
1.5 The Cayley transform 6
2. Some elementary properties of hyponormal and semi-
hyponormal operators 6
2.1 Some elementary properties 6
2.2 The joint point spectrum 9
2.3 The joint approximate point spectrum 11
2.4 Berberian's technique 15
2.5 Completely nonnormal operators 18
3. Homotopy properties of the spectrum and the
spectral cutting 19
3.1 A homotopy property of the spectrum 19
3.2 The angular cutting of the spectrum 21
3.3 The rectangular cutting of the spectrum 24
3.4 The spectral mapping for similarity
transformations 25
CHAPTER 2. Symbols 27
1. The definition of symbols and their basic
properties 27
1.1 Definition and elementary properties 27
1.2 Decomposition properties of the spectrum 31
2. The symbols of a hyponormal operator and the
polar symbols of a semi-hyponormal operator 32
2.1 Definition of r operators and their
relations with symbols 32
VIII
2.2 Sufficient conditions for the existence of
the symbols 33
2.3 Symbols of hyponormal operators 36
3. The projection of the spectrum and the spectral
radius 38
3.1 Rectangular and polar decomposition of the
spectrum 38
3.2 Spectral radius 41
3.3 Spectral mapping theorem for the Cayley
transformation 44
CHAPTER 3. Singular integral models 47
1. A class of singular integral operators 47
1.1 The space of vector-valued square-integrable
functions 47
1.2 Vector-valued Fourier series 50
1.3 A class of semi-hyponormal operators 52
1.4 Singular integrals with Cuachy kernel on
the real line 55
1.5 A class of hyponormal operators 56
2. Function mOdels of unitary operators and of
operators commuting with a unitary operator 58
2.1 Function models of unitary operators 58
2.2 Function models of operators commuting
with a unitary operator 61
3. Models of operators in SHU and hyponormal
operators 65
3.1 The singular integral model of an operator
in SHU 65
3.2 Model of a hyponormal operator 68
4. The function model of a semi-hyponormal operator 72
4.1 Model of unilateral shifts 72
4.2 Singular integral model 77
CHAPTER 4. Relations between the spectra of semi-hyponormal
operators and those of the general polar symbols 80
1. Spectra of the general symbols 80
1.1 The spectrum of a normal multiplication
operator 80
1.2 Union of spectra of the general symbols 82
1.3 An estimate for a class of resolvents 83
IX
2. Some lemmas 84
2.1 Properties of vector-valued measurable
functions 84
2.2 Properties of sets with positive measure 85
2.3 A dense subset in the space of square
integrable functions 86
2.4 Point spectrum (Jp(T*) in the case
of one-dimensional ~ 87
3. Spectra of hyponormal operators 89
3.1 Statement of the theorem and the proof for
(J(T) c reT) 89
3.2 Proof of reT) C (J(T) 91
4. Spectra of semi-hyponormal operators 95
CHAPTER 5. Mosaics and characteristic functions 98
1. Riemann-Hilbert problems 98
1.1 Determining functions 98
1.2 The analytic function R(.,.) 100
1.3 Riemann-Hilbert problem 100
1.4 Semi-hyponormal operators case 101
2. The Mosaics 102
2.1 Some lemmas 102
2.2 The mosaic of a hyponormal operator 105
2.3 Mosaic of a semi-hyponormal operator 108
3. Determining sets and Putnam inequality 109
3.1 The determining set of a mosaic 109
3.2 Putnam inequality 113
3.3 Rectangular cutting of a hyponormal
operator 115
4. Characteristic functions 118
4.1 Another class of determining functions 118
4.2 Characteristic functions 119
5. Toeplitz operator related to a semi-hyponormal
operator 121
5.1 Toeplitz operators 121
5.2 Operator defined by the characteristic
function 123
x
CHAPTER 6. Spectral mapping 127
1. Functional transformations of hyponormal
operators 127
1.1 A class of functional transformations 127
1.2 Some class of functions 128
1.3 Functional transformation of hyponormal
operators 130
2. Spectral mapping theorems of hyponormal operators 131
2.1 Statement of the problem 131
2.2 Some lemmas 132
2.3 Spectral mapping theorems 133
3. Spectral mapping theorems of semi-hyponormal
operators 138
3.1 Some classes of functional transformations 139
3.2 Spectral mapping theorems 140
3.3 Another spectral mapping theorem 143
3.4 Angular cutting of semi-hyponormal operators 149
4. Estimate of resolvents 152
4.1 Estimate of resolvents of operators given
by functional transformations of a hypo-
normal operator 152
4.2 Estimate of resolvent of a semi-hyponormal
operator 153
5. Quasi-hyponormal operators 155
5.1 Generalization of the Putnam inequality 155
5.2 Class of quasi-hyponormal operators 156
5.3 Properties of quasi-hyponormal operators 158
CHAPTER 7. Pincus principal functions, traces and
determinants 160
1. Traces 160
1.1 The definition of the trace 160
1.2 Properties of the trace 161
2. Pincus principal functions and the trace formula 164
2.1 Hyponormal operators case 164
2.2 Semi-hyponormal operators 170
3. The trace formula of a ·nearly normal operator 170
3.1 Collapsing bilinear functionals 170
