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10.1090/mmono/018
Translations o f
MATHEMATICAL
MONOGRAPHS
Volume 18
Introduction t o th e
Theory o f Linear
Nonselfadjoint Operator s
in Hilbert Spac e
I. C. Gohberg
M. G. Krem
American Mathematical Societ y
Providence, Rhode Island
k/WDEO
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M. u. roxBEPr
M. T. KPEflH
BBE^EHHE B TEOPHIO JIHHEttHMX
HECAMOCOIIPHJKEHHblX OnEPATOPO B
B THJIBEPTOBHOM nPOCTPAHCTB E
Translated from the Russian by
A. Feinstein
H3flaTejibCTBO «Hayica»
TjiaBHaH PeaaKUHH
OH3HKo-MaTeMaTHHecKott JlHTepaTypw
MocKBa 1965
2000 Mathematics Subject Classification. Primar y 47-02;
Secondary 47Bxx, 47D50.
Library of Congress Card Number 67-22348
Standard Book Number 0-8218-1568-7
International Standard Serial Number 0065-9282
Copying and reprinting. Individua l readers of this publication, and nonprofit libraries acting
for them, are permitted to make fair use of the material, such as to copy a chapter for use
in teaching or research. Permissio n is granted to quote brief passages from this publication in
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
(including abstracts) is permitted only under license from the American Mathematical Society.
Requests for such permission should be addressed to the Assistant to the Publisher, American
Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Request s can also
be made by e-mail to reprint-penaissionfiams.org.
© Copyright 1969 by the American Mathematical Society. All rights reserved.
Translation authorized by the
Ail-Union Agency for Author's Rights, Moscow.
Printed in the United States of America.
The American Mathematical Society retains all rights
except those granted to the United States Government.
@ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at URL: http://www.ams.org/
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TABLE OF CONTENTS
Introduction xii i
Chapter I. Genera l Theorems on Bounded Nonselfadjoint Operators . . 1
§1. Notatio n and some known results 1
§2. Norma l points of a bounded operator 8
§3. Stabilit y of the root multiplicities 1 3
§4. Som e spectral properties of completely continuous operators . . 15
§5. A theore m o n holomorphi c operator-function s an d it s
corollaries 1 9
Chapter II. s-Numberso f Completely Continuous Operators 2 4
§1. Minima x propertie s o f th e eigenvalue s o f selfadjoin t com -
pletely continuous operators 2 4
§2. 5-number s o f completel y continuou s operator s an d thei r
simplest properties 2 6
§3. Inequalitie s relating the s-numbers, eigenvalues and diagonal
elements of completely continuous operators 3 3
§4. Inequalitie s fo r th e s-number s o f sum s an d product s o f
completely continuous operators 4 6
§5. Som e generalizations of the preceding inequalities 5 1
§6. Inequalitie s fo r th e eigenvalue s o f linea r operator s wit h
completely continuous imaginary component 5 6
§7. s-number s of bounded operators 5 9
Chapter III. Symmetricaily-norme d Ideal s of the Rin g of Bounde d
Linear Operators . 6 5
§1. Two-side d ideals of the ring of bounded linear operators 6 6
§2. Symmetricaily-norme d ideals 6 8
§3. Symmetri c norming functions 7 1
§4. Symmetricaily-norme d ideal s generate d b y a symmetri c
norming function 8 0
§5. A criterion for an operator to belong to an s.n. ideal ©* 8 3
§6. Separabl e s.n. ideals 8 7
§7. The symmetricaily-normed ideals © 9 1
p
§8. Nuclea r operators 9 5
§9. Hilbert-Schmid t operators 10 6
V
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CONTENTS
VI
§10. Tests for the nuclearity o f integral operator s an d formula s
for calculating the trace 11 2
§11. Function s adjoint to s.n. functions 12 5
§12. Symmetrically-normed ideal s adjoin t t o separabl e sym -
metrically -normed ideals 12 8
§13. The three lines theorem for operator-functions which wander
in © spaces 13 6
p
§14. The symmetrically-normed ideals ©„ and©}?' 13 9
§15. The symmetrically-normed ideal s © , an d thei r connectio n
with ©,, and ©,(,0) 14 5
§16. Another interpolation theorem 15 0
§17. Conical norms in the real Banach spaces of operators @ 15 2
Chapter IV. Infinit e Determinants and Related Analytic Methods . . 156
§1. The characteristic determinant of a nuclear operator 15 6
§2. Regularize d characteristi c determinant s fo r th e operator s
from© 16 6
p
§3. Perturbatio n determinants 17 1
§4. A lemma o n th e growt h o f th e perturbatio n determinan t
of a dissipative operator 17 5
§5. A theorem on the perturbation determinant of a dissipative
operator 17 8
§6. The determinants D . (\) an d D (\) fo r a dissipativ e
AiA AgttA
operator A with nuclear imaginary component 18 0
§7. Dissipativ e Volterr a operator s wit h nuclea r imaginar y
component 18 3
§8. Nondissipative operators with nuclear imaginary component . 188
§9. An asymptoti c propert y o f th e spectru m o f a n operato r
with nuclear imaginary component 19 8
§10. A theorem o n Volterra operators with a finite-dimensional
imaginary component 20 3
§11. Furthe r theorem s o n relation s betwee n th e Hermitia n
components of Volterra operators 20 9
Chapter V. Theorem s on the Completeness of the System of Root
Vectors 22 2
§1. Lemma s on dissipative operators 22 2
§2. Tests for the completeness of the system of root vectors for
dissipative operators with nuclear imaginary component.... 22 6
§3. Tests for the completeness of the system of roo t vector s of
a contraction operator 23 1
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CONTENTS vi i
§4. Theorem s o n test s fo r th e completenes s o f th e system s o f
root vector s of dissipative operator s with nuclea r imaginar y
component 23 7
§5. Estimatio n of th e growt h o f th e resolvent s o f operator s o f
various classes 24 2
§6. Theorem s on the completeness of the systems of root vector s
of higher classes of operators 24 5
§7. Tw o lemmas on the resolvents of normal operators 25 2
§8. Theorem s on the completeness o f the system of root vector s
of a weakly perturbed selfadjoint operator 25 6
§9. Theorem s o n th e multipl e completenes s o f th e syste m o f
eigenvectors and associated vectors of an operator bundle . . . 265
§10. Test s fo r th e completenes s o f th e system s o f roo t vector s
of unbounded operators 27 5
§11. Asymptoti c properties of the spectrum of a weakly perturbe d
positive operator 27 8
§12. Selfadjoin t quadratic bundles 29 1
Chapter VI . Bases . Test s fo r th e Existenc e o f Bases , Consistin g
of Root Vectors of a Dissipative Operator 30 6
§1. Base s of a Hilbert space 30 6
§2. Base s equivalent to orthonormal bases (Riesz bases) 30 9
§3. Base s quadratically close to orthonormal bases (Bari bases) . . 319
§4. Test s for the existence of a basis, consisting o f eigenvector s
of a dissipative operator 32 8
§5. Basi s of subspaces 33 2
§6. Test s fo r th e existenc e o f a basi s consistin g o f th e roo t
subspaces of a dissipative operator 34 5
Bibliography 35 2
Subject Index 36 3
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AUTHORS' PREFACE TO THE ENGLISH EDITION
We were happy to learn that immediately after th e publication of our
book, the American Mathematical Society had undertaken the publication
of an English edition.
We wish to thank the translation editor, Dr. S. H.Gould, for the energy
and initiative which he has displayed.
We wish to express our gratitude to Dr. A. Feinstein, who took upon
himself the task of translating thi s book an d who has fulfilled i t with
considerable responsibility i n the shortest time. We owe to him, as well
as to Professor T. Ando, the correction o f a number o f inaccuracies, a
list of which would, t o ou r surprise, prov e t o b e rathe r long .
February 23,1967. I.C . Gohberg, M.G. Krein
ix
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