Table Of ContentA Short Introduction
to Perturbation Theory
for Linear Operators
Tosio Kato
A Short Introduction
to Perturbation Theory
for Linear Operators
Springer-Verlag
New York Heidelberg Berlin
Tosio Kato
Department of Mathematics
University of California
Berkeley, CA 94720
USA
AMS Subject Classifications (1980): 46-01, 34-01, 34010, 47-01, 47A55
Library of Congress Cataloging in Publication Data
Kato, Tosio, 1917-
A short introd~ction to perturbation theory for linear operators
"Slightly expanded reproduction of the first two chapters (plus introduction) of ...
Perturbation theory for linear operators" - Pref.
Bibliography: p.
Includes index.
1. Linear operators. 2. Perturbation
(Mathematics) I. Title
QA329.2.K3725 1982 515.7'246 82-10505
This book is a slightly expanded version of Chapters 1 and 2 of:
T. Kato, Perturbation Theory for Linear Operators 2nd ed.
(Grundlehren der mathematischen Wissenschaften 132).
Springer-Verlag, Berlin, 1980.
C 1982 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover I st edition 1982
All rights reserved.
No part of this book may be translated or reproduced in any form without written
permission from Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, USA.
Typesetting by Briihlsche Universitatsdruckerei (FRG).
Printing and binding by R. R. Donnelley & Sons, Harrisonburg, VA.
9 8 7 6 5 4 3 2 1
ISBN-13: 978-1-4612-5702-8 e-ISBN-13: 978-1-4612-5700-4
DOl: 10.1007/978-1-4612-5700-4
Preface
This book is a slightly expanded reproduction of the first two chapters
(plus Introduction) of my book Perturbation Theory tor Linear Operators,
Grundlehren der mathematischen Wissenschaften 132, Springer 1980.
Ever since, or even before, the publication of the latter, there have been
suggestions about separating the first two chapters into a single volume.
I have now agreed to follow the suggestions, hoping that it will make
the book available to a wider audience.
Those two chapters were intended from the outset to be a comprehen
sive presentation of those parts of perturbation theory that can be treated
without the topological complications of infinite-dimensional spaces. In
fact, many essential and. even advanced results in the theory have non
trivial contents in finite-dimensional spaces, although one should not
forget that some parts of the theory, such as those pertaining to scatter
ing. are peculiar to infinite dimensions.
I hope that this book may also be used as an introduction to linear
algebra. I believe that the analytic approach based on a systematic use
of complex functions, by way of the resolvent theory, must have a strong
appeal to students of analysis or applied mathematics, who are usually
familiar with such analytic tools.
In addition to minor local improvements and modifications throughout
the two chapters, the following new sections and paragraphs have been
added in the new version: in Chapter One. § 4.7 on product formulas,
§ 6.11 on dissipative operators and contraction semigroups, and § 7 on
positive matrices; in Chapter Two, §§ 4.7-4.9 on the extended treatment
of analytic perturbation theory, § 6.6 on nonsymmetric perturbation of
symmetric operators, and § 7 on perturbation of (essentially) nonnegative
matrices.
The numbering of chapters, sections, paragraphs, theorems, lemmas,
etc. remains the same as in the larger volume. Thus "I~§ 2.3" denotes the
third paragraph of the second section of Chapter One, and "Lemma 1-2.3"
is Lemma 2.3 of Chapter One; the chapter number is omitted when
referred to within the same chapter. Due to the particular genesis of the
new version, however, some irregularities occur in the numbering of
newly introduced theorems, lemmas, etc. Thus, for example, II-§ 5.7
contains new Theorems s.13a-s.13c and a new Remark S.13d.
References are given by numbers in square brackets, such as (1) for
books and [1] for monographs. They retain the same numbers as in the
larger volume (unless they are new), although only those referred to in
VI Preface
the present volume are listed in the Bibliography. Thus, for example,
only the papers [1,2, 3, 6, 9, 12, 13J are listed under T. Kato.
Another irregularity is that in a few places, particularly in the Intro
duction, there remain references to the later chapters not contained in
the present version. But I thought it more convenient to retain, rather
than eliminate, such formal imperfections.
I would like to thank the staff of Springer-Verlag for their encourage
ment and help in conceiving this new book.
Berkeley TOSIO KATO
January, 1982
Contents
page
Introduction XI
Chapter One
Operator theory in finite-dimensional vector spaces
§ 1. Vector spaces and normed vector spaces 1
1. Basic notions 1
2. Bases 2
3. Linear manifolds 3
4. Convergence and norms 4
5. Topological notions in a normed space 6
6. Infinite series of vectors 7
7. Vector-valued functions 8
§ 2. Linear forms and the adjoint space . 10
1. Linear forms 10
2. The adjoint space 11
3. The adjoint basis 12
4. The adjoint space of a normed space. 13
5. The convexity of balls 14
6. The second adjoint space. 15
§ 3. Linear operators . 16
1. Definitions. Matrix representations 16
2. Linear operations on operators 18
3. The algebra of linear operators 19
4. Projections. Nilpotents . 20
5. Invariance. Decomposition 22
6. The adjoint operator. 23
§ 4. Analysis with operators . 25
1. Convergence and norms for operators 25
2. The norm of 1'" . 27
3. Examples of norms 28
4. Infinite series of operators 30
5. Operator-valued functions 32
6. Pairs of projections 33
7. Product formulas 35
§ 5. The eigenvalue problem. 36
1. Definitions 36
2. The resolvent . 38
3. Singularities of the resolvent 40
4. The canonical form of an operator . 42
S. The adjoint problem. 45
6. Functions of an operator . 46
7. Similarity transformations 48
VIII Contents
§ 6. Operators in unitary spaces 48
1. Unitary spaces 48
2. The adjoint space . . 50
3. Orthonormal families 51
4. Linear operators 52
5. Symmetric forms and symmetric operators 54
6. Unitary, isometric and normal operators 55
7. Projections . . . . . . 57
8. Pairs of projections 58
9. The eigenvalue problem 60
10. The minimax principle. 62
11. Dissipative operators and contraction semigroups 64
§ 7. Positive matrices ..,........... 65
1. Definitions and notation . . . . . . . . . . 65
2. The spectral properties of nonnegative matrices 66
3. Semigroups of nonnegative operators. 68
4. Irreducible matrices . . . . 69
5. Positivity and dissipativity. . . . . 70
Chapter Two
Perturbation theory in a finite-dimensional space 72
§ 1. Analytic perturbation of eigenvalues 72
1. The problem . . . . . . . . 72
2. Singularities of the eigenvalues . 74
3. Perturbation of the resolvent . . 76
4. Perturbation of the eigenprojections and eigennilpotents 77
5. Singularities of the eigenprojections 80
6. Remarks and examples. . . 81
7. The case of T(,,) linear in" 83
8. Summary 84
§ 2. Perturbation series 85
1. The total projection for the A-group 85
2. The weighted mean of eigenvalues . 88
3. The reduction process . . . . . . 92
4. Formulas for higher approximations 94
5. A theorem of MOTZKIN-TAUSSKY 96
6. The ranks of the coefficients of the perturbation series 97
§ 3. Convergence radii and error estimates 99
1. Simple estimates . . . . . . . 99
2. The method of majorizing series. 100
3. Estimates on eigenvectors 102
4. Further error estimates 104
5. The special case of a normal unperturbed operator. 105
6. The enumerative method . . . . . . . . . . . . 108
§ 4. Similarity transformations of the eigenspaces and eigenvectors. 109
1. Eigenvectors . . . . . . . . . . 109
2. Transformation functions. . . . . . . . . . . . . 110
3. Solution of the differential equation . . . . . . . . 114
4. The transformation function and the reduction process 115
Contents IX
5. Simultaneous transformation for several projections 116
6. Diagonalization of a holomorphic matrix function 117
7. Geometric eigenspaces (eigenprojections) . . . . . 118
8. Proof of Theorems 4.8,4.9 . . . . . . . . . . . 120
9. Remarks on projection families and transformation functions 122
§ 5. Non-analytic perturbations . . . . . . . . . . . . . 123
1. Continuity of the eigenvalues and the total projection 123
2. The numbering of the eigenvalues . . . . . . 125
3. Continuity of the eigenspaces and eigenvectors 127
4. Differentiability at a point . . . . . . . . . 128
5. Differentiability in an interval .... . . . 130
6. Asymptotic expansion of the eigenvalues and eigenvectors 132
7. Operators depending on several parameters . 133
8. The eigenvalues as functions of the operator 136
§ 6. Perturbation of symmetric operators . . . . . 138
1. Analytic perturbation of symmetric operators 138
2. Orthonormal families of eigenvectors. . . . 140
3. Continuity and differentiability . . . . . . 141
4. The eigenvalues as functions of the symmetric operator 143
5. Applications. A theorem of LIDSKII . . . . . . . 143
6. Nonsymmetric perturbation of symmetric operators 145
§ 7. Perturbation of (essentially) nonnegative matrices 146
1. Monotonicity of the principal eigenvalue 146
2. Convexity of the principal eigenvalue 147
Bibliography . 149
Notation index 153
Author index. 155
Subject index 157
Introduction
Throughout this book, "perturbation theory" means "perturbation
theory for linear operators". There are other disciplines in mathematics
called perturbation theory, such as the ones in analytical dynamics
(celestial mechanics) and in nonlinear oscillation theory. All of them
are based on the idea of studying a system deviating slightly from a
simple ideal system for which the complete solution of the problem
under consideration is known; but the problems they treat and the tools
they use are quite different. The theory for linear operators as developed
below is essentially independent of other perturbation theories.
Perturbation theory was created by RAYLEIGH and SCHRODINGER (d.
SZ.-NAGY [1]). RAYLEIGH gave a formula for computing the natural fre
quencies and modes of a vibrating system deviating slightly from a
simpler system which admits a complete determination of the frequencies
and modes (see RAYLEIGH [1), §§ 90, 91). Mathematically speaking,
the method is equivalent to an approximate solution of the eigenvalue
problem for a linear operator slightly different from a simpler operator
for which the problem is completely solved. SCHRODINGER developed a
similar method, with more generality and systematization, for the
eigenvalue problems that appear in quantum mechanics (see SCHRODIN
GER [ID, [IJ).
These pioneering works were, however, quite formal and mathe
matically incomplete. It was tacitly assumed that the eigenvalues and
eigenvectors (or eigenfunctions) admit series expansions in the small
parameter that measures the deviation of the "perturbed" operator
from the "unperturbed" one; no attempts were made to prove that the
series converge.
It was in a series of papers by RELLICH that the question of convergence
was finally settled (see RELLICH [1]-[5J; there were some attempts at the
convergence proof prior to RELLICH, but they were not conclusive; see e. g.
WILSON [lJ). The basic results of RELLICH, which will be described in
greater detail in Chapters II and VII, may be stated in the following
way. Let T(,,) be a bounded selfadjoint operator in a Hilbert space H,
depending on a real parameter" as a convergent power series
+ " + ,,2 + ....
(1) T(,,) = T T(l) T(S)
Suppose that the unperturbed operator T = T (0) has an isolated eigen
value A. (isolated from the rest of the spectrum) with a finite multi
plicity m. Then T(,,) has exactly m eigenvalues Pi(")' j = 1, ..., m
XII Introduction
(multiple eigenvalues counted repeatedly) in the neighborhood of A for
sufficiently small ,,,', and these eigenvalues can be expanded into con
vergent series
(2) 1'1(") = A + "1'~1) + ,,1I1'~1) + ... , j = 1, " . "' m.
The associated eigenvectors CPi (,,) of T (,,) can also be chosen as con
vergent series
(3) CPi (,,) = CPI + " cp~l) + "I cp~a) + ... , j = 1, " . "' m ,
satisfying the orthonormality conditions
(4)
where the CPI form an orthonormal family of eigenvectors of T for the
eigenvalue A.
These results are exactly what were anticipated by RAYLEIGH, SCHRO
DINGER and other authors, but to prove them is by no mean simple.
Even in the case in which H is finite-dimensional, so that the eigen
value problem can be dealt with algebraically, the proof is not at all
trivial. In this case it is obvious that the 1'1(") are branches of aIge
broidal functions of ", but the possibility that they have a branch
,,=
point at 0 can be eliminated only by using the selfadjointness of
T(,,). In fact, the eigenvalues of a selfadjoint operator are real, but a
function which is a power series in some fractional power ~/' of" cannot
be real for both positive and negative values of ", unless the series reduces
to a power series in ". To prove the existence of eigenvectors satisfying
(3) and (4) is much less simple and requires a deeper analysis.
Actually RELLICH considered a more general case in which T (,,) is an
unbounded operator; then the series (1) required new interpretations,
which form a substantial part of the theory. Many other problems related
to the one above were investigated by RELLICH, such as estimates for the
convergence radii, error estimates, simultaneous consideration of all the
eigenvalues and eigenvectors and the ensuing question of uniformity,
and non-analytic perturbations.
RELLICH'S fundamental work stimulated further studies on similar
and related problems in the theory of linear operators. One new de
velopment was the creation by FRIEDRICHS of the perturbation theory of
continuous spectra (see FRIEDRICHS [2]), which proved extremely
important in scattering theory and in quantum field theory. Here an
entirely new method had to be developed, for the continuous spectrum
is quite different in character from the discrete spectrum. The main
problem dealt with in FRIEDRICHS'S theory is the similarity of T(,,) to T,
that is, the existence of a non-singular operator W(,,) such that T(,,)
= We,,) TW(,,)-l.
