Table Of Content

A 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.