Spectral Theory of Non-Self-Adjoint Two-Point Differential Operators

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John Locker

American Mathematical Society Editorial Board Georgia Benkart Michael Loss Peter Landweber Tudor Ratiu, Chair

1991 Mathematics Subject Classification. Primary 34L05, 47E05; Secondary 34B27, 34L10, 34L20, 47A53.

ABSTRACT. This monograph develops the spectral theory of an nth order non-self-adjoint two- point differential operator L in the complex Hilbert space L2[0,1]. The differential operator L is determined by an nth order formal differential operator £ and by n linearly independent boundary values B\,... , Bn- The mathematical foundation is laid in the first part, Chapters 1-2, where the spectral theory is developed for closed linear operators and Fredholm operators in Hilbert spaces. An important completeness theorem is established for the Hilbert-Schmidt discrete operators. The operational calculus plays a major role in this general theory. In the second part, Chapters 3-6, the spectral theory of the differential operator L is developed. Expressing L in the form L = T + S, where T is the principal part of L determined by the nth order derivative and S is the part determined by the lower order derivatives, the spectral theory of T is developed first using operator theory, and then the spectral theory of L is developed by treating L as a perturbation of T. The spectral theory of L closely mirrors that of its principal part T. Regular and irregular boundary values are allowed for T, while only regular boundary values are considered for L. The main features of the spectral theory for L and T include the following: asymptotic formulas for the characteristic determinant and Green's function; classification of the boundary values as either regular, irregular, or degenerate; calculation of the eigenvalues and the corresponding algebraic multiplicities and ascents; calculation of the associated family of projections, which project onto the generalized eigenspaces; growth rates for the resolvent, thereby demonstrating the completeness of the generalized eigenfunctions; uniform bounds on the family of all finite sums of the associated projections; and expansions of functions in L2[0,1] in series of generalized eigenfunctions of L and T.

Library of Congress Cataloging-in-Publication Data Locker, John. Spectral theory of non-self-adjoint two-point differential operators / John Locker. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 73) Includes bibliographical references and index. ISBN 0-8218-2049-4 (alk. paper) 1. Nonselfadjoint operators. 2. Spectral theory (Mathematics) I. Title. II. Series. QA329.2.L65 1999 515'.7246~dc21 99-44328 CIP

Copying and reprinting. Individual 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. Permission 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 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. Requests can also be made by e-mail to reprint-permissionOams.org. © 2000 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ 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/ 10 9 8 7 6 5 4 3 2 1 05 04 03 02 01 00 Dedicated to my father and mother,

Harold Roy Locker and Helen Jeanette Locker Contents

Preface ix Chapter 1. Unbounded Linear Operators 1 1. Introduction 1 2. Closed Linear Operators 5 3. Analytic Vector-Valued Functions 9 4. Spectral Theory 21 5. Poles of the Resolvent 35 Chapter 2. Fredholm Operators 41 1. Basic Properties 41 2. Spectral Theory for Fredholm Operators 44 3. Spectral Theory for Index Zero 58 4. Hilbert-Schmidt Operators 64 5. Quasi-Nilpotent Hilbert-Schmidt Operators 70 6. A Hilbert-Schmidt Completeness Theorem 78 Chapter 3. Introduction to the Spectral Theory of Differential Operators 83 1. An Overview 83 2. Sobolev Spaces 87 3. The Characteristic Determinant and Eigenvalues 89 4. Algebraic Multiplicities 92 Chapter 4. Principal Part of a Differential Operator 97 1. The Principal Part T 97 2. The Characteristic Determinant of T 98 3. The Green's Function of XI - T 103 4. Alternate Representations 107 5. The Boundary Values: Case n = 2v 110 6. The Boundary Values: Case n = 2v — 1 118 7. The Eigenvalues: Case n = 2v 128 8. The Eigenvalues: Case n = 2v — 1 146 9. Completeness of the Generalized Eigenfunctions 181 Chapter 5. Projections and Generalized Eigenfunction Expansions 193 1. The Associated Projections: n — 2v 193 2. The Associated Projections: n = 2v — 1 201 3. Expansions in the Generalized Eigenfunctions 206 viii CONTENTS

Chapter 6. Spectral Theory for General Differential Operators 211 1. The Resolvents of T and L 211 2. The Operator SR\(T) and Completeness 213 3. Background Theory of Projections 217 4. The Spectral Theory of L: n = 2v, Case 1 225 5. The Spectral Theory of L: n = 2z/, Case 2 232 6. The Spectral Theory of L: n = 2u - 1, Case 1 239 Bibliography 247

Index 249 Preface

This monograph is a sequel to my earlier book on functional analysis and two-point differential operators [24]. In the previous work we developed the basic structure of an nth order differential operator L in the Hilbert space L2[a,b] that is determined by an nth order formal differential operator £ and by independent n boundary values Bi,... ,Bn defined on the Sobolev space H [a,b]. As such L has the structure of a Fredholm operator, as does the adjoint L*, which is the differential operator determined by the formal adjoint £* and by adjoint boundary values B{,... ,£*. The Green's function and generalized Green's function are characterized in the third and fourth chapters of [24]. The current work is divided into two parts, with Chapters 1 and 2 comprising the first part where the foundations of the spectral theory are laid in a general Hilbert space setting. Chapter 1 introduces the closed linear operators, analytic vector-valued and operator-valued functions, Cauchy's Theorem, and Taylor series and Laurent series expansions. For the special case of a bounded linear operator, the operational calculus is developed; it is one of major tools used to study the spectral theory. Turning to the spectral theory, we introduce the resolvent set, spectrum, and resolvent of a closed linear operator, illustrating these ideas with two-point differential operators. Since our emphasis is on the non-self-adjoint operators, we introduce the ascent and descent of an operator, and then the generalized eigenspace and algebraic multiplicity corresponding to an eigenvalue. Of special importance is the section on poles of the resolvent. Chapter 2 introduces the Fredholm operators, with differential operators again serving as models. Upon defining the nullity, defect, and index of a Fredholm operator and forming the generalized inverse, the basic theorems for products and perturbations are discussed, and the spectral theory is then studied in detail. The local and global behavior of the algebraic multiplicity and ascent are determined, and the spectrum is characterized. Special emphasis is placed on the spectral theory for index zero, where again poles of the resolvent play a major role. In this chapter the expansion problem for a vector in terms of the generalized eigenvectors is discussed for the first time. After reviewing the Hilbert-Schmidt operators, a very powerful completeness theorem is presented for the Hilbert-Schmidt discrete operators; this theorem is a key component of the second part of the monograph. Since most of the mathematics in the first part is well-known, we omit most proofs and simply give references to the literature. The sole exception is the material dealing directly with the spectral theory, where the results are presented in detail. We hope that this approach allows the reader to get more quickly to the main topic of this book: the spectral theory of non-self-adjoint two-point differential operators. The second part consists of Chapters 3 through 6, where the spectral theory of two-point differential operators is developed. In Chapter 3 the two-point differential

ix x PREFACE operator L is introduced in the Hilbert space L2[a, 6], and an overview of its spectral theory is given. Fundamental to this discussion is the Sobolev space Hn[a,b] and its associated Sobolev structure. Since L is a Fredholm operator of index zero, we are able to characterize the spectrum of L using the general results of Chapter 2. The characteristic determinant D is defined in its initial form, and the eigenvalues of L are shown to be the zeros of D. A key result is that the algebraic multiplicity of an eigenvalue is equal to its order as a zero of D. The spectral theory in Chapters 4-6 is set in the Hilbert space L2[0,1], and the differential operator L is expressed in the form

L = T + S, where T is the principal part of L determined by the nth order derivative and S is the part determined by the lower order derivatives. The differential operator T is of great interest in its own right; it serves as a model for the general spectral theory of differential operators. The spectral theory of T is established in Chapters 4 and 5. Included are the following topics: (i) Asymptotic formulas for the characteristic determinant A of T and for the Green's function G( •; •; A) of XI — T. These quantities are simpler when expressed in terms of the p variable where A = pn. (ii) Classification of the boundary values B\,... , Bn determining T as being either regular, irregular, or degenerate, depending on the form of the characteristic determinant. (iii) Calculation of the eigenvalues of T by calculating the zeros of A. For the eigenvalues asymptotic formulas are derived, and the corresponding algebraic multiplicities and ascents are determined. (iv) Calculation of the family of projections V associated with T, and formation of the corresponding subspaces Soc(T) and Moo(T). The projections in V map L2[0,1] onto the generalized eigenspaces of T, and the subspace S'oo(T) consists of all functions in 1? [0,1] that can be expressed in a series of generalized eigenfunctions of T. (v) Development of decay rates for the resolvent R\(T) along rays from the origin, thereby showing that

S^(T)=L2[0,1] and M^T) = {0}. This is accomplished using the completeness theorem of Chapter 2; it shows that the generalized eigenfunctions of T are complete in L2[0,1]; and it is valid for both regular and irregular boundary values. (vi) Demonstrating that the family of all finite sums of the projections in V is uniformly bounded in norm. Here it is assumed that the boundary values are regular. (vii) Establishing that S'00(T) is a closed subspace when the boundary values are regular, in which case each function in L2[0,1] can be expanded in a series of generalized eigenfunctions of the differential operator T. For the special case n = 2 with irregular boundary values, it is known that the projections in V are unbounded and S^iT} is a proper dense subspace of L2[0,1]. See [22, 23]. The situation for nth order T is unknown; we conjecture that it is identical to the second order case. PREFACE XI

Chapter 6 develops the spectral theory for the general differential operator L determined by regular boundary values. The spectral theory of L mirrors that of its principal part T. In developing this spectral theory perturbation techniques are used. Very little is known about the case of general L subject to irregular boundary values, but the case n — 2 has been recently analyzed in the series [26—29]. The spectral theory of two-point differential operators was begun by Birkhoff in his two papers [3, 4] of 1908, where he introduced regular boundary values for the first time. It was continued by Stone [38, 39] with the initial work on irregular boundary values, and by Hoffman [12] in his thesis which examined second order differential operators under irregular boundary values. Much of the spectral theory for regular boundary values is also given in Naimark [31]. In Chapter XIX of their treatise [6], Dunford and Schwartz give a modern operator theoretic development of the spectral theory for regular boundary values; it includes the L2-expansion of functions in terms of the generalized eigenfunctions. Benzinger [2] and Schultze [36] have studied Riesz summability of eigenfunction expansions in the case of special classes of irregular boundary values. These references are but a few in the extensive literature on the spectral theory of differential operators. They represent the work that the author is most familiar with, and that has most directly influenced his own research. Each of these references contains a bibliography which can be used as a guide to the literature (see especially [6, pp. 2371-2374]). The author's research in this area is contained in the references [18—30]; much of it is coauthored with Patrick Lang. Let us briefly discuss the relationship between Chapter XIX of Dunford and Schwartz [6] and this monograph. First, their treatment of the spectral theory of differential operators is based on the theory of unbounded spectral operators, which they earlier develop in Chapters XV-XVIII. They consider only regular boundary values. Our approach is based on the theory of Fredholm operators and on the characteristic determinant and the Green's function; it uses only basic operator theory. We consider not only regular boundary values, but also include the irregular boundary values wherever possible. Second, the multiple eigenvalue case is introduced in [6, p. 2324] as Case l.B where (3 = ±1, but it is never mentioned again. An explanation for this is given in [25] for the case of the formal differential operator — (d/dt)2 subject to regular boundary conditions. For a special class of regular boundary values, it is shown that the associated projections are unbounded, and hence, the theory of spectral operators can not be used in their study. However, by using a pairwise grouping of the projections, we are able to produce a family of uniformly bounded projections, and these differential operators do have a complete spectral theory which closely resembles that of spectral operators. These ideas are generalized here to include the nth order case (see Case 2 that appears in §7 of Chapter 4 and in §§1 and 3 of Chapter 5). Third, the completeness theory appearing on pp. 2334 and 2341 of [6] is incom plete because the basis functions <7/c(£, /i), k = 0,1,... , n — 1, are not bounded on the sectors A and Ai, A2. We correct this problem by altering the bases and sectors to produce the boundedness needed to estimate the Green's function and resolvent. In establishing completeness of the generalized eigenfunctions, we use Theorem 6.2 of Chapter 2. This result is Corollary XI.6.31 of [6], which surprisingly is never used by them in their treatment of completeness. xii PREFACE

Fourth, for the case n odd, we show that the Second Regularity Hypothesis For Odd Order Case [6, p. 2337] is a direct consequence of the First Regularity Hypothesis For Odd Order Case [6, p. 2336], thereby improving the results in [6]. We also make a slight improvement in the Regularity Hypothesis For Even Order Case [6, p. 2322]. For the case n even, we employ only one basis and one sector of angular opening 2n/n, so the theory does not have to be worked out twice on two different sectors of angular opening ir/n. Acknowledgments. I would like to take this opportunity to thank my friend and coauthor, Patrick Lang, for his many contributions to our joint work over the last seventeen years. His inspiration and hard work have had a profound influence on this monograph. And to my wife Georgia, my companion and friend of forty years, I express my sincerest thanks and gratitude. Your patience, positive attitude, and words of encouragement have made this book possible.

John Locker Bibliography

1. Shmuel Agmon, Lectures on elliptic boundary value problems, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1965. 2. Harold E. Benzinger, Green's function for ordinary differential operators, Journal of Differ ential Equations 7 (1970), 478-496. 3. George D. Birkhoff, Boundary value and expansion problems of ordinary linear differential equations, Transactions of the American Mathematical Society 9 (1908), 373-395. 4. , On the asymptotic character of the solutions of certain linear differential equations containing a parameter, Transactions of the American Mathematical Society 9 (1908), 219— 231. 5. Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York, 1955. 6. Nelson Dunford and Jacob T. Schwartz, Linear operators, I, II, III, Wiley-Interscience, New York, 1958, 1963, 1971. 7. I. C. Gohberg and M. G. Krein, The basic propositions on defect numbers, root numbers and indices of linear operators, Uspekhi Mat. Nauk. 12, 2(74) (1957), 43-118, Translated in Amer. Math. Soc. Transl., vol. 13, Ser. 2, 1960, pp. 185-264. 8. , Introduction to the theory of linear nons elf adjoint operators, Translations of Math ematical Monographs, vol. 18, American Mathematical Society, Providence, Rhode Island, 1969. 9. Seymour Goldberg, Unbounded linear operators: Theory and applications, McGraw-Hill Book Company, New York, 1966. 10. Paul R. Halmos, Finite-dimensional vector spaces, Springer-Verlag, New York, 1974. 11. G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, second ed., Cambridge University Press, Cambridge, 1952. 12. Stephen Hoffman, Second-order linear differential operators defined by irregular boundary conditions, Ph.D. thesis, Yale University, 1957. 13. Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cam bridge, 1985. 14. Shmuel Kaniel and Martin Schechter, Spectral theory for Fredholm operators, Communications on Pure and Applied Mathematics 16 (1963), 423-448. 15. L. V. Kantorovich and G. P. Akilov, Functional analysis, second ed., Pergamon Press, Oxford, 1982. 16. Tosio Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, Journal d'Analyse Mathematique 6 (1958), 261-322. 17. , Perturbation theory for linear operators, second ed., Springer-Verlag, Berlin, 1976. 18. Patrick Lang and John Locker, Spectral decomposition of a Hilbert space by a Fredholm op erator, Journal of Functional Analysis 79 (1988), 9-17. 19. , Spectral representation of the resolvent of a discrete operator, Journal of Functional Analysis 79 (1988), 18-31. 20. , Denseness of the generalized eigenvectors of an H-S discrete operator, Journal of Functional Analysis 82 (1989), 316-329. 21. , Spectral theory for a differential operator: Characteristic determinant and Green's function, Journal of Mathematical Analysis and Applications 141 (1989), 405-423. 22. , Spectral theory of two-point differential operators determined by —D2. I. Spectral properties, Journal of Mathematical Analysis and Applications 141 (1989), 538-558. 23. , Spectral theory of two-point differential operators determined by — D2. II. Analysis of cases, Journal of Mathematical Analysis and Applications 146 (1990), 148-191.

247 248 BIBLIOGRAPHY

24. John Locker, Functional analysis and two-point differential operators, Pitman Research Notes in Mathematics, vol. 144, Longmans, Harlow, Essex, 1986. 25. , The nonspectral Birkhoff-regular differential operators determined by — D2, Journal of Mathematical Analysis and Applications 154 (1991), 243-254. 26. , The spectral theory of second order two-point differential operators: I. A priori estimates for the eigenvalues and completeness, Proceedings of the Royal Society of Edinburgh 121A (1992), 279-301. 27. , The spectral theory of second order two-point differential operators: II. Asymptotic expansions and the characteristic determinant, Journal of Differential Equations 114 (1994), 272-287. 28. , The spectral theory of second order two-point differential operators: III. The eigen values and their asymptotic formulas, Rocky Mountain Journal of Mathematics 26 (1996), 679-706. 29. , The spectral theory of second order two-point differential operators: IV. The associ ated projections and the subspace Soo(L), Rocky Mountain Journal of Mathematics 26 (1996), 1473-1498. 30. John Locker and Patrick Lang, Eigenfunction expansions for the nonspectral differential op erators determined by —D2, Journal of Differential Equations 96 (1992), 318-339. 31. M. A. Naimark, Linear differential operators, I, GITTL, Moscow, 1954, English transl., Ungar, New York, 1967. 32. Bruce P. Palka, An introduction to complex function theory, Springer-Verlag, New York, 1991. 33. Walter Rudin, Real and complex analysis, third ed., McGraw-Hill, New York, 1987. 34. , Functional analysis, second ed., McGraw-Hill, New York, 1991. 35. Martin Schechter, Principles of functional analysis, Academic Press, New York, 1971. 36. Bernd Schultze, Strongly irregular boundary value problems, Proceedings of the Royal Society of Edinburgh 82A (1979), 291-303. 37. J. Schwartz, Perturbations of spectral operators, and applications. I. Bounded perturbations, Pacific Journal of Mathematics 4 (1954), 415-458. 38. M. H. Stone, A comparison of the series of Fourier and Birkhoff, Transactions of the American Mathematical Society 28 (1926), 695-761. 39. , Irregular differential systems of order two and the related expansion problems, Trans actions of the American Mathematical Society 29 (1927), 23-53. 40. , Linear transformations in Hilbert space and their applications to analysis, American Mathematical Society Colloquium Publications, vol. XV, American Mathematical Society, New York, 1932. 41. J. Tamarkin, Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions, Mathematische Zeitschrift 27 (1927), 1-54. 42. Angus E. Taylor and David C. Lay, Introduction to functional analysis, second ed., Krieger Publishing Company, Malabar, Florida, 1980. 43. E. C. Titchmarsh, The theory of functions, second ed., Oxford University Press, Oxford, 1939. 44. Philip W. Walker, A nonspectral Birkhoff-regular differential operator, Proceedings of the American Mathematical Society 66 (1977), 187-188. 45. S. Yakubov, Completeness of root functions of regular differential operators, Pitman Mono graphs and Surveys in Pure and Applied Mathematics, vol. 71, Longman Scientific & Techni cal, Harlow, Essex, 1994. Index

Adjoint operator, 2 Characteristic determinant A, 27, 98, 100, Algebraic multiplicity, 33 101-102, 107, 117, 128, 133, 136, 140, differential operator, 84, 86, 92-95, 98, 143, 146, 152, 155, 162, 168, 175, 193, 101 201, 212, 229-231, 237, 244 n even, 133, 134, 136, 146, 232, 238 asymptotic formula, 98, 115, 122 n odd, 152, 155, 168, 181, 245 n even, 100 Fredholm operator, 48, 56-58 n odd, 100 Ascent, 30, 31 order of zero, 101 differential operator, 84, 92, 98 zero of, 98, 100, 102 n even, 133, 134, 136, 146, 232, 238 Characteristic determinant Ao, 107, 108, n odd, 152, 155, 168, 181, 245 146, 148, 154, 157, 165, 171, 178, 201 Fredholm operator, 57-59 asymptotic formula, 126 Ascent factor Closed Graph Theorem, 6, 21, 26, 32, 79 Fredholm operator, 48, 54 Closed linear operator, 6 examples, 7 Boundary coefficient matrix, 97, 211 reduced by pair of subspaces, 28 Boundary values, 9, 83, 88, 97, 105, 109, 211 Closed Range Theorem, 8 adjoint, 9, 83 Closure of an operator, 8 degenerate, 98, 116, 118, 124 examples, 8 irregular, 87, 98, 111, 116, 119, 124, 128, Cofactor, 106, 109, 116, 123, 127 136, 146, 181 Compact linear operator, 34 n even, 110-118 ascent and descent, 34 n odd, 118-128 Completeness theorem, 64, 80, 185, 191, 217 normalized, 97, 117, 124, 211 Continuity of inversion, 19 order of, 97 Continuous spectrum, 21 regular, 87, 98, 111, 116, 119, 124, 128, Cramer's rule, 106, 109 146, 181, 193, 201, 206, 212, 213, 225, Curve 232, 239 closed, 13 B2{H), Hilbert-Schmidt operators, 64 rectifiable, 13 Banach algebra, 65, 66 simple, 13 inner product, 66 trace of, 13 Caratheodory's Inequality, 77 Cycle Carleman Inequality, 67, 68, 70, 74, 77, 80 homologous to zero, 20 finite-dimensional range, 71 surrounds the spectrum, 20 finite-dimensions, 69 quasi-nilpotent Hilbert-Schmidt operator, Defect, 41, 48, 51, 52 77, 78 Descent, 30, 31 Carleman theory, 64, 67 Differential operator, 8, 83, 211 Cauchy domain, 14, 18, 36, 71 adjoint operator, 83 Cauchy Integral Formula, 15 algebraic multiplicity, see Algebraic Cauchy's Theorem, 14, 18, 20, 71, 86 multiplicity, differential operator Chain basis, 92-95 ascent, see Ascent, differential operator Characteristic determinant D, 86, 91, basis for solution space, 91, 93, 94, 99, 92-95, 101, 102 101, 103, 107, 212 order of zero, 92, 95 boundary coefficient matrix, see zero of, 91 Boundary coefficient matrix 250 INDEX

boundary values, see Boundary spectrum, see Spectrum, differential values operator characteristic determinant, see Dunford, N., and J. T. Schwartz, 1, 64, 117 Characteristic determinant completeness of generalized eigenfunctions, Eigenspace, 33 98, 216, 217 differential operator, 92 n even, 181-185, 232, 239 Eigenvalue, 21 n odd, 185-191, 245 algebraic multiplicity, 33 eigenvalue, see Eigenvalue, differential differential operator, 91, 92, 95, 100, 101, operator 111, 119 examples, 3-5, 8, 26, 27, 46, 85, 218 apriori estimates, 130, 139, 149, 160, expansion in generalized eigenfunctions, 173 98 asymptotic formula, 98, 134, 136, 142, n 146, 155, 168, 181, 232, 238, 245 H -expansion, 210 n even, 128-146, 228-232, 235-239 L2-expansion, 207-209, 232, 239, 245 n odd, 146-181, 242-245 n even, 206-208 geometric multiplicity, 33 n odd, 208-209 Eigenvector, 21 family of projections associated with e-neighbor hood, 19 operator, 86 integral representation, 86, 194, 196, Formal adjoint differential operator, 9, 83 197, 203, 204, 226, 240 Formal differential operator, 9, 83, 87, 88, n even, 193-200, 206, 228, 232 97, 211 n odd, 201-206, 208, 242, 245 Fredholm operator, 28, 41 uniformly bounded, 98, 200, 206, 207, examples, 9, 45, 46 232, 239, 245 family of projections associated with formal differential operator, see operator, 60 Formal differential operator operator is T-bounded, 43, 44 Fredholm operator, 84, 88 operator is T-compact, 43, 44 Fredholm operator of index zero, 84, 85, perturbation of, 44 213 powers of, 43, 50-52 Fredholm set, 84 product of, 43 Green's function, see Green's function spectral family, 60 index, see Index, differential operator Fredholm set, 44, 56-58, 61, 63 logarithmic case, 136, 146, 168, 180, 183, Function of an operator, 18 184, 186, 190 maximal operator, 5, 84 Generalized eigenspace, 33 minimal operator, 5, 84 differential operator, 84, 92 modified family of projections associated Fredholm operator, 60 with operator, 195, 208, 236, 238 Hilbert-Schmidt discrete operator, 79 Generalized inverse, 42, 43 integral representation, 195, 233 Geometric multiplicity, 33 uniformly bounded, 200, 239 differential operator, 92 multiple eigenvalues, 135, 232, 238 Gohberg, I. C, and M. G. Krein, 46 n even, Case 1, 133, 134, 181-185, Goldberg, S., 1 194-200, 206-208, 213-215, 225-232 Graph norm, 6, 41, 43, 50, 88, 210 n even, Case 2, 135, 135-136, 181-185, Graph of an operator, 6 195, 199, 200, 208, 213-215, 232-239 Green's formula, 84 n even, Case 3, 136, 137-146, 181-185 Green's function, 85, 86, 103-110, 181, 183, n odd, Case 1, 146, 147-155, 185-191, 186, 187, 193, 201, 202, 213 201-206, 208, 209, 215-216, 239-245 asymptotic formula, 98, 183—184, n odd, Case 2, 155, 156-168, 185-191 188-189 n odd, Case 3, 168, 169-181, 185-191 partial derivatives, 214-215 principal part, 87, 97, 212, 217, 225, 232, 238, 239, 245 H-S discrete operator, 78 resolvent, see Resolvent, differential Hadamard's Inequality, 67, 74 operator Hellinger-Toeplitz Theorem, 3 Sobolev space, see Hn[a, 6], Sobolev space Hilbert-Schmidt discrete operator, 64, 78, spectral theory, see Spectral theory 80, 181, 216 INDEX 251

adjoint operator, 79 Moo, 218-225 Fredholm operator, 79 Moo, 221-225 Fredholm set, 79 Moo(L), 86, 216, 217 Hilbert-Schmidt norm, 64, 65 Moo(T), 61, 62, 79, 80, 98, 181, 185, 191 Hilbert-Schmidt operator, 64 Moo(L), 229, 232, 236, 239, 242, 245 compact operator, 66 Moo(T), 206-209 examples, 64, 65 finite-dimensional range, 65, 66, 70-74 nth roots of unity, 98, 110, 118 operational calculus, 67, 71 Naimark, M. A., 117, 124 quasi-nilpotent, 64, 70, 74-78, 80 Nilpotent operator, 92 trace product, 67, 74, 75 Null space, 1 Hn[a,b], Sobolev space, 83 Nullity, 41, 48, 51, 52, 54 i7n-Sobolev structure, 88 #n-structure, 88, 90 Open Mapping Theorem, 88 Horizontal strip, 130, 150 Operational calculus, 17, 19, 35, 73 equivalence in, 19 Index, 41, 45, 47, 51, 58 Operator-valued function, 10 differential operator, 84 analytic, 10 Hilbert-Schmidt discrete operator, 79 continuity of, 10 Index zero, 58—63 different iable, 10 Integral of operator-valued function V, family of projections, 60, 63, 98, 194, existence, uniqueness, 12 195, 200, 202, 206-209 Integral of vector-valued function, 11 Poo, projection, 219-225 existence, uniqueness, 11 po, integer, 98 Integral operator, 65, 85, 104, 105, 108 Palka, B. P., 20 Invariant subspace, 28 Parseval's formula, 65 Inverse graph, 6 Phragmen-Lindelof Theorem, 81 Inverse operator, 2 Point spectrum, 21 Isolated essential singularity of resolvent, 39 Pole of resolvent, 39, 60 ascent and descent, 39 Kaniel, S., and M. Schechter, 52 generalized eigenspace, 39 Kato, T., 1 Polynomial operator, 30, 31 Krein, M. G., see Gohberg, I. C, and M. G. Principal part, see Differential operator, Krein principal part Principle of Uniform Boundedness, 198, 218 Laurent series, see Vector-valued function, Product space, 5 Laurent series or Resolvent, Laurent Projections, see Differential operator, family series of projections associated with operator Lay, D. C., see Taylor, A. E., and D. C. Lay or modified family of projections Line integral along curve, 13 associated with operator Linear operator, 1 disjoint, 217, 226, 229, 233, 236, 240, 242 bounded, 1 double sequence of, 218, 220 densely defined, 1 perturbations of, 217-225 equality of, 1 uniformly bounded, 221, 225 extension of, 2 Punctured logarithmic strip, 142, 144, 163, from H to Hi, 1 167, 176, 179 in#, 1 Punctured sector, 132, 150, 153, 213, 226, onE, 1 233, 240 on H to ifi, 1 powers of, 30 Q, family of projections, 86, 228, 232, 236, product of, 2, 29 238, 242, 245 restriction of, 2 Qoo, projection, 221-225 scalar multiple, 2 sum of, 2, 29 Ray in plane, 111, 119, 181, 184, 189, 190, unbounded, 1 216 Liouville's Theorem, 77 Residual spectrum, 21 Locker, J., 1 Resolvent, 21, 24 Logarithmic strip, 137, 156, 158, 169, 172 adjoint of, 26 252 INDEX

differential operator, 85, 98, 103, 105, Uniform operator norm, 1, 89, 90 106, 110, 181, 193, 211-213, 226, 233, Vandermonde matrix, 103 240 Vector-valued function, 10 growth rate, 183-185, 188-191, 213-217 analytic, 10 Taylor series, 89 continuity of, 10 Laurent series, 35, 39, 59 different iable, 10 Resolvent equation, 21 higher-order derivatives, 14 Resolvent set, 21, 24 integrable, 11 differential operator, 85 Laurent series, 16-17 Fredholm operator, 59 Taylor series, 14-16 Riesz-Schauder theory, 66 Right shift operator, 45 Wronskian, 91 Rouche's Theorem, 133, 135, 142, 144, 152, 154, 163, 167, 176, 179 Rudin, W., 20

SRX{T), 212-216, 229-231, 237-238, 243-245 Schechter, M., 1; see also Kaniel, S., and M. Schechter Schwartz, J. T., see Dunford, N., and J. T. Schwartz Sector in plane, 111, 118, 119, 128, 146 Self-adjoint operator, 3 Soo, 218-225 Soo, 221-225 Soo(L), 86, 216, 217 Soo(T), 61, 63, 79, 80, 98, 181, 185, 191 Soo(L), 229, 232, 236, 239, 242, 245 Soo(T), 206-209 Sobolev spaces, 87-89 sp(L), 86, 216, 217 sp(T), 61, 181, 185, 191, 207 Spectral Mapping Theorem, 20, 35, 72, 73, 75 multiplicity of eigenvalues, 72, 73 Spectral theory differential operator, 98 n even, Case 1, 225-232 n even, Case 2, 232-239 n odd, Case 1, 239-245 Spectrum, 21, 24 compact resolvent, 25 continuity of, 19, 78 differential operator, 85, 86, 134, 136, 146, 155, 168, 180, 216, 232, 238, 245 examples, 26-28 Fredholm operator, 59, 60 spectrum of adjoint, 60 Hilbert-Schmidt discrete operator, 79 Hilbert-Schmidt operator, 66 isolated point of, 35 Stone, M. H., 4 Symmetric operator, 3

Taylor, A. E., and D. C. Lay, 1 Taylor series, see Vector-valued function, Taylor series