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73 John Locker, Spectral theory of non-self-adjoint two-point differential operators, 2000 72 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, 2000 71 Lajos Pukanszky, Characters of connected Lie groups, 1999 70 Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems

and differential equations, 1999 69 C. T. C. Wall (A. A. Ranicki, Editor) , Surgery on compact manifolds, second edition,

1999 68 David A. Cox and Sheldon Katz , Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N . Wallach, Continuous cohomology, discrete subgroups, and

representations of reductive groups, second edition, 2000 66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra,

1999 64 Rene A. Carmona and Boris Rozovskii , Editors, Stochastic partial differential

equations: Six perspectives, 1999 63 Mark Hovey, Model categories, 1999 62 Vladimir I. Bogachev, Gaussian measures, 1998 61 W . Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic

algebra for ordinary differential and quasi-differential operators, 1999 60 Iain Raeburn and Dana P. Wil l iams, Morita equivalence and continuous-trace

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groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in

analysis, 1997 53 Andreas Kriegl and Peter W . Michor, The convenient setting of global analysis, 1997 52 V . A. Kozlov, V. G. Maz'ya, and J. Rossmann, Elliptic boundary value problems in

domains with point singularities, 1997 51 Jan Maly and Wil l iam P. Ziemer, Fine regularity of solutions of elliptic partial

differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential

equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert , Integer-valued polynomials, 1997 47 A. D . Elmendorf, I. Kriz, M. A. Mandel l , and J. P. May (with an appendix by

M. Cole) , Rings, modules, and algebras in stable homotopy theory, 1997 46 S tephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and A d a m O. Hausknecht , Cogroups and co-rings in

categories of associative rings, 1996 44 J. Amoros , M. Burger, K. Corlette , D . Kotschick, and D . Toledo, Fundamental

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http://dx.doi.org/10.1090/surv/073

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

John Locker

American Mathematical Society

Editorial Board Georgia Benkar t Michael Loss Pe te r Landweber Tudor Rat iu , Chair

1991 Mathematics Subject Classification. P r i m a r y 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-Publicat ion D a t a 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

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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 C O N T E N T S

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 boundary values Bi,... ,Bn defined on the Sobolev space Hn[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 P R E F A C E

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)=L 2 [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 incom­plete 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 P R E F A C E

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 Differ­ential 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 Math­ematical 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 eigen­values 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 associ­ated 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 op­erators 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 & Techni­cal, Harlow, Essex, 1994.

Index

Adjoint operator, 2 Algebraic multiplicity, 33

differential operator, 84, 86, 92-95, 98, 101 n even, 133, 134, 136, 146, 232, 238 n odd, 152, 155, 168, 181, 245

Fredholm operator, 48, 56-58 Ascent, 30, 31

differential operator, 84, 92, 98 n even, 133, 134, 136, 146, 232, 238 n odd, 152, 155, 168, 181, 245

Fredholm operator, 57-59 Ascent factor

Fredholm operator, 48, 54

Boundary coefficient matrix, 97, 211 Boundary values, 9, 83, 88, 97, 105, 109, 211

adjoint, 9, 83 degenerate, 98, 116, 118, 124 irregular, 87, 98, 111, 116, 119, 124, 128,

136, 146, 181 n even, 110-118 n odd, 118-128 normalized, 97, 117, 124, 211 order of, 97 regular, 87, 98, 111, 116, 119, 124, 128,

146, 181, 193, 201, 206, 212, 213, 225, 232, 239

B2{H), Hilbert-Schmidt operators, 64 Banach algebra, 65, 66 inner product, 66

Caratheodory's Inequality, 77 Carleman Inequality, 67, 68, 70, 74, 77, 80

finite-dimensional range, 71 finite-dimensions, 69 quasi-nilpotent Hilbert-Schmidt operator,

77, 78 Carleman theory, 64, 67 Cauchy domain, 14, 18, 36, 71 Cauchy Integral Formula, 15 Cauchy's Theorem, 14, 18, 20, 71, 86 Chain basis, 92-95 Characteristic determinant D, 86, 9 1 ,

92-95, 101, 102 order of zero, 92, 95 zero of, 91

Characteristic determinant A, 27, 98, 100, 101-102, 107, 117, 128, 133, 136, 140, 143, 146, 152, 155, 162, 168, 175, 193, 201, 212, 229-231, 237, 244

asymptotic formula, 98, 115, 122 n even, 100 n odd, 100 order of zero, 101 zero of, 98, 100, 102

Characteristic determinant Ao, 107, 108, 146, 148, 154, 157, 165, 171, 178, 201

asymptotic formula, 126 Closed Graph Theorem, 6, 21, 26, 32, 79 Closed linear operator, 6

examples, 7 reduced by pair of subspaces, 28

Closed Range Theorem, 8 Closure of an operator, 8

examples, 8 Cofactor, 106, 109, 116, 123, 127 Compact linear operator, 34

ascent and descent, 34 Completeness theorem, 64, 80, 185, 191, 217 Continuity of inversion, 19 Continuous spectrum, 21 Cramer's rule, 106, 109 Curve

closed, 13 rectifiable, 13 simple, 13 trace of, 13

Cycle homologous to zero, 20 surrounds the spectrum, 20

Defect, 41, 48, 51, 52 Descent, 30, 31 Differential operator, 8, 83, 211

adjoint operator, 83 algebraic multiplicity, see Algebraic

multiplicity, differential operator ascent, see Ascent, differential operator basis for solution space, 91, 93, 94, 99,

101, 103, 107, 212 boundary coefficient matrix, see

Boundary coefficient matrix

250 INDEX

boundary values, see Boundary values

characteristic determinant, see Characteristic determinant

completeness of generalized eigenfunctions, 98, 216, 217 n even, 181-185, 232, 239 n odd, 185-191, 245

eigenvalue, see Eigenvalue, differential operator

examples, 3-5, 8, 26, 27, 46, 85, 218 expansion in generalized eigenfunctions,

98 Hn-expansion, 210 L2-expansion, 207-209, 232, 239, 245 n even, 206-208 n odd, 208-209

family of projections associated with operator, 86 integral representation, 86, 194, 196,

197, 203, 204, 226, 240 n even, 193-200, 206, 228, 232 n odd, 201-206, 208, 242, 245 uniformly bounded, 98, 200, 206, 207,

232, 239, 245 formal differential operator, see

Formal differential operator Fredholm operator, 84, 88 Fredholm operator of index zero, 84, 85,

213 Fredholm set, 84 Green's function, see Green's function index, see Index, differential operator logarithmic case, 136, 146, 168, 180, 183,

184, 186, 190 maximal operator, 5, 84 minimal operator, 5, 84 modified family of projections associated

with operator, 195, 208, 236, 238 integral representation, 195, 233 uniformly bounded, 200, 239

multiple eigenvalues, 135, 232, 238 n even, Case 1, 133, 134, 181-185,

194-200, 206-208, 213-215, 225-232 n even, Case 2, 135, 135-136, 181-185,

195, 199, 200, 208, 213-215, 232-239 n even, Case 3, 136, 137-146, 181-185 n odd, Case 1, 146, 147-155, 185-191,

201-206, 208, 209, 215-216, 239-245 n odd, Case 2, 155, 156-168, 185-191 n odd, Case 3, 168, 169-181, 185-191 principal part, 87, 97, 212, 217, 225, 232,

238, 239, 245 resolvent, see Resolvent, differential

operator Sobolev space, see Hn[a, 6], Sobolev space spectral theory, see Spectral theory

spectrum, see Spectrum, differential operator

Dunford, N., and J. T. Schwartz, 1, 64, 117

Eigenspace, 33 differential operator, 92

Eigenvalue, 21 algebraic multiplicity, 33 differential operator, 91, 92, 95, 100, 101,

111, 119 apriori estimates, 130, 139, 149, 160,

173 asymptotic formula, 98, 134, 136, 142,

146, 155, 168, 181, 232, 238, 245 n even, 128-146, 228-232, 235-239 n odd, 146-181, 242-245

geometric multiplicity, 33 Eigenvector, 21 e-neighbor hood, 19

Formal adjoint differential operator, 9, 83 Formal differential operator, 9, 83, 87, 88,

97, 211 Fredholm operator, 28, 41

examples, 9, 45, 46 family of projections associated with

operator, 60 operator is T-bounded, 43, 44 operator is T-compact, 43, 44 perturbation of, 44 powers of, 43, 50-52 product of, 43 spectral family, 60

Fredholm set, 44, 56-58, 61, 63 Function of an operator, 18

Generalized eigenspace, 33 differential operator, 84, 92 Fredholm operator, 60 Hilbert-Schmidt discrete operator, 79

Generalized inverse, 42, 43 Geometric multiplicity, 33

differential operator, 92 Gohberg, I. C , and M. G. Krein, 46 Goldberg, S., 1 Graph norm, 6, 41, 43, 50, 88, 210 Graph of an operator, 6 Green's formula, 84 Green's function, 85, 86, 103-110, 181, 183,

186, 187, 193, 201, 202, 213 asymptotic formula, 98, 183—184,

188-189 partial derivatives, 214-215

H-S discrete operator, 78 Hadamard's Inequality, 67, 74 Hellinger-Toeplitz Theorem, 3 Hilbert-Schmidt discrete operator, 64, 78,

80, 181, 216

INDEX 251

adjoint operator, 79 Fredholm operator, 79 Fredholm set, 79

Hilbert-Schmidt norm, 64, 65 Hilbert-Schmidt operator, 64

compact operator, 66 examples, 64, 65 finite-dimensional range, 65, 66, 70-74 operational calculus, 67, 71 quasi-nilpotent, 64, 70, 74-78, 80 trace product, 67, 74, 75

Hn[a,b], Sobolev space, 83 i7n-Sobolev structure, 88 # n - s t ruc ture , 88, 90 Horizontal strip, 130, 150

Index, 4 1 , 45, 47, 51, 58 differential operator, 84 Hilbert-Schmidt discrete operator, 79

Index zero, 58—63 Integral of operator-valued function

existence, uniqueness, 12 Integral of vector-valued function, 11

existence, uniqueness, 11 Integral operator, 65, 85, 104, 105, 108 Invariant subspace, 28 Inverse graph, 6 Inverse operator, 2 Isolated essential singularity of resolvent, 39

Kaniel, S., and M. Schechter, 52 Kato, T., 1 Krein, M. G., see Gohberg, I. C , and M. G.

Krein

Laurent series, see Vector-valued function, Laurent series or Resolvent, Laurent series

Lay, D. C., see Taylor, A. E., and D. C. Lay Line integral along curve, 13 Linear operator, 1

bounded, 1 densely defined, 1 equality of, 1 extension of, 2 from H to Hi, 1 i n # , 1 o n E , 1 on H to i f i , 1 powers of, 30 product of, 2, 29 restriction of, 2 scalar multiple, 2 sum of, 2, 29 unbounded, 1

Liouville's Theorem, 77 Locker, J., 1 Logarithmic strip, 137, 156, 158, 169, 172

Moo, 218-225 Moo, 221-225 Moo(L), 86, 216, 217 Moo(T), 61, 62, 79, 80, 98, 181, 185, 191 Moo(L), 229, 232, 236, 239, 242, 245 Moo(T), 206-209

n th roots of unity, 98, 110, 118 Naimark, M. A., 117, 124 Nilpotent operator, 92 Null space, 1 Nullity, 4 1 , 48, 51, 52, 54

Open Mapping Theorem, 88 Operational calculus, 17, 19, 35, 73

equivalence in, 19 Operator-valued function, 10

analytic, 10 continuity of, 10 different iable, 10

V, family of projections, 60, 63, 98, 194, 195, 200, 202, 206-209

Poo, projection, 219-225 po, integer, 98 Palka, B. P., 20 Parseval's formula, 65 Phragmen-Lindelof Theorem, 81 Point spectrum, 21 Pole of resolvent, 39, 60

ascent and descent, 39 generalized eigenspace, 39

Polynomial operator, 30, 31 Principal part, see Differential operator,

principal part Principle of Uniform Boundedness, 198, 218 Product space, 5 Projections, see Differential operator, family

of projections associated with operator or modified family of projections associated with operator

disjoint, 217, 226, 229, 233, 236, 240, 242 double sequence of, 218, 220 perturbations of, 217-225 uniformly bounded, 221, 225

Punctured logarithmic strip, 142, 144, 163, 167, 176, 179

Punctured sector, 132, 150, 153, 213, 226, 233, 240

Q, family of projections, 86, 228, 232, 236, 238, 242, 245

Qoo, projection, 221-225

Ray in plane, 111, 119, 181, 184, 189, 190, 216

Residual spectrum, 21 Resolvent, 21, 24

adjoint of, 26

252 INDEX

Uniform operator norm, 1, 89, 90

Vandermonde matrix, 103 Vector-valued function, 10

analytic, 10 continuity of, 10 different iable, 10 higher-order derivatives, 14 integrable, 11 Laurent series, 16-17 Taylor series, 14-16

Wronskian, 91

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

differential operator, 85, 98, 103, 105, 106, 110, 181, 193, 211-213, 226, 233, 240 growth rate, 183-185, 188-191, 213-217 Taylor series, 89

Laurent series, 35, 39, 59 Resolvent equation, 21 Resolvent set, 21, 24

differential operator, 85 Fredholm operator, 59

Riesz-Schauder theory, 66 Right shift operator, 45 Rouche's Theorem, 133, 135, 142, 144, 152,

154, 163, 167, 176, 179 Rudin, W., 20