This book provides a comprehensive exploration of the spectral analysis of unitary and self-adjoint operators, along with the theory of extensions of symmetric operators. It begins with foundational topics such as the trigonometric moment problem, analytic functions, and integral representations of operators, progressing to more advanced concepts like the Cayley transform, spectral types, and canonical forms of self-adjoint operators. The text also delves into the theory of extensions, covering deficiency indices, Neumann formulas, and Krein’s resolvent formula. Appendices extend the discussion to generalized extensions, spectral functions, and differential operators, including regular and singular cases, with practical examples and inversion formulas. The book is a rigorous mathematical treatise aimed at understanding the structure, spectra, and extensions of operators in functional analysis.

Translated from the Russian by Merlynd Nestell

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CONTENTS

Chapter VI. THE SPECTRAL ANALYSIS OF UNITARY AND SELF-ADJOINT OPERATORS

1. The Trigonometric Moment Problem 1
2. Analytic Functions with Values in a Half-plane 5
3. The Theorem of Bochner 11
4. The Resolution of the Identity 14
5. The Integral Representation of a Unitary Operator 16
6. Operators Represented by Stieltjes Integrals 22
7. The Integral Representation of a Group of Unitary Operators 29
8. The Integral Representation of the Resolvent of a Self-Adjoint Operator 31
9. The Integral Representation of Self-Adjoint Operators 36
10. The Cayley Transform 42
11. The Spectra of Self-Adjoint and Unitary Operators 46
12. The Simple Spectrum 50
13. Spectral Types 56
14. The Multiple Spectrum 59
15. The Canonical Form of a Self-Adjoint Operator with a Spectrum of Finite Multiplicity 60
16. Some Remarks about Unitary Invariants of Self-Adjoint Operators 65
17. Some Remarks about Functions of Self-Adjoint Operators 74
18. Commutative Operators 76
19. Rings of Bounded Self-Adjoint Operators 80
20. Examples 84

Chapter VII. THEORY OF EXTENSIONS OF SYMMETRIC OPERATORS

21. Deficiency Indices 91
22. Further Remarks on the Cayley Transform 94
23. The Neumann Formulas 97
24. Simple Symmetric Operators 101
25. The Structure of Maximal Operators 103
26. Spectra of Self-Adjoint Extensions of Symmetric Operators 107
27. The Formula of Krein for the Resolvent of the Self-Adjoint Extensions of a Symmetric Operator 110
28. Semi-Bounded Operators 114
29. Some Remarks about the General Theory of Extensions 119

Appendix I. GENERALIZED EXTENSIONS AND GENERALIZED SPECTRAL FUNCTIONS OF SYMMETRIC OPERATORS

1. Generalized Resolution of the Identity. Naimark’s Theorem 121
2. Self-Adjoint Extensions to Larger Spaces and Spectral Functions of Symmetric Operators 126
3. Spectral Functions of Symmetric Operators and Generalized Resolvents 133
4. The Formula of Krein for Generalized Resolvents 139
5. Quasi-Self-Adjoint Extensions and the Characteristic Function of a Symmetric Operator 146

Appendix II. DIFFERENTIAL OPERATORS

1. Self-Adjoint Differential Expressions 162
2. Regular Differential Operators 166
3. Self-Adjoint Extensions of a Regular Differential Operator 168
4. Singular Differential Operators 170
5. Self-Adjoint Extensions of a Singular Differential Operator 174
6. The Resolvents of Self-Adjoint Extensions 177
7. Inversion Formulas Related to Differential Operators of the Second Order 186
8. Generalization to Differential Operators of Arbitrary Order 200
9. Examples 204

INDEX 216