Helmberg / Lauwerier / Koiter Introduction to Spectral Theory in Hilbert Space

North-Holland Series in Applied Mathematics and Mechanics
1. Auflage 2014
ISBN: 978-1-4831-6417-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 6, 362 Seiten, Web PDF

Reihe: North-Holland Series in Applied Mathematics and Mechanics

North-Holland Series in Applied Mathematics and Mechanics, Volume 6: Introduction to Spectral Theory in Hilbert Space focuses on the mechanics, principles, and approaches involved in spectral theory in Hilbert space. The publication first elaborates on the concept and specific geometry of Hilbert space and bounded linear operators. Discussions focus on projection and adjoint operators, bilinear forms, bounded linear mappings, isomorphisms, orthogonal subspaces, base, subspaces, finite dimensional Euclidean space, and normed linear spaces. The text then takes a look at the general theory of linear operators and spectral analysis of compact linear operators, including spectral decomposition of a compact selfadjoint operator, weakly convergent sequences, spectrum of a compact linear operator, and eigenvalues of a linear operator. The manuscript ponders on the spectral analysis of bounded linear operators and unbounded selfadjoint operators. Topics include spectral decomposition of an unbounded selfadjoint operator and bounded normal operator, functions of a unitary operator, step functions of a bounded selfadjoint operator, polynomials in a bounded operator, and order relation for bounded selfadjoint operators. The publication is a valuable source of data for mathematicians and researchers interested in spectral theory in Hilbert space.

Autoren/Hrsg.

Helmberg, Gilbert

Lauwerier, H. A.

Koiter, W. T.

Inhaltsverzeichnis

1;Front Cover;1
2;Introduction to Spectral Theory in Hilbert Space;3
3;Copyright Page;5
4;Table of Contents;12
5;Dedication;6
6;EDITORIAL NOTE;7
7;PREFACE;8
8;CHAPTER I. The concept of Hilbert space;16
8.1;§ 1. Finite-dimensional Euclidean space;16
8.2;§ 2. Inner product spaces;21
8.3;§ 3. Normed linear spaces;28
8.4;§ 4. The Hilbert space c*;38
8.5;§ 5. £2 Hilbert spaces;42
9;CHAPTER II. Specific geometry of Hilbert space;51
9.1;§ 6. Subspaces;51
9.2;§ 7. Orthogonal subspaces;55
9.3;§ 8. Base;62
9.4;§ 9. Polynomial bases in £2 spaces;70
9.5;§ 10. Isomorphisms;82
10;CHAPTER III. Bounded linear operators;86
10.1;§ 11. Bounded linear mappings;86
10.2;§ 12. Linear operators;92
10.3;§ 13. Bilinear forms;103
10.4;§ 14. Adjoint operators;110
10.5;§ 15. Projection operators;117
10.6;§ 16. The Fourier-Plancherel operator;123
11;CHAPTER IV. General theory of linear operators;132
11.1;§17. Adjoint operators (general case);132
11.2;§ 18. Differentiation operators in £2 spaces;139
11.3;§ 19. Multiplication operators in £2 spaces;146
11.4;§ 20. Closed linear operators;154
11.5;§ 21. Invariant subspaces of a linear operator;159
11.6;§ 22. Eigenvalues of a linear operator;165
11.7;§ 23. The spectrum of a linear operator;172
11.8;§ 24. The spectrum of a selfadjoint operator;182
12;CHAPTER V. Spectral analysis of compact linear operators;192
12.1;§ 25. Compact linear operators;192
12.2;§ 26. Weakly converging sequences;199
12.3;§ 27. The spectrum of a compact linear operator;205
12.4;§ 28. The spectral decomposition of a compact selfadjoint operator;212
12.5;§ 29. Fredholm integral equations;223
13;CHAPTER VI. Spectral analysis of bounded linear operators;234
13.1;§ 30. The order relation for bounded selfadjoint operators;234
13.2;§ 31. Polynomials in a bounded linear operator;243
13.3;§ 32. Continuous functions of a bounded selfadjoint operator;245
13.4;§ 33. Step functions of a bounded selfadjoint operator;259
13.5;§ 34. The spectral decomposition of a bounded selfadjoint operator;266
13.6;§ 35. Functions of a unitary operator;281
13.7;§ 36. The spectral decomposition of a unitary operator;287
13.8;§ 37. The spectral decomposition of a bounded normal operator;295
14;CHAPTER VII. Spectral analysis of unbounded selfadjoint operators;303
14.1;§ 38. The Cayley transform;303
14.2;§ 39. The spectral decomposition of an unbounded selfadjoint operator;307
15;APPENDIX A. The graph of a linear operator;325
16;APPENDIX B. Riemann-Stieltjes and Lebesgue integration;331
16.1;Bl. Weierstrass' approximation theorem;331
16.2;B2. Riemann-Stieltjes integration;332
16.3;B3. Lebesgue measurable sets;333
16.4;B4. Lebesgue measure;334
16.5;B5. Lebesgue measurable functions;335
16.6;B6. Lebesgue integrable functions;337
16.7;B7. Properties of the Lebesgue integral;339
16.8;B8. Fubini's theorem;341
16.9;B9. Absolutely continuous functions;342
16.10;BIO. Differentiation under the integral sign;343
17;Bibliography;345
18;Index of symbols;349
19;Subject index;355