Introduction to Hilbert Space and the Theory of Spectral Multiplicity

Overview

2013 Reprint of 1951 Edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. The subject matter of the book is funneled into three chapters: [1] The geometry of Hubert space; [2] the structure of self-adjoint and normal operators; [3] and multiplicity theory for a normal operator. For the last, an expert knowledge of measure theory is indispensable. Indeed, multiplicity theory is a magnificent measure-theoretic tour de force. The subject matter of the first two chapters ...
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Overview

2013 Reprint of 1951 Edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. The subject matter of the book is funneled into three chapters: [1] The geometry of Hubert space; [2] the structure of self-adjoint and normal operators; [3] and multiplicity theory for a normal operator. For the last, an expert knowledge of measure theory is indispensable. Indeed, multiplicity theory is a magnificent measure-theoretic tour de force. The subject matter of the first two chapters might be said to constitute an introduction to Hilbert space, and for these, an a priori knowledge of classic measure theory is not essential. Paul Richard Halmos (1916-2006) was a Hungarian-born American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). He was also recognized as a great mathematical expositor.
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Product Details

  • ISBN-13: 9781614274711
  • Publisher: Martino Fine Books
  • Publication date: 9/10/2013
  • Pages: 118
  • Product dimensions: 6.00 (w) x 9.00 (h) x 0.28 (d)

Table of Contents

Preface 3
0 Prerequisites and Notation 8
Ch. I The Geometry of Hilbert Space
1 Linear Functionals 11
2 Bilinear Functionals 12
3 Quadratic Forms 12
4 Inner Product and Norm 13
5 The Inequalities of Bessel and Schwarz 14
6 Hilbert Space 16
7 Infinite Sums 17
8 Conditions for Summability 19
9 Examples of Hilbert Spaces 20
10 Subspaces 21
11 Vectors in and out of Subspaces 23
12 Orthogonal Complements 24
13 Vector Sums 25
14 Bases 27
15 A Non-closed Vector Sum 28
16 Dimension 29
17 Boundedness 31
18 Bounded Bilinear Functionals 32
Ch. II The Algebra of Operators
19 Operators 35
20 Examples of Operators 36
21 Inverses 37
22 Adjoints 38
23 Invariance 40
24 Hermitian Operators 41
25 Normal and Unitary Operators 42
26 Projections 43
27 Projections and Subspaces 44
28 Sums of Projections 46
29 Products and Differences of Projections 47
30 Infima and Suprema of Projections 49
31 The Spectrum of an Operator 50
32 Compactness of Spectra 52
33 Transforms of Spectra 53
34 The Spectrum of a Hermitian Operator 54
35 Spectral Heuristics 56
36 Spectral Measures 58
37 Spectral Integrals 60
38 Regular Spectral Measures 61
39 Real and Complex Spectral Measures 63
40 Complex Spectral Integrals 65
41 Description of the Spectral Subspaces 66
42 Characterization of the Spectral Subspaces 68
43 The Spectral Theorem for Hermitian Operators 69
44 The Spectral Theorem for Normal Operators 71
Ch. III The Analysis of Spectral Measures
45 The Problem of Unitary Equivalence 74
46 Multiplicity Functions in Finite-dimensional Spaces 76
47 Measures 77
48 Boolean Operations on Measures 79
49 Multiplicity Functions 80
50 The Canonical Example of a Spectral Measure 83
51 Finite-dimensional Spectral Measures 84
52 Simple Finite-dimensional Spectral Measures 85
53 The Commutator of a Set of Projections 87
54 Pairs of Commutators 88
55 Columns 89
56 Rows 90
57 Cycles 91
58 Separable Projections 92
59 Characterizations of Rows 93
60 Cycles and Rows 95
61 The Existence of Rows 96
62 Orthogonal Systems 97
63 The Power of a Maximal Orthogonal System 99
64 Multiplicities 100
65 Measures from Vectors 102
66 Subspaces from Measures 104
67 The Multiplicity Function of a Spectral Measure 106
68 Conclusion 108
References 109
Bibliography 112
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