Hilbert Space: Compact Operators and the Trace Theorem

Cover......Page 1
Title......Page 2
Copyright......Page 3
Dedication......Page 4
Aknowledgment......Page 5
Contents......Page 6
Introduction......Page 10
0. THE INEQUALITIES OF IT ALL ......Page 12
I. PRELIMINARIES ......Page 14
Remarks, Exercises and Hints ......Page 20
1.4 Theore lli. (Cauchy-Schwarz-Bunyakovsky Inequality) ......Page 16
II.2 Theorem. (Bessel's Inequality). ......Page 22
11.3 Corollary to Bessel's inequality. ......Page 24
11.8 The Riesz-Fischer Theorem. ......Page 28
11.11 Zorn's Lemma, ......Page 30
11.14 Parseval's equality. ......Page 31
Remarks, Exercises and Hints ......Page 33
III. ISOMORPHISMS AND ISOMETRIES ......Page 36
Remarks, Exercises and Hints ......Page 38
IV. BOUNDED LINEAR OPERATORS ON HILBERT SPACE ......Page 39
IV.4 Riesz Representation Theorem. ......Page 42
Remarks, Exercises, and Hints ......Page 44
V. ELEMENTARY SPECTRAL THEORY ......Page 47
V.3 C. Neumann expansion. ......Page 48
Remarks, Exercises, Hints ......Page 58
VI. SELF-ADJOINT OPERATORS ......Page 60
Remarks, Exercises, and Hints ......Page 65
VII. COMPACT OPERATORS ......Page 71
VII.4 Theorem. (Schur). ......Page 73
Remarks, Exercises and Hints ......Page 77
APPENDIX A: COMPACT INTEGRAL OPERATORS ......Page 80
VIII. SQUARE ROOTS ......Page 81
VIII.6 Polar Decomposition Theorem. ......Page 84
VIII.8 The Schmidt Decomposition Theorem. ......Page 86
Remarks, Exercises, and Hints ......Page 88
IX. THE WEAK WEYL INEQUALITY ......Page 89
IX.2 (Holder's Inequality). ......Page 90
IX.3 (Minkowski's inequality). ......Page 91
B.l Hadamard's Inequality. ......Page 98
B.3 Weyl's Inequality. ......Page 100
Remarks, Exercises and Hints ......Page 101
X. HILBERT-SCHMIDT AND TRACE CLASS OPERATORS ......Page 104
Remarks, Exercises and Hints ......Page 113
Grothendieck-Pietsch Theorem: ......Page 116
XI. THE LIDSKIJ TRACE THEOREM ......Page 120
XI.3 Theorem. (Hardy's inequality). ......Page 122
XI.4 Theorem. (Konig) ......Page 123
XI,5 Lemma, (Localization of Eigenvalues). ......Page 124
XI.7 Theorem (Konig). ......Page 126
XI.ll (Lidskij Trace Theorem). ......Page 131
1. (Horn) ......Page 133
APPENDIX C: LOCALIZATION OF EIGENVALUES ......Page 135
C.1 Localization of Eigenvalues. ......Page 136
BIBLIOGRAPHY ......Page 137
INDEX OF NOTATION ......Page 138
INDEX OF TERMS ......Page 141
Back Cover......Page 143