Linear Operators: Spectral Theory and Some Other Applications

Cover......Page 1
Title Page......Page 2
Copyright......Page 3
Preface......Page 4
Notation......Page 7
CONTENTS......Page 8
1.1. HILBERT FUNCTIONAL SPACE aa(f)......Page 12
1.2. THE RIESZ-FISCHER THEOREM ; THE SEPARABILITY OF THE SPACE a2(f)......Page 18
1.3. ORTHONORMAL SETS IN as(f)......Page 25
1.4. METRIC FUNCTIONS ; ABSTRACT HILBERT SPACE......Page 33
1.5. ORTHONORMAL SETS IN 1......Page 41
EXAMPLES ON CHAPTER 1......Page 46
2.1. INTRODUCTION OF INFINITE MATRICES INTO THE "NEW " QUANTUM THEORY......Page 50
2.2. THE HARMONIC OSCILLATOR, BY HEISENBERG'S METHOD......Page 55
2.3. PERTURBATION THEORY, BY HEISENBERG'S METHOD......Page 60
2.4. SCHRODINGIER'S WAVE EQUATION, AND THE CALCULATION OF THE HEISENBERG MATRICES FROM THE EIGENFUNCTIONS......Page 68
2.5. THE HARMONIC OSCILLATOR AND PERTURBATION THEORY, BY SCHRODING}ER'S METHOD......Page 75
2.6. CONTINUOUS MATRICES AND THE SPECTRUM......Page 78
2.7. REMARKS ON LINEAR OPERATORS......Page 81
3.1. ADJOINTS, UNITARY OPERATORS......Page 93
3.2. BOUNDED LINEAR OPERATORS......Page 100
3.3. IDEMPOTENT LINEAR OPERATORS ; PROJECTORS......Page 102
3.4. EXTENSION OF LINEAR AND BILINEAR FUNCTIONS......Page 108
3.5. OPERATOR OF A FUNCTION ; MAXIMAL AND HYPERMAXIMAL (SBLF-ADJOINT) OPERATORS......Page 120
3.6. THE DEFICIENCY INDICES......Page 131
3.7. A NOTE ON REDUCIBILITY OF OPERATORS......Page 143
4.1. THE EIOENVALUE PROBLEM......Page 145
4.2. THE POINT SPECTRUM AND THE CONTINUOUS SPECTRUM......Page 153
4.3. EXAMPLES. FORMAL RULES FOR HERMITIAN OPERATORS......Page 157
4.4. VON NEUMANN'S SPECTRAL RESOLUTION THEOREM ; THE CAYLEY TRANSFORMATION......Page 171
4.5. SOLUBILITY OF THE EIGENVALUE PROBLEM FOR HYPERMAXIMAL OPERATORS......Page 177
4.6. THE SPECTRAL RESOLUTION FOR UNITARY OPERATORS......Page 179
5.1. STONE'S DEFINITIONS OF POINT, CONTINUOUS, AND RESIDUAL SPECTRA......Page 191
5.2. EXAMPLES ON STONE'S DEFINITIONS......Page 194
5.3. THE THEORY OF LENGYEL......Page 199
5.4. BOCHNER'S THEOREM ON POSITIVE-DEFINITE FUNCTIONS......Page 216
5.5. THE THEORY or COOPER......Page 225
5.6. THE EQUIVALENCE OF Two HYPOTHESES......Page 235
5.7. THE THEORY OF RIESZ AND LORCH......Page 248
5.8. THE THEORY OF LENGYEL AND STONE FOR THE SPECIAL CASE OF BOUNDED OPERATORS......Page 258
5.9. SOME REMARKS ON THE REMAINING SPECTRAL THEORIES......Page 270
6.1. CONTINUITY OF TRANSFORMATIONS AND FUNCTIONALS......Page 283
6.2. LINEAR FUNCTIONALS......Page 293
6.8. HOMOEOMORPHIC SEQUENCE SPACES......Page 302
6.4. MATRIX SPACES......Page 308
6.5. MATRIX RINGS......Page 322
6.6. PROJECTIVE CONVERGENCE AND LIMIT IN MATRIX SPACES......Page 335
6.7. PROJECTIVE CONVERGENCE IN A MATRIX RING......Page 345
6.8. STRONG PROJECTIVE-CONVERGENCE IN A PERFECT MAXIMUM RING......Page 351
6.9. NORMAL BOUNDS OF MATRICES......Page 354
EXAMPLES ON CHAPTER 6......Page 356
7.1. THE HAHN-BANACH THEOREM AND SOME ALGEBRAIC PRELIMINARIES......Page 358
7.2. BANACH ALGEBRAS AND BANACH FIELDS......Page 365
7.3. QUOTIENT ALGEBRAS MODULO IDEALS AND MAXIMAL IDEALS, WITH APPLICATIONS......Page 372
7.4. THE STRONG TOPOLOGY ON THE SPECTRUM OF A BANACH ALGEBRA......Page 380
7.5. POSITIVE RADON MEASURES......Page 392
7.6. BANACH ALGEBRAS WITH INVOLUTION, AND APPLICATIONS......Page 413
7.7 BANACH ALGEBRAS WITHOUT UNIT ELEMENT......Page 421
7.7. WIENER'S THEOREM ON THE CLOSURE OF TRANSLATIONS......Page 426
APPENDIX (A) TO CHAPTER 7......Page 430
APPENDIX (B) TO CHAPTER 7......Page 432
EXAMPLES ON CHAPTER 7......Page 434
REFERRED TO IN THE TEXT......Page 439
INDEX OF NAMES......Page 457
GENERAL INDEX......Page 460