| Introduction to the Theory of Linear Operators | p. 1 |
| Introduction | p. 1 |
| Generalities about Unbounded Operators | p. 2 |
| Adjoint, Symmetric and Self-adjoint Operators | p. 5 |
| Spectral Theorem | p. 13 |
| Functional Calculus | p. 15 |
| L[superscript 2] Spectral Representation | p. 22 |
| Stone's Theorem, Mean Ergodic Theorem and Trotter Formula | p. 29 |
| One-Parameter Semigroups | p. 35 |
| References | p. 40 |
| Introduction to Quantum Statistical Mechanics | p. 41 |
| Quantum Mechanics | p. 42 |
| Classical Mechanics | p. 42 |
| Quantization | p. 46 |
| Fermions and Bosons | p. 53 |
| Quantum Statistical Mechanics | p. 54 |
| Density Matrices | p. 54 |
| Boltzmann Gibbs | p. 57 |
| References | p. 67 |
| Elements of Operator Algebras and Modular Theory | p. 69 |
| Introduction | p. 70 |
| Discussion | p. 70 |
| Notations | p. 71 |
| C*-algebras | p. 71 |
| First definitions | p. 71 |
| Spectral analysis | p. 73 |
| Representations and states | p. 79 |
| Commutative C*-algebras | p. 83 |
| Appendix | p. 84 |
| von Neumann algebras | p. 86 |
| Topologies on B(H) | p. 86 |
| Commutant | p. 89 |
| Predual, normal states | p. 90 |
| Modular theory | p. 92 |
| The modular operators | p. 92 |
| The modular group | p. 96 |
| Self-dual cone and standard form | p. 100 |
| References | p. 105 |
| Quantum Dynamical Systems | p. 107 |
| Introduction | p. 107 |
| The State Space of a C*-algebras | p. 110 |
| States | p. 110 |
| The GNS Representation | p. 119 |
| Classical Systems | p. 123 |
| Basics of Ergodic Theory | p. 123 |
| Classical Koopmanism | p. 127 |
| Quantum Systems | p. 130 |
| C*-Dynamical Systems | p. 132 |
| W*-Dynamical Systems | p. 139 |
| Invariant States | p. 141 |
| Quantum Dynamical Systems | p. 142 |
| Standard Forms | p. 147 |
| Ergodic Properties of Quantum Dynamical Systems | p. 153 |
| Quantum Koopmanism | p. 161 |
| Perturbation Theory | p. 165 |
| KMS States | p. 168 |
| Definition and Basic Properties | p. 168 |
| Perturbation Theory of KMS States | p. 178 |
| References | p. 180 |
| The Ideal Quantum Gas | p. 183 |
| Introduction | p. 184 |
| Fock space | p. 185 |
| Bosons and Fermions | p. 185 |
| Creation and annihilation operators | p. 188 |
| Weyl operators | p. 191 |
| The C*-algebras CAR[subscript F] (H), CCR[subscript F] (H) | p. 194 |
| Leaving Fock space | p. 197 |
| The CCR and CAR algebras | p. 198 |
| The algebra CAR (D) | p. 199 |
| The algebra CCR (D) | p. 200 |
| Schrodinger representation and Stone - von Neumann uniqueness theorem | p. 203 |
| Q-space representation | p. 207 |
| Equilibrium state and thermodynamic limit | p. 209 |
| Araki-Woods representation of the infinite free Boson gas | p. 213 |
| Generating functionals | p. 214 |
| Ground state (condensate) | p. 217 |
| Excited states | p. 222 |
| Equilibrium states | p. 224 |
| Dynamical stability of equilibria | p. 228 |
| References | p. 233 |
| Topics in Spectral Theory | p. 235 |
| Introduction | p. 236 |
| Preliminaries: measure theory | p. 238 |
| Basic notions | p. 238 |
| Complex measures | p. 238 |
| Riesz representation theorem | p. 240 |
| Lebesgue-Radon-Nikodym theorem | p. 240 |
| Fourier transform of measures | p. 241 |
| Differentiation of measures | p. 242 |
| Problems | p. 247 |
| Preliminaries: harmonic analysis | p. 248 |
| Poisson transforms and Radon-Nikodym derivatives | p. 249 |
| Local L[superscript p] norms, 0 < p < 1 | p. 253 |
| Weak convergence | p. 253 |
| Local L[superscript p]-norms, p > 1 | p. 254 |
| Local version of the Wiener theorem | p. 255 |
| Poisson representation of harmonic functions | p. 256 |
| The Hardy class H[superscript infinity] (C[subscript +]) | p. 258 |
| The Borel transform of measures | p. 261 |
| Problems | p. 263 |
| Self-adjoint operators, spectral theory | p. 267 |
| Basic notions | p. 267 |
| Digression: The notions of analyticity | p. 269 |
| Elementary properties of self-adjoint operators | p. 269 |
| Direct sums and invariant subspaces | p. 272 |
| Cyclic spaces and the decomposition theorem | p. 273 |
| The spectral theorem | p. 273 |
| Proof of the spectral theorem-the cyclic case | p. 274 |
| Proof of the spectral theorem-the general case | p. 277 |
| Harmonic analysis and spectral theory | p. 279 |
| Spectral measure for A | p. 280 |
| The essential support of the ac spectrum | p. 281 |
| The functional calculus | p. 281 |
| The Weyl criteria and the RAGE theorem | p. 283 |
| Stability | p. 285 |
| Scattering theory and stability of ac spectra | p. 286 |
| Notions of measurability | p. 287 |
| Non-relativistic quantum mechanics | p. 290 |
| Problems | p. 291 |
| Spectral theory of rank one perturbations | p. 295 |
| Aronszajn-Donoghue theorem | p. 296 |
| The spectral theorem | p. 298 |
| Spectral averaging | p. 299 |
| Simon-Wolff theorems | p. 300 |
| Some remarks on spectral instability | p. 301 |
| Boole's equality | p. 302 |
| Poltoratskii's theorem | p. 304 |
| F. & M. Riesz theorem | p. 308 |
| Problems and comments | p. 309 |
| References | p. 311 |
| Index of Volume I | p. 313 |
| Information about the other two volumes | |
| Contents of Volume II | p. 318 |
| Index of Volume II | p. 321 |
| Contents of Volume III | p. 323 |
| Index of Volume III | p. 327 |
| Table of Contents provided by Blackwell. All Rights Reserved. |
