Linear Operators and their Spectra

E. Brian Davies0521866294, 9780521866293, 9780511286513

This wide ranging but self-contained account of the spectral theory of non-self-adjoint linear operators is ideal for postgraduate students and researchers, and contains many illustrative examples and exercises. Fredholm theory, Hilbert-Schmidt and trace class operators are discussed, as are one-parameter semigroups and perturbations of their generators. Two chapters are devoted to using these tools to analyze Markov semigroups. The text also provides a thorough account of the new theory of pseudospectra, and presents the recent analysis by the author and Barry Simon of the form of the pseudospectra at the boundary of the numerical range. This was a key ingredient in the determination of properties of the zeros of certain orthogonal polynomials on the unit circle. Finally, two methods, both very recent, for obtaining bounds on the eigenvalues of non-self-adjoint Schrodinger operators are described. The text concludes with a description of the surprising spectral properties of the non-self-adjoint harmonic oscillator.

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Table of contents :
Cover……Page 1
Half-title……Page 3
Series-title……Page 4
Title……Page 5
Copyright……Page 6
Contents……Page 7
Preface……Page 11
1.1 Banach spaces……Page 15
1.2 Bounded linear operators……Page 26
1.3 Topologies on vector spaces……Page 33
1.4 Differentiation of vector-valued functions……Page 37
1.5 The holomorphic functional calculus……Page 41
2.1 Lp spaces……Page 49
2.2 Operators acting on Lp spaces……Page 59
2.3 Approximation and regularization……Page 68
2.4 Absolutely convergent Fourier series……Page 74
3.1 The Fourier transform……Page 81
3.2 Sobolev spaces……Page 91
3.3 Bases of Banach spaces……Page 94
3.4 Unconditional bases……Page 104
4.1 The spectral radius……Page 113
4.2 Compact linear operators……Page 116
4.3 Fredholm operators……Page 130
4.4 Finding the essential spectrum……Page 138
5.1 Bounded operators……Page 149
5.2 Polar decompositions……Page 151
5.3 Orthogonal projections……Page 154
5.4 The spectral theorem……Page 157
5.5 Hilbert-Schmidt operators……Page 165
5.6 Trace class operators……Page 167
5.7 The compactness of f(Q)g(P)……Page 174
6.1 Basic properties of semigroups……Page 177
6.2 Other continuity conditions……Page 191
6.3 Some standard examples……Page 196
7.1 Norm continuity……Page 204
7.2 Trace class semigroups……Page 208
7.3 Semigroups on dual spaces……Page 211
7.4 Differentiable and analytic vectors……Page 215
7.5 Subordinated semigroups……Page 219
8.1 Elementary properties of resolvents……Page 224
8.2 Resolvents and semigroups……Page 232
8.3 Classification of generators……Page 241
8.4 Bounded holomorphic semigroups……Page 251
9.1 Pseudospectra……Page 259
9.2 Generalized spectra and pseudospectra……Page 265
9.3 The numerical range……Page 278
9.4 Higher order hulls and ranges……Page 290
9.5 Von Neumann’s theorem……Page 299
9.6 Peripheral point spectrum……Page 301
10.1 Long time growth bounds……Page 310
10.2 Short time growth bounds……Page 314
10.3 Contractions and dilations……Page 321
10.4 The Cayley transform……Page 324
10.5 One-parameter groups……Page 329
10.6 Resolvent bounds in Hilbert space……Page 335
11.1 Perturbations of unbounded operators……Page 339
11.2 Relatively compact perturbations……Page 344
11.3 Constant coefficient differential operators on the half-line……Page 349
11.4 Perturbations: semigroup based methods……Page 353
11.5 Perturbations: resolvent based methods……Page 364
12.1 Definition of Markov operators……Page 369
12.2 Irreducibility and spectrum……Page 373
12.3 Continuous time Markov chains……Page 376
12.4 Reversible Markov semigroups……Page 380
12.5 Recurrence and transience……Page 383
12.6 Spectral theory of graphs……Page 388
13.1 Aspects of positivity……Page 394
13.2 Invariant subsets……Page 400
13.3 Irreducibility……Page 404
13.4 Renormalization……Page 407
13.5 Ergodic theory……Page 409
13.6 Positive semigroups on C(X)……Page 413
14.1 Introduction……Page 422
14.2 Bounds on the numerical range……Page 423
14.3 Bounds in one space dimension……Page 426
14.4 The essential spectrum of Schrödinger operators……Page 434
14.5 The NSA harmonic oscillator……Page 438
14.6 Semi-classical analysis……Page 441
References……Page 450
Index……Page 460