Linear Operators and their Spectra

Linear Operators and their Spectra

by E. Brian Davies
     
 

This wide-ranging and 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… See more details below

Overview

This wide-ranging and 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 zero of certain orthogonal polynomials on the unit circle. Finally, two methods, both very recent, for obtaining bounds on the eigenvalues of non-self-adjoint Schrodingers 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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Product Details

ISBN-13:
9780521866293
Publisher:
Cambridge University Press
Publication date:
04/30/2007
Series:
Cambridge Studies in Advanced Mathematics Series, #106
Pages:
464
Product dimensions:
5.98(w) x 8.98(h) x 1.06(d)

Table of Contents


Preface     ix
Elementary operator theory     1
Banach spaces     1
Bounded linear operators     12
Topologies on vector spaces     19
Differentiation of vector-valued functions     23
The holomorphic functional calculus     27
Function spaces     35
L[superscript p] spaces     35
Operators acting on L[superscript p] spaces     45
Approximation and regularization     54
Absolutely convergent Fourier series     60
Fourier transforms and bases     67
The Fourier transform     67
Sobolev spaces     77
Bases of Banach spaces     80
Unconditional bases     90
Intermediate operator theory     99
The spectral radius     99
Compact linear operators     102
Fredholm operators     116
Finding the essential spectrum     124
Operators on Hilbert space     135
Bounded operators     135
Polar decompositions     137
Orthogonal projections     140
The spectral theorem     143
Hilbert-Schmidt operators     151
Trace classoperators     153
The compactness of f(Q)g(P)     160
One-parameter semigroups     163
Basic properties of semigroups     153
Other continuity conditions     577
Some standard examples     182
Special classes of semigroup     190
Norm continuity     190
Trace class semigroups     194
Semigroups on dual spaces     197
Differentiable and analytic vectors     201
Subordinated semigroups     205
Resolvents and generators     210
Elementary properties of resolvents     210
Resolvents and semigroups     218
Classification of generators     227
Bounded holomorphic semigroups     237
Quantitative bounds on operators     245
Pseudospectra     245
Generalized spectra and pseudospectra     251
The numerical range     264
Higher order hulls and ranges     276
Von Neumann's theorem     285
Peripheral point spectrum     287
Quantitative bounds on semigroups     296
Long time growth bounds     296
Short time growth bounds     300
Contractions and dilations      307
The Cayley transform     310
One-parameter groups     315
Resolvent bounds in Hilbert space     321
Perturbation theory     325
Perturbations of unbounded operators     325
Relatively compact perturbations     330
Constant coefficient differential operators on the half-line     335
Perturbations: semigroup based methods     339
Perturbations: resolvent based methods     350
Markov chains and graphs     355
Definition of Markov operators     355
Irreducibility and spectrum     359
Continuous time Markov chains     362
Reversible Markov semigroups     366
Recurrence and transience     369
Spectral theory of graphs     374
Positive semigroups     380
Aspects, of positivity     380
Invariant subsets     386
Irreducibility     390
Renormalization     293
Ergodic theory     395
Positive semigroups on C(X)     399
NSA Schrodinger operators     408
Introduction     408
Bounds on the numerical range     409
Bounds in one space dimension      412
The essential spectrum of Schrodinger operators     420
The NSA harmonic oscillator     424
Semi-classical analysis     427
References     436
Index     446

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