# Linear Operators and their Spectra

Cambridge University Press, Apr 26, 2007 - Mathematics
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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### Contents

 1 1 2 35 3 67 4 99 5 135 53 Orthogonal projections 140 Prove also that the second inequality becomes an equality if 153 6 163
 9 245 10 296 Since t 0 is arbitrary this implies that Rez 300 and a oneparameter unitary group Ut on such that 310 11 325 rMeafMA7 340 12 355 From now on we often do not specify whether an 359

 Problem 6311 Formulate and prove a vectorvalued version of Theorem 189 7 190 Proof We start with the observation that 196 define two oneparameter semigroup s on which commute with 209 8 210
 jLewdxa if dx a n 377 13 380 14 408 25 427

### Popular passages

Page 3 - This rxxik follows the convention that inner products are linear in the first variable and conjugate linear in the second variable. 2Some books use the words "completely continuous" in place of "compact
Page 2 - X, then there exists a continuous function f: X — >- [0, 1] such that f(x) - 0 for all x € A and f(x) - 1 for all x € B.
Page 14 - Every continuous linear operator A from the Banach space B^ to the Banach space #2 defines an adjoint operator A , which takes II -^ into fij, by the equation (A*t) (/) = l(Af) (fe Bv I e B2'). The characteristics of A" induced by those of A are well known.