Linear Operators and their Spectra
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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53 Orthogonal projections
Prove also that the second inequality becomes an equality if
Since t 0 is arbitrary this implies that Rez
and a oneparameter unitary group Ut on such that
From now on we often do not specify whether an
Problem 6311 Formulate and prove a vectorvalued version of Theorem
Proof We start with the observation that
define two oneparameter semigroup s on which commute with
jLewdxa if dx a n
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algebra analytic function applying assume Banach space boundary conditions bounded linear operator bounded operator closed linear closed operator closure compact operator computation constant continuous function convex convolution Corollary countable deduce denote depends differential operator Dom(Z domain DomH DomZ eigenvalue equation equivalent essential spectrum Example exists following theorem formula Fourier transform Fredholm operator functions f given Hence Hilbert space identity implies integral kernel invariant invertible irreducible isometric L2RN Lemma linear subspace Lp spaces LpRN LpX dx Markov operator Markov semigroup matrix measure non-negative non-zero norm continuous norm convergent numerical range obtain one-one one-parameter contraction semigroup one-parameter group one-parameter semigroup one-parameter semigroup Tt operator acting orthogonal perturbation polynomial Problem Proof Prove pseudospectra satisfies Schrödinger operators self-adjoint operator sequence Spec SpecA spectral theorem SpecZ subset Suppose theory topology unbounded unitary vector
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 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.