Linear Operators and their Spectra

Abstract

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.

Robust error estimates for approximations of non-self-adjoint eigenvalue problems

“…A recent reference is [17], see in particular [17,Example 13.4.4]. For classical references we point the reader to the monographs [24,42] and the references therein.…”

“…If the curve C contains several semisimple eigenvalues (and no others) then dim Ran(S(C)) = r, and we call r the joint algebraic multiplicity of the eigenvalues inside C. If in addition C d encloses only the conjugates of the eigenvalues which were enclosed by C, then dim Ran(S d (C d )) = r as well. Finally, according to [17,Theorem 9.2.19], the norm S(C) is an appropriate measure of the local sensitivity of a semisimple eigenvalue enclosed by C. It can be characterized as S(C) = lim z→λ |z − λ| (z − A) −1 .…”

“…More details on the spectral theory of non-self-adjoint operators can be found in [17,32] and the classical reference [24]. The problem (1)-under standard restrictions on the coefficients-provides an important example of a more general class of non-self-adjoint eigenvalue problems in a Hilbert space for which a Riesz basis (defined below) can be constructed from associated eigenvectors, see Example 11 and [17,24,42] for further discussion and references.…”

A Bound on the Pseudospectrum for a Class of Non-normal Schrödinger Operators

“…By closedness of the set of compact operators and Lemma 3.3 we only have to show that R n (τ ) is compact for every n (cf. Theorem 7.1.4 in [Dav07]). Since furthermore…”

The Semigroup and the Inverse of the Laplacian on the Heisenberg Group

“…While the semigroup and the inverse can be studied in the framework of functional analysis as explained in [3,4,5,8,9,16], the results and methods in this paper are based on explicit formulas in hard analysis and are related to the works in [1,2,6,7,10,12,14,15].…”