Essential numerical ranges for linear operator pencils.

Abstract

We introduce concepts of essential numerical range for the linear operator pencil λ↦A−λB⁠. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem Tx=λx into the pencil problem BTx=λBx for suitable choices of B⁠, we can obtain nonconvex spectral enclosures for T and, in the study of truncation and projection methods, confine spectral pollution to smaller sets than with hitherto known concepts. We apply the results to various block operator matrices. In particular, Theorem 4.12 presents substantial improvements over previously known results for Dirac operators while Theorem 4.5 excludes spectral pollution for a class of nonselfadjoint Schrödinger operators which has not been possible to treat with existing methods.