Marco Marletta (Cardiff)

The essential numerical range of operators and pencils

Abstract:
We discuss several alternative characterisations of the essential numerical range We(T) for an unbounded operator T on a domain in a Hilbert space. Unlike the bounded case, studied by Stampfli and Williams in the 1960s, these definitions are no longer equivalent. We examine how they differ and which one(s) are of greatest use for applications. One of these definitions turns out to be the natural non-selfadjoint replacement for the extended essential spectrum used by Davies, and by Levitin and Shargorodsky, to analyse spectral pollution when approximating self-adjoint operators using projection methods. We also examine generalisations to linear operator pencils and generalised Morawetz tricks. These allow us to establish, e.g., substantial improvements of a result of Lewin and Sere on spectral pollution for Dirac systems.

This is joint work with Sabine Boegli (Durham) and Christiane Tretter (Bern).