Marco Marletta

The essential numerical range for unbounded linear operators and pencils

The concept of essential numerical range was introduced in the 1960s in the context of Calkin algebras of bounded operators, independently by Filmore and by Stampfli and Williams, and studied also by Pokrzywa.
We introduce the concept of essential numerical range W_e(T) for unbounded Hilbert space operators T and study its fundamental properties including possible equivalent characterizations and perturbation results. Many of the properties known for the bounded case do not carry over to the unbounded case, and new interesting phenomena arise which we illustrate by some striking examples. A key feature of the essential numerical range We(T) is that it captures spectral pollution in a unified and minimal way when approximating T by projection methods or domain truncation methods for PDEs. We also discuss the concept of essential numerical range for a linear pencil A-\lambda B, which allows us to introduce a new abstract version of a trick devised by Morawetz for scattering problems. We present applications involving a-priori spectral enclosures for Maxwell and Dirac equations.
This is joint work with Sabine Boegli (Durham), Francesco Ferraresso (Cardiff) and Christiane Tretter (Bern).