Marco Marletta: The essential numerical range for unbounded linear operators

Location

FR4 (Oskar Klein), Albanova

Schedule

15:15–15:30 SMC prizes for excellent master theses the academic year 2023/2024 will be announced and awarded.
15:30–16:30 Colloquium lecture by Marco Marletta.
16:30–17:00 SMC social get together with refreshments.

Abstract

The numerical range of a linear operator on a domain in a Hilbert space is a convex set in the complex plane which, under mild hypotheses, contains the spectrum of the operator. In the 1960s the concept of essential numerical range was introduced for bounded linear operators, giving a way to enclose the essential spectrum and (thanks to work of Pokrzywa in the 1970s) study phenomena such as spectral pollution. 

This talk presents my recent work with Bögli and Tretter in which we study essential numerical ranges for unbounded linear operators and for operator pencils (when the numerical range need not be convex). We show that using these concepts one can even improve some results from the selfadjoint case, without assuming selfadjointness. For the case of linear pencils we show that an abstract version of a PDE trick of Morawetz can even be used to characterise the approximate point spectrum precisely by intersecting numerical ranges. Applications are given to Stokes problems, Dirac equations and (if time permits) some Maxwell systems.