PDE Seminar Abstracts

Spectrum and Asymptotics of Contraction Semigroups

Jochen Glück
University of Ulm, Germany
Friday 13 November 2015, 11:00-12:00, Carslaw Room 829 (AGR)

Abstract

It is well-known that positivity of an operator semigroup has important consequences for its spectrum and thus for its asymptotic behaviour. However, it might come as a surprise that similar conclusions can still be made if we replace the positivity assumption by another geometric invariance condition, namely contractivity.

In this talk we present several theorems which show how contractivity of an operator semigroup can affect its spectrum if the underlying Banach space satisfies certain geometric conditions. As a consequence we obtain the surprising result that a contractive, eventually norm-continuous ${C}_{0}$-semigroup on a real ${L}^{p}$-space automatically converges as $t\to \infty$ if $p\in \left(1,\infty \right)$ and $p\ne 2$.