Spectral and Asymptotic Properties of Contractive Semigroups on Non-Hilbert Spaces

TL;DR

This paper investigates the spectral and long-term behavior of contractive semigroups on non-Hilbert spaces, demonstrating conditions under which these semigroups exhibit trivial spectra and strong convergence.

Contribution

It provides new insights into the spectral properties and asymptotic behavior of contractive semigroups on non-Hilbert spaces like L^p, extending classical results.

Findings

Semigroup generators have trivial spectrum on the imaginary axis under certain conditions.

Contractive, eventually norm continuous semigroups on L^p spaces converge strongly for p not in {1,2,∞}.

Geometry of the unit ball influences spectral and asymptotic properties.

Abstract

We analyse $C_0$-semigroups of contractive operators on real-valued $L^p$-spaces for $p \not= 2$ and on other classes of non-Hilbert spaces. We show that, under some regularity assumptions on the semigroup, the geometry of the unit ball of those spaces forces the semigroup's generator to have only trivial (point) spectrum on the imaginary axis. This has interesting consequences for the asymptotic behaviour as $t \to \infty$. For example, we can show that a contractive and eventually norm continuous $C_0$-semigroup on a real-valued $L^p$-space automatically converges strongly if $p \not\in \{1,2,\infty\}$.