Stability of uniformly eventually positive $C_0$-semigroups on $L_p$-spaces

TL;DR

This paper provides an elementary proof that the growth bound of positive $C_0$-semigroups on $L_p$-spaces equals the spectral bound of their generator, and extends this to uniformly eventually positive semigroups.

Contribution

It offers a simplified proof of Weis's theorem and generalizes it to uniformly eventually positive semigroups on $L_p$-spaces.

Findings

Growth bound equals spectral bound for positive $C_0$-semigroups.

Extension of the equality to uniformly eventually positive semigroups.

Elementary proof approach.

Abstract

We give a short and elementary proof of the theorem of Lutz Weis that the growth bound of a positive $C_0$-semigroup on $L_p(\mu)$ equals the spectral bound of its generator. In addition, we generalise the result to the case of uniformly eventually positive semigroups.