Stability of (eventually) positive semigroups on spaces of continuous functions

TL;DR

This paper provides a concise proof demonstrating that for positive $C_0$-semigroups on spaces of continuous functions, the spectral bound equals the growth bound, extending to eventually positive semigroups and clarifying the role of the underlying space.

Contribution

It introduces a new, simplified proof technique that works even when the space lacks constant functions, and extends the result to eventually positive semigroups.

Findings

Spectral and growth bounds coincide for positive $C_0$-semigroups.

The proof is transparent and applicable even without constant functions in the space.

Extension of the result to eventually positive semigroups.

Abstract

We present a new and very short proof of the fact that, for positive $C_0$-semigroups on spaces of continuous functions, the spectral and the growth bound coincide. Our argument, inspired by an idea of Vogt, makes the role of the underlying space completely transparent and also works if the space does not contain the constant functions - a situation in which all earlier proofs become technically quite involved. We also show how the argument can be adapted to yield the same result for semigroups that are only eventually positive rather than positive.