# On a Convergence Theorem for Semigroups of Positive Integral Operators

## TL;DR

This paper presents a simplified proof of Greiner's theorem on the strong convergence of positive contractive semigroups on L^p spaces, emphasizing the role of integral operators and positive fixed points.

## Contribution

The authors provide a more concise and elegant proof of a key convergence theorem for positive semigroups, extending the asymptotic theory with a streamlined approach.

## Key findings

- Proof simplifies Greiner's convergence theorem
- Highlights importance of integral operators and fixed points
- Suggests potential for broader generalizations

## Abstract

We give a new and very short proof of a theorem of Greiner asserting that a positive and contractive $C_0$-semigroup on an $L^p$-space is strongly convergent in case that it has a strictly positive fixed point and contains an integral operator. Our proof is a streamlined version of a much more general approach to the asymptotic theory of positive semigroups developed recently by the authors. Under the assumptions of Greiner's theorem, this approach becomes particularly elegant and simple. We also give an outlook on several generalisations of this result.