# Convergence of Positive Operator Semigroups

## TL;DR

This paper introduces new algebraic conditions ensuring the strong convergence of positive operator semigroups over time, extending classical results without requiring continuity assumptions.

## Contribution

It combines the Jacobs, de Leeuw, and Glicksberg splitting theorem with algebraic methods to generalize convergence results for a broad class of semigroup representations.

## Key findings

- Established new convergence conditions for positive operator semigroups.
- Unified various existing theorems under a broader algebraic framework.
- Removed the need for continuity or regularity assumptions in convergence proofs.

## Abstract

We present new conditions for semigroups of positive operators to converge strongly as time tends to infinity. Our proofs are based on a novel approach combining the well-known splitting theorem by Jacobs, de Leeuw and Glicksberg with a purely algebraic result about positive group representations. Thus we obtain convergence theorems not only for one-parameter semigroups but for a much larger class of semigroup representations. Our results allow for a unified treatment of various theorems from the literature that, under technical assumptions, a bounded positive $C_0$-semigroup containing or dominating a kernel operator converges strongly as $t \to \infty$. We gain new insights into the structure theoretical background of those theorems and generalise them in several respects; especially we drop any kind of continuity or regularity assumption with respect to the time parameter.