Long--Term Analysis of Positive Operator Semigroups via Asymptotic Domination

TL;DR

This paper investigates the long-term behavior of positive operator semigroups on ordered Banach spaces, focusing on how asymptotic domination influences properties like almost periodicity, mean ergodicity, and strong convergence.

Contribution

It introduces new conditions under which asymptotic domination ensures inheritance of ergodic properties and strong convergence, extending existing results to broader classes of Banach spaces.

Findings

Almost periodicity and mean ergodicity are inherited under asymptotic domination.

Asymptotic domination of orbits by a positive vector often implies strong convergence.

Results apply to both discrete and continuous time semigroups, generalizing prior work.

Abstract

We consider positive operator semigroups on ordered Banach spac\-es and study the relation of their long time behaviour to two different domination properties. First, we analyse under which conditions almost periodicity and mean ergodicity of a semigroup $\mathcal{T}$ are inherited by other semigroups which are asymptotically dominated by $\mathcal{T}$. Then, we consider semigroups whose orbits asymptotically dominate a positive vector and show that this assumption is often sufficient to conclude strong convergence of the semigroup as time tends to infinity. Our theorems are applicable to time-discrete as well as time-continuous semigroups. They generalise several results from the literature to considerably larger classes of ordered Banach spaces.