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

Research paper by Jochen Glück, Manfred P. H. Wolff

Indexed on: 14 Feb '18Published on: 14 Feb '18Published in: arXiv - Mathematics - Functional Analysis



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.