# Almost Interior Points in Ordered Banach Spaces and the Long--Term Behaviour of Strongly Positive Operator Semigroups

## TL;DR

This paper surveys almost interior points in ordered Banach spaces and investigates the long-term behavior of strongly positive operator semigroups, providing criteria for their convergence.

## Contribution

It introduces a generalization of quasi-interior points and extends convergence results for strongly positive semigroups beyond Banach lattices.

## Key findings

- Criteria for strong or norm convergence of semigroups
- Generalization of known results to broader classes of spaces
- Connection between almost interior points and long-term semigroup behavior

## Abstract

The first part of this article is a brief survey of the properties of so-called almost interior points in ordered Banach spaces. Those vectors can be seen as a generalization of ``functions which are strictly positive almost everywhere'' on $L^p$-spaces and of ``quasi-interior points'' in Banach lattices. In the second part we study the long--term behaviour of strongly positive operator semigroups on ordered Banach spaces; these are semigroups which, in a sense, map every non-zero positive vector to an almost interior point. Using the Jacobs--de Leeuw--Glicksberg decomposition together with the theory presented in the first part of the paper we deduce sufficiency criteria for such semigroups to converge (strongly or in operator norm) as time tends to infinity. This generalises known results for semigroups on Banach lattices as well as on normally ordered Banach spaces with unit.