Indexed on: 10 Jan '19Published on: 10 Jan '19Published in: arXiv - Mathematics - Functional Analysis
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. Almost interior points appear in numerous articles and monographs, but many of their properties are scattered throughout the literature. Here we gather an overview of their basic properties, their relations to other concepts and their appearance in an operator theoretic context as well as a discussion of several examples and an open problem which we consider to be central to their theory. 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.