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Almost interior points in ordered Banach spaces and the long-term behaviour of strongly positive operator semigroups

Volume 254 / 2020

Jochen Glück, Martin R. Weber Studia Mathematica 254 (2020), 237-263 MSC: Primary 46B40; Secondary 47D06, 47B65. DOI: 10.4064/sm190111-18-10 Published online: 30 March 2020


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 generalisation 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.


  • Jochen GlückInstitut für Angewandte Analysis
    Universität Ulm
    89069 Ulm, Germany
  • Martin R. WeberInstitut für Analysis
    Technische Universität Dresden
    01062 Dresden, Germany

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