Order boundedness and order continuity properties of positive operator semigroups

DOI10.2989/16073606.2023.2287831arXiv2212.00076OpenAlexW4392594241MaRDI QIDQ6132176

Author name not available (Why is that?)

Publication date: 18 April 2024

Published in: Quaestiones Mathematicae (Search for Journal in Brave)

Abstract: Relatively uniformly continuous (ruc) semigroups were recently introduced and studied by Kandi'c, Kramar-Fijavv{z}, and the second-named author, in order to make the theory of one-parameter operator semigroups available in the setting of vector lattices, where no norm is present in general. In this article, we return to the more standard Banach lattice setting - where both ruc semigroups and

C_{0}

-semigroups are well-defined concepts - and compare both notions. We show that the ruc semigroups are precisely those positive

C_{0}

-semigroups whose orbits are ordered bounded for small times. We then relate the main result of the paper to three different topics: (i) equality of the spectral and the growth bound for positive

C_{0}

-semigroups; (ii) a uniform order boundedness principle which holds for all operator families between Banach lattices; and (iii) a description of unbounded order convergence in terms of almost everywhere convergence for nets which have an uncountable index set containing a co-final sequence.

Full work available at URL: https://arxiv.org/abs/2212.00076

zbMATH Keywords

maximal inequality order convergence positive \(C_{0}\)-semigroup relatively uniform convergence ru-continuous semigroup

Mathematics Subject Classification ID

One-parameter semigroups and linear evolution equations (47D06) Banach lattices (46B42) Positive linear operators and order-bounded operators (47B65) Ordered topological linear spaces, vector lattices (46A40)