Order boundedness and order continuity properties of positive operator semigroups

TL;DR

This paper compares relatively uniformly continuous semigroups with classical $C_0$-semigroups in Banach lattices, establishing order boundedness characterizations and exploring implications for spectral theory and convergence properties.

Contribution

It characterizes ruc semigroups as positive $C_0$-semigroups with order bounded orbits and links this to spectral bounds, order boundedness principles, and unbounded order convergence.

Findings

ruc semigroups are exactly positive $C_0$-semigroups with order bounded orbits

spectral bound equals growth bound for certain positive $C_0$-semigroups

unbounded order convergence relates to almost everywhere convergence

Abstract

Relatively uniformly continuous (ruc) semigroups were recently introduced and studied by Kandi\'c, Kramar-Fijav\v{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 order bounded for small times. We then relate this result 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.