Order boundedness and order continuity properties of positive operator semigroups

Abstract

Relatively uniformly continuous (ruc) semigroups were recently introduced and studied by Kandic, Kramar-Fijavˇz, 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 C0-semigroups  are well-defined concepts – and compare both notions. We show that the ruc semigroups are precisely those positive C0- 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 C0-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.