Michael Kaplin | Searcharxiv

arXiv2022

Order boundedness and order continuity properties of positive operator semigroups

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.

math.FA↗

arXiv2018

Relatively Uniformly Continuous Semigroups on Vector Lattices

In this paper we study continuous semigroups of positive operators on general vector lattices equipped with the relative uniform topology $τ_{ru}$. We introduce the notions of strong continuity with respect to $τ_{ru}$ and relative uniform continuity for semigroups. These notions allow us to study semigroups on non-locally convex spaces such as $L^p(\mathbb{R})$ for $0<p<1$ and non-complete spaces such as $Lip(\mathbb{R})$, $UC(\mathbb{R})$, and $C_c(\mathbb{R})$. We show that the (left) translation semigroup on the real line, the heat semigroup and some Koopman semigroups are relatively uniformly continuous on a variety of spaces.

math.FA↗