Relatively Uniformly Continuous Semigroups on Vector Lattices

TL;DR

This paper investigates continuous semigroups of positive operators on vector lattices with the relative uniform topology, introducing new notions of continuity to analyze semigroups on non-locally convex and non-complete spaces.

Contribution

It introduces the concepts of strong and relative uniform continuity for semigroups on vector lattices, expanding the analysis to non-locally convex and incomplete spaces.

Findings

Translation, heat, and Koopman semigroups are relatively uniformly continuous on various spaces.

The paper extends semigroup theory to non-locally convex spaces like $L^p$ for $0<p<1$.

It develops new notions of continuity suited for non-complete and non-locally convex spaces.

Abstract

In this paper we study continuous semigroups of positive operators on general vector lattices equipped with the relative uniform topology $\tau_{ru}$. We introduce the notions of strong continuity with respect to $\tau_{ru}$ 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.