Generation of relatively uniformly continuous Semigroups on Vector Lattices

TL;DR

This paper establishes a Hille-Yosida type theorem characterizing generators of relatively uniformly continuous positive semigroups on vector lattices, expanding the theoretical framework for such semigroups.

Contribution

It introduces new notions of uniform continuity, differentiability, and integrability on vector lattices and provides necessary and sufficient conditions for operators to generate these semigroups.

Findings

Characterization of generators of relatively uniformly continuous positive semigroups

Introduction of new concepts of uniform continuity, differentiability, and integrability on vector lattices

Main theorem providing necessary and sufficient conditions for generator operators

Abstract

In this paper we prove a Hille-Yosida type theorem for relatively uniformly continuous positive semigroups on vector lattices. We introduce the notions of relatively uniformly continuous, differentiable, and integrable functions on $\mathbb{R}_+$. These notions allow us to study the generators of relatively uniformly continuous semigroups. Our main result provides sufficient and necessary conditions for an operator to be the generator of an exponentially order bounded, relatively uniformly continuous, positive semigroup.