On equicontinuity and tightness of bi-continuous semigroups

TL;DR

This paper investigates the properties of bi-continuous semigroups, focusing on minimal conditions for equicontinuity and tightness within locally convex topologies, and clarifies their interrelations through examples.

Contribution

It provides new minimal assumptions for equicontinuity and tightness in bi-continuous semigroups and clarifies their relationships using topological vector space techniques.

Findings

Established minimal conditions for tightness and equicontinuity.

Clarified relationships between different notions of these properties.

Applied results to semigroups on spaces of bounded continuous functions.

Abstract

In contrast to classical strongly continuous semigroups, the study of bi-continuous semigroups comes with some freedom in the properties of the associated locally convex topology. This paper aims to give minimal assumptions in order to recover typical features like tightness and equicontinuity with respect to the mixed topology as well as to carefully clarify on mutual relations between previously studied variants of these notions. The abstract results -- exploiting techniques from topological vector spaces -- are thoroughly discussed by means of several example classes, such as semigroups on spaces of bounded continuous functions.