On equicontinuity and tightness of bi-continuous semigroups

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.