Adjoint bi-continuous semigroups and semigroups on the space of measures

TL;DR

This paper explores the properties of adjoint bi-continuous semigroups on Banach spaces, particularly on spaces of measures and continuous functions, establishing their relationships and characterizations under various topologies.

Contribution

It introduces a framework for adjoint bi-continuous semigroups, characterizes them on measure and function spaces, and compares different classes of semigroups on C(K).

Findings

Bi-continuous semigroups on M(K) are adjoints of those on C(K).

On Polish spaces, bi-continuous and equicontinuous semigroups coincide.

The equivalence of semigroup classes fails for non-Polish spaces.

Abstract

For a given bi-continuous semigroup T on a Banach space X we define its adjoint on an appropriate closed subspace X^o of the norm dual X'. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology (X^o,X). An application is the following: For K a Polish space we consider operator semigroups on the space C(K) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(K) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(K) are precisely those that are adjoints of a bi-continuous semigroups on C(K). We also prove that the class of bi-continuous semigroups on C(K) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if K is not Polish space this is not the case.