Adjoint bi-continuous semigroups and semigroups on the space of measures
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.
Cite
@article{arxiv.0805.4280,
title = {Adjoint bi-continuous semigroups and semigroups on the space of measures},
author = {Bálint Farkas},
journal= {arXiv preprint arXiv:0805.4280},
year = {2010}
}