Continuity and Equicontinuity of Transition Semigroups

We study continuity and equicontinuity of semigroups on norming dual pairs with respect to topologies defined in terms of the duality. In particular, we address the question whether continuity of a semigroup already implies (local/quasi) equicontinuity. We apply our results to transition semigroups and show that, under suitable hypothesis on E, every transition semigroup on Cb(E) which is continuous with respect to the strict topology β0 is automatically quasi-equicontinuous with respect to that topology. We also give several characterizations of β0-continuous semigroups on Cb(E) and provide a convenient condition for the transition semigroup of a Banach space valued Markov process to be β0-continuous.