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 $C_b(E)$ which is continuous with respect to the strict topology $\beta_0$ is automatically quasi-equicontinuous with respect to that topology. We also give several characterizations of $\beta_0$-continuous semigroups on $C_b(E)$ and provide a convenient condition for the transition semigroup of a Banach space valued Markov process to be $\beta_0$-continuous.