Continuity and Equicontinuity of Transition Semigroups

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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.