C 0 -sequentially equicontinuous semigroups

Abstract

We present and apply a theory of one-parameter $C_0$-semigroups of linear operators in locally convex spaces. Replacing the notion of equicontinuity considered by the literature with the weaker notion of sequential equicontinuity, we prove the basic results of the classical theory of $C_0$-equicontinuous semigroups: we show that the semigroup is uniquely identified by its generator and we provide a generation theorem in the spirit of the celebrated Hille–Yosida theorem. Then we particularize the theory in some functional spaces and identify two locally convex topologies that allow us to gather—under a unified framework—various notions of $C_0$-semigroups introduced by some authors to deal with Markov transition semigroups. Finally, we apply the results to transition semigroups associated to stochastic differential equations (SDEs).