$C_0$-sequentially equicontinuous semigroups: theory and applications

TL;DR

This paper develops a theory of $C_0$-semigroups in locally convex spaces using sequential equicontinuity, establishing fundamental results and applying them to transition semigroups in functional spaces.

Contribution

It introduces a weaker notion of equicontinuity for $C_0$-semigroups, proving core theorems and unifying various existing notions in a single framework.

Findings

Established a generation theorem akin to Hille-Yosida.

Unified different $C_0$-semigroup notions under two topologies.

Applied the theory to transition semigroups in functional spaces.

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 to gather under a unified framework various notions $C_0$-semigroup introduced by some authors to deal with Markov transition semigroup. Finally, we apply the results to transition semigroups.