The Hille-Yosida generation theorem for almost surely bounded $C_{0}$--semigroups of continuous module homomorphisms$^1$

TL;DR

This paper extends the Hille-Yosida theorem to almost surely bounded $C_{0}$-semigroups of continuous module homomorphisms, providing new characterizations and counterexamples to highlight the importance of boundedness conditions.

Contribution

It introduces a generalized Hille-Yosida theorem for almost surely bounded $C_{0}$-semigroups of module homomorphisms, including new characterizations and a necessary boundedness condition.

Findings

Characterization of almost surely bounded $C_{0}$-semigroups

Establishment of a generalized Hille-Yosida theorem

Counterexample demonstrating the necessity of boundedness

Abstract

In this paper, we first study some properties peculiar to $C_{0}$--semigroups of continuous module homomorphisms and give a characterization for such a $C_{0}$--semigroup to be almost surely bounded. Then, based on these, we establish the Hille-Yosida generation theorem for almost surely bounded $C_{0}$--semigroups of continuous module homomorphisms, which generalizes some known results. Moreover, the counterexample constructed in this paper also shows that it is necessary to require the almost sure boundedness for such $C_{0}$--semigroups.