The closedness of the generator of a semigroup

TL;DR

This paper investigates conditions under which the generator of a semigroup of bounded operators on a Banach space is closed, using Laplace transforms and Pettis integrability to analyze weak topology continuity.

Contribution

It provides new sufficient conditions for the closedness of the generator of a semigroup with respect to various weak topologies, utilizing Laplace transforms and Pettis integrability.

Findings

Sufficient conditions for the generator to be closed under weak topologies

Use of Laplace transforms to analyze semigroup generators

Establishment of Pettis integrability criteria for vector-valued functions

Abstract

We study semigroups of bounded operators on a Banach space such that the members of the semigroup are continuous with respect to various weak topologies and we give sufficient conditions for the generator of the semigroup to be closed with respect to the topologies involved. The proofs of these results use the Laplace transforms of the semigroup. Thus we first give sufficient conditions for Pettis integrability of vector valued functions with respect to scalar measures.