A Pettis-Type Integral and Applications to Transition Semigroups

TL;DR

This paper develops a new Pettis-type integral framework for norming dual pairs, providing conditions for integrability and applications to transition semigroups, including their Laplace transforms and uniqueness properties.

Contribution

It introduces a novel Pettis-type integral on norming dual pairs and applies it to analyze semigroups related to transition processes, extending existing theories.

Findings

Established a sufficient condition for Pettis-type integrability.

Provided criteria for semigroups' Laplace transforms to be kernel operators.

Identified conditions for the uniqueness of semigroups via their Laplace transforms.

Abstract

Motivated by applications to transition semigroups, we introduce the notion of a norming dual pair and study a Pettis-type integral on such pairs. In particular, we establish a sufficient condition for integrability. We also introduce and study a class of semigroups on such dual pairs which are an abstract version of transition semigroups. Using our results, we give conditions ensuring that a semigroup consisting of kernel operators has a Laplace transform which also consists of kernel operators. We also provide conditions under which a semigroup is uniquely determined by its Laplace transform.