Direct integrals of strongly continuous operator semigroups

TL;DR

This paper develops a theoretical framework for direct integrals of strongly continuous operator semigroups on Hilbert spaces, extending recent approaches for sums of Banach spaces and semigroups.

Contribution

It introduces existence, characterization, and decomposability results for direct integrals of $C_0$-semigroups on Hilbert spaces, paralleling recent work on Banach spaces.

Findings

Established conditions for the existence of direct integral semigroups.

Characterized asymptotic behavior of these semigroups.

Analyzed decomposability properties of the direct integral structures.

Abstract

The goal of this article is to develop a theory for direct integrals of $C_0$-semigroups on Hilbert spaces parallel to the recent approach by Lachowicz and Moszy\'nski for direct sums of Banach spaces, diagonal operators, and semigroups. In it we deal with the existence and characterisation of semigroups, asymptotic rates, and questions of decomposability.