On perturbed substochastic semigroups in abstract state spaces

TL;DR

This paper develops a generalized honesty theory for perturbed substochastic semigroups across various abstract state spaces, unifying existing approaches and introducing a novel semigroup-perturbation perspective.

Contribution

It extends honesty theory to general state spaces, including noncommutative spaces, and introduces a new perturbation approach that is shown to be equivalent to existing methods.

Findings

Unified honesty theory for various state spaces

Introduction of a new semigroup-perturbation honesty approach

Revisiting classical generation theorems with new insights

Abstract

The object of this paper is twofold: In the first part, we unify and extend the recent developments on honesty theory of perturbed substochastic semigroups (on $L^{1}(\mu)$-spaces or noncommutative $L^{1}$ spaces) to general state spaces; this allows us to capture for instance a honesty theory in preduals of abstract von Neumann algebras or subspaces of duals of abstract $C^{\ast }$-algebras. In the second part of the paper, we provide another honesty theory (a semigroup-perturbation approach) independent of the previous resolvent-perturbation approach and show the equivalence of the two approaches. This second viewpoint on honesty is new even in $L^{1}(\mu)$ spaces. Several fine properties of Dyson-Phillips expansions are given and a classical generation theorem by T. Kato is revisited.