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Zeitschrift für Analysis und ihre Anwendungen

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Volume 30, Issue 4, 2011, pp. 457–495
DOI: 10.4171/ZAA/1444

Published online: 2011-09-29

On Perturbed Substochastic Semigroups in Abstract State Spaces

Luisa Arlotti[1], B. Lods[2] and M. Mokhtar-Kharroubi[3]

(1) Università di Udine, Italy
(2) Università degli Studi di Torino, Italy
(3) Université de Franche-Comté, Besançon, France

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.

Keywords: Substochastic semigroups, additive norm, total mass carried by a trajectory, Dyson-Phillips expansion

Arlotti Luisa, Lods B., Mokhtar-Kharroubi M.: On Perturbed Substochastic Semigroups in Abstract State Spaces. Z. Anal. Anwend. 30 (2011), 457-495. doi: 10.4171/ZAA/1444