On perturbed substochastic semigroups in abstract state spaces

Luisa Arlotti (1) , Bertrand Lods (2) , Mustapha Mokhtar-Kharroubi (3)

1 Università degli Studi di Udine - University of Udine [Italie]
2 LMBP - Laboratoire de Mathématiques Blaise Pascal
3 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

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.

Mots clés: ubstochastic semigroups; additive norm; total mass carried by a trajectory; Dyson-Phillips expansion.; Dyson-Phillips expansion

Domaines: Equations aux dérivées partielles [math.AP]

Luisa Arlotti, Bertrand Lods, Mustapha Mokhtar-Kharroubi. On perturbed substochastic semigroups in abstract state spaces. 2009. ⟨hal-00395363⟩