The closedness of the generator of a semigroup

Abstract

Abstract. We study semigroups of bounded operators on a Banach space such that the members of the semigroup are continuous with respect to various weak topologies and we give sufficient conditions for the generator of the semigroup to be closed with respect to the topologies involved. The proofs of these results use the Laplace transforms of the semigroup. Thus we first give sufficient conditions for Pettis integrability of vector valued functions with respect to scalar measures. MotivationThe motivation for this article comes from the area of quantum Markov semigroups (QMSs) and the well known problem of describing the form of the generator of a general QMS. A semigroup on a Banach space X is a family (T t ) t≥0 of bounded operators on X satisfying T 0 = 1 and T t T s = T t+s for t, s ≥ 0.A QMS on a von Neumann algebra M is a semigroup (T t ) t≥0 on M such that each T t is completely positive, σ-weakly-σ-weakly continuous, and satisfies T t (1) = 1, and the map t → T t A is σ-weakly continuous for each A ∈ M. This definition was finalized in 1976 by Lindblad [17] who was able to describe the form of the generator of the semigroup in the case that the von Neumann algebra M is semifinite and the map t → T t is norm continuous.

Completeness in the Mackey topology by norming subspaces

“…This research line was motivated initially by Kunze's paper [11] on vector integration (cf. [1,15]). Bonet and Cascales [3] exploited some results of [7] to prove that if X contains a subspace isomorphic to ℓ 1 (c), then there is a norming and norm-closed subspace Y ⊂ X * for which (X, µ(X, Y )) is not complete.…”

On integration in banach spaces and total sets

“…For instance, Diestel and Faires (see [12,Corollary 1.3]) proved that Gelfand and Pettis integrability coincide for any strongly measurable function f : Ω → X * whenever X * contains no subspace isomorphic to ℓ ∞ , while Musia l (see [28,Theorem 4]) showed that one can give up strong measurability if the assumption on X is strengthened to being w * -sequentially dense in X * * . More recently, Γ-integrability has been studied in [2,25] in connection with semigroups of operators. This paper is organized as follows.…”