Bi-stetige Halbgruppen auf Raeumen mit zwei Topologien: Theorie und Anwendungen

Abstract

In order to deal with semigroups on Banach spaces which are not strongly continuous we introduce the concept of bi-continuous semigroups on spaces with two topologies. To that purpose we consider Banach spaces with an aditional locally convex topology tau which is coarser than the norm topology and such that the topological dual (X, tau)' is norming for X endowed with its norm. On such spaces we define bi-continuous semigroups as semigroups consisting of bounded linear operators which are locally bi-equicontinuous tau and such that the orbit maps are tau-continuous. We show that these semigroups allow, as in the theory of strongly continuous semigroups, a systematic theory including Hille-Yosida and Trotter-Kato type theorems. A long series of applications including semigroups induced by flows, the Ornstein-Uhlenbeck semigroup on C_b(H), adjoint semigroups, and implemented semigroups, shows the flexibility and strength of our theory.

In order to deal with semigroups on Banach spaces which are not strongly continuous we introduce the concept of bi-continuous semigroups on spaces with two topologies. To that purpose we consider Banach spaces with an aditional locally convex topology tau which is coarser than the norm topology and such that the topological dual (X, tau)' is norming for X endowed with its norm. On such spaces we define bi-continuous semigroups as semigroups consisting of bounded linear operators which are locally bi-equicontinuous tau and such that the orbit maps are tau-continuous. We show that these semigroups allow, as in the theory of strongly continuous semigroups, a systematic theory including Hille-Yosida and Trotter-Kato type theorems. A long series of applications including semigroups induced by flows, the Ornstein-Uhlenbeck semigroup on C_b(H), adjoint semigroups, and implemented semigroups, shows the flexibility and strength of our theory.