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

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URI: http://nbn-resolving.de/urn:nbn:de:bsz:21-opus-2366
http://hdl.handle.net/10900/48156
Dokumentart: Dissertation
Date: 2001
Language: German
Faculty: 7 Mathematisch-Naturwissenschaftliche Fakultät
Department: Sonstige - Mathematik und Physik
Advisor: Nagel, Rainer
Day of Oral Examination: 2001-02-14
DDC Classifikation: 510 - Mathematics
Keywords: Stark Stetige Halbgruppen , Funktionalanalysis
Other Keywords: Trotter-Kato Approximationstheoreme , Trotter Produktformel , Bi-stetige Halbgruppe , Ornstein-Uhlenbeck Halbgruppe
Hille-Yosida Theorem , Trotter-Kato Approximation , Bi-continuous semigroup , Ornstein-Uhlenbeck semigroup
License: Publishing license including print on demand
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Inhaltszusammenfassung:

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.

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.

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