A Pettis-type integral and applications to transition semigroups

Kunze, Markus

doi:10.1007/s10587-011-0065-3

Cited by 31 publications

(60 citation statements)

References 26 publications

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“…So fix a closed set

F \subset Ω

. By [, Thm. 6.3] there exists a countable set

M \subset C_{b} (F)

such that for all measures

μ \in M (F)

μ \neq 0

, there exists

f \in M

with

⟨ μ, f ⟩ \neq 0

.…”

Section: The Lattice Of Kernel Operatorsmentioning

confidence: 99%

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On the lattice structure of kernel operators

Gerlach

Kunze

2014

Math. Nachr.

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Abstract. Consider the lattice of bounded linear operators on the space of Borel measures on a Polish space. We prove that the operators which are continuous with respect to the weak topology induced by the bounded measurable functions form a sublattice that is lattice isomorphic to the space of transition kernels. As an application we present a purely analytic proof of Doob's theorem concerning stability of transition semigroups.

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“…So fix a closed set

F \subset Ω

. By [, Thm. 6.3] there exists a countable set

M \subset C_{b} (F)

such that for all measures

μ \in M (F)

μ \neq 0

, there exists

f \in M

with

⟨ μ, f ⟩ \neq 0

.…”

Section: The Lattice Of Kernel Operatorsmentioning

confidence: 99%

“…The following characterization of operators of this form follows from Propositions 3.1 and 3.5 of [15].…”

Section: The Lattice Of Kernel Operatorsmentioning

confidence: 99%

On the lattice structure of kernel operators

Gerlach

Kunze

2014

Math. Nachr.

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“…The dual of C b (Ω) (as a Banach space) is isomorphic to the space M(βΩ) of all bounded Baire measures on the Stone-Čech compactification βΩ of Ω, and the isomorphism is given by ϕ(f ) = βΩ f dµ. (One can represent a continuous linear function even by regular Borel measures, in this respect we refer to Knowles [21] and Mařík [25]. )…”

Section: Semigroups On the Space Continuous Functions And Of Measuresmentioning

confidence: 99%

Adjoint bi-continuous semigroups and semigroups on the space of measures

Farkas

2011

Czech Math J

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For a given bi-continuous semigroup (T (t)) t 0 on a Banach space X we define its adjoint on an appropriate closed subspace X • of the norm dual X ′ . Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X • , X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C b (Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak * -topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C b (Ω). We also prove that the class of bi-continuous semigroups on C b (Ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if Ω is not a Polish space this is not the case.

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“…Bi-continuity has the drawback, however, that it is a non-topological notion. Kunze [23,24] studies semigroups of which he assumes that the resolvent can be given in integral form. His notions are topological, and he gives a Hille-Yosida theorem for equi-continuous semigroups.…”

Section: Introductionmentioning

confidence: 99%

Strongly continuous and locally equi-continuous semigroups on locally convex spaces

Kraaij

2015

Semigroup Forum

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We consider locally equi-continuous strongly continuous semigroups on locally convex spaces (X, τ ) that are also equipped with a 'suitable' auxiliary norm. We introduce the set N of τ -continuous semi-norms that are bounded by the norm. If (X, τ ) has the property that N is closed under countable convex combinations, then a number of Banach space results can be generalised in a straightforward way. Importantly, we extend the Hille-Yosida theorem. We relate our results to those on bicontinuous semigroups and show that they can be applied to semigroups on (C b (E), β) and (B(H), β) for a Polish space E and a Hilbert space H and where β is their respective strict topology.

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A Pettis-type integral and applications to transition semigroups

Cited by 31 publications

References 26 publications

On the lattice structure of kernel operators

On the lattice structure of kernel operators

Adjoint bi-continuous semigroups and semigroups on the space of measures

Strongly continuous and locally equi-continuous semigroups on locally convex spaces

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