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

Title:

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

Creator:

Farkas, Bálint

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:4e7c0660-e137-4c89-9532-cf80adc7ed25
uuid:4e7c0660-e137-4c89-9532-cf80adc7ed25

Subject:

not strongly continuous semigroups, bi-continuous semigroups, adjoint semigroup, mixed-topology, strict topology, and one-parameter semigroups on the space of measures

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

309-322

Source:

Czechoslovak Mathematical Journal | 2011 Volume:61 | Number:2

Harvested from:

CDK

Metadata only:

false