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

Farkas, Bálint. "Adjoint bi-continuous semigroups and semigroups on the space of measures." Czechoslovak Mathematical Journal 61.2 (2011): 309-322. <http://eudml.org/doc/197213>.

@article{Farkas2011,
abstract = {For a given bi-continuous semigroup $(T(t))_\{t\ge 0\}$ on a Banach space $X$ we define its adjoint on an appropriate closed subspace $X^\circ $ of the norm dual $X^\{\prime \}$. 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.},
author = {Farkas, Bálint},
journal = {Czechoslovak Mathematical Journal},
keywords = {not strongly continuous semigroups; bi-continuous semigroups; adjoint semigroup; mixed-topology; strict topology; one-parameter semigroups on the space of measures; not strongly continuous semigroup; bi-continuous semigroup; adjoint semigroup; mixed topology; strict topology; one-parameter semigroups on the space of measures},
language = {eng},
number = {2},
pages = {309-322},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Adjoint bi-continuous semigroups and semigroups on the space of measures},
url = {http://eudml.org/doc/197213},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Farkas, Bálint
TI - Adjoint bi-continuous semigroups and semigroups on the space of measures
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 2
SP - 309
EP - 322
AB - For a given bi-continuous semigroup $(T(t))_{t\ge 0}$ on a Banach space $X$ we define its adjoint on an appropriate closed subspace $X^\circ $ of the norm dual $X^{\prime }$. 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.
LA - eng
KW - not strongly continuous semigroups; bi-continuous semigroups; adjoint semigroup; mixed-topology; strict topology; one-parameter semigroups on the space of measures; not strongly continuous semigroup; bi-continuous semigroup; adjoint semigroup; mixed topology; strict topology; one-parameter semigroups on the space of measures
UR - http://eudml.org/doc/197213
ER -