ADJOINT BI-CONTINUOUS SEMIGROUPS AND SEMIGROUPS ON THE SPACE OF MEASURES

作者：Farkas Balint^*

来源：Czechoslovak Mathematical Journal, 2011, 61(2): 309-322.

DOI：10.1007/s10587-011-0076-0

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

For a given bi-continuous semigroup (T(t))(t >= 0) on a Banach space X we de fine its adjoint on an appropriate closed subspace X-o of the norm dual X'. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology sigma (X-o, X). We give the following application: For Omega a Polish space we consider operator semigroups on the space C-b(Omega) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Omega) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Omega) are precisely those that are adjoints of bi-continuous semigroups on C-b(Omega). We also prove that the class of bi-continuous semigroups on 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.

• 出版日期2011-6