# Article

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Keywords:
not strongly continuous semigroups; bi-continuous semigroups; adjoint semigroup; mixed-topology; strict topology; one-parameter semigroups on the space of measures
Summary:
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.
References:
[1] Albanese, A. A., Manco, V., Lorenzi, L.: Mean ergodic theorems for bi-continuous semigroups. Semigroup Forum 82 (2011), 141-171. DOI 10.1007/s00233-010-9260-z | MR 2753839
[2] Albanese, A. A., Mangino, E.: Trotter-Kato theorems for bi-continuous semigroups and applications to Feller semigroups. J. Math. Anal. Appl. 289 (2004), 477-492. DOI 10.1016/j.jmaa.2003.08.032 | MR 2026919 | Zbl 1071.47043
[3] Alber, J.: On implemented semigroups. Semigroup Forum 63 (2001), 371-386. DOI 10.1007/s002330010082 | MR 1851817 | Zbl 1041.47028
[4] Alexandroff, A. D.: Additive set-functions in abstract spaces. Mat. Sb. N. Ser. 13 (1943), 169-238. MR 0012207 | Zbl 0060.13502
[5] Cerrai, S.: A Hille-Yosida theorem for weakly continuous semigroups. Semigroup Forum 49 (1994), 349-367. DOI 10.1007/BF02573496 | MR 1293091
[6] Dorroh, J. R., Neuberger, J. W.: Lie generators for semigroups of transformations on a Polish space. Electronic J. Diff. Equ. 1 (1993), 1-7. MR 1234797 | Zbl 0807.54033
[7] Dorroh, J. R., Neuberger, J. W.: A theory of strongly continuous semigroups in terms of Lie generators. J. Funct. Anal. 136 (1996), 114-126. DOI 10.1006/jfan.1996.0023 | MR 1375155 | Zbl 0848.22005
[8] Edgar, G. A.: Measurability in a Banach space, II. Indiana Univ. Math. J. 28 (1979), 559-579. DOI 10.1512/iumj.1979.28.28039 | MR 0542944 | Zbl 0418.46034
[9] Es-Sarhir, A., Farkas, B.: Perturbation for a class of transition semigroups on the Hölder space $C^\theta_{b, loc}(H)$. J. Math. Anal. Appl. 315 (2006), 666-685. DOI 10.1016/j.jmaa.2005.04.024 | MR 2202608 | Zbl 1097.47042
[9] Es-Sarhir, A., Farkas, B.: Perturbation for a class of transition semigroups on the Hölder space $C^\theta_{b, loc}(H)$. J. Math. Anal. Appl. 315 (2006), 666-685. DOI 10.1016/j.jmaa.2005.04.024 | MR 2202608 | Zbl 1097.47042

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