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

Abstract

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.

On equicontinuity and tightness of bi-continuous semigroups

“…Example. We use the example of a τ co -bi-continuous semigroup (T (t)) t≥0 on the space C b (Ω) of bounded R-valued continuous functions on Ω from [31,Example 4.1,p. 320] where Ω ∶= [0, ω 1 ) is equipped with the order topology and ω 1 is the first uncountable ordinal.…”

“…Let βΩ be the Stone-Čech compactification of Ω, i.e. βΩ = [0, ω 1 ], and choose x ∈ βΩ as constructed in [31,Example 4 [31,Example 4.1,p. 320].…”

“…320]. Further, the operators T (t) for t > 0 are not τ cocontinuous on ⋅ ∞ -bounded sets by [31,Example 4.1,p. 320] and thus the semigroup (T (t)) t≥0 is not locally ( ⋅ ∞ , τ co )-equitight by Proposition 3.16 below.…”

“…Then there is a τ co -bi-continuous semigroup (T (t)) t≥0 on C b (Ω) such that T ○ (t) = S(t) for all t ≥ 0.Proof. The proof is almost the same as for[31, Theorem 3.6, p. 318]. We only need to replace the applications of[31, Theorem 3.5, p. 318] and[31, Theorem 3.1 c), p. 316-317] by Theorem 4.4 and Remark 3.24, respectively, if Ω is a hemicompact Hausdorff k R -space.…”

Towards a Koopman theory for dynamical systems on completely regular spaces

“…It follows from Proposition 2.10 that Koopman semigroups associated with jointly continuous semiflows on compactly generated completely regular spaces X form a bi-continuous semigroup with respect to the norm and the compact-open topologies. It should also be noted that in some cases the class of bi-continuous semigroups on C b ( X ) with respect to the compact-open topology is strictly larger than the one of locally equicontinuous, strongly continuous semigroups with respect to the mixed topology, see [39, Example 4.1]. However, if X is a Polish space or a σ -compact, locally compact space, then both classes coincide, see [39, Proposition 1.6, Theorem 3.1], [40, Remark 2.5], a basis for this being the fundamental work [30] of Sentilles on strict topologies.…”