Weighted composition semigroups on spaces of continuous functions and their subspaces

Abstract

This paper is dedicated to weighted composition semigroups on spaces of continuous functions and their subspaces. We consider semigroups induced by semiflows and semicocycles on Banach spaces $\mathcal{F}(\Omega )$ of continuous functions on a Hausdorff space $\Omega$ such that the norm-topology is stronger than the compact-open topology like the Hardy spaces, the weighted Bergman spaces, the Dirichlet space, the Bloch type spaces, the space of bounded Dirichlet series and weighted spaces of continuous or holomorphic functions. It was shown by Gallardo-Gutiérrez, Siskakis and Yakubovich that there are no non-trivial norm-strongly continuous weighted composition semigroups on Banach spaces $\mathcal{F}(\mathbb{D})$ of holomorphic functions on the open unit disc $\mathbb{D}$ such that $H^{\infty }\subset \mathcal{F}(\mathbb{D})\subset {\mathcal{B}}_1$ where $H^{\infty }$ is the Hardy space of bounded holomorphic functions on $\mathbb{D}$ and ${\mathcal{B}}_1$ the Bloch space. However, we show that there are non-trivial weighted composition semigroups on such spaces which are strongly continuous w.r.t. the mixed topology between the norm-topology and the compact-open topology. We study such weighted composition semigroups in the general setting of Banach spaces of continuous functions and derive necessary and sufficient conditions on the spaces involved, the semiflows and semicocycles for strong continuity w.r.t. the mixed topology and as a byproduct for norm-strong continuity as well. Moreover, we give several characterisations of their generator and their space of norm-strong continuity.

References in corpus (14)

• Continuity and Equicontinuity of Transition Semigroups
• Strongly continuous and locally equi-continuous semigroups on locally convex spaces
• Intermediate and extrapolated spaces for bi-continuous semigroups
• Composition Semigroups on BMOA and $H^{\infty }$
• On generators of $C_0$-semigroups of composition operators
• On equicontinuity and tightness of bi-continuous semigroups
• Semigroups of composition operators and Integral operators in BMOA-type spaces
• Subordination for sequentially equicontinuous equibounded $C_0$-semigroups
• Final state observability estimates and cost-uniform approximate null-controllability for bi-continuous semigroups
• Towards a Koopman theory for dynamical systems on completely regular spaces
• Holomorphic semigroups and Sarason's characterization of vanishing mean oscillation
• A note on the Lumer--Phillips theorem for bi-continuous semigroups
• Dirichlet-to-Robin Operators via Composition Semigroups
• Mixed topologies on Saks spaces of vector-valued functions