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This paper is dedicated to weighted composition semigroups on spaces ofcontinuous functions and their subspaces. We consider semigroups induced bysemiflows and semicocycles on Banach spaces $\mathcal{F}(\Omega)$ of continuousfunctions on a Hausdorff space $\Omega$ such that the norm-topology is coarserthan the compact-open topology like the Hardy spaces, the weighted Bergmanspaces, the Dirichlet space, the Bloch type spaces and weighted spaces ofcontinuous or holomorphic functions. It was shown by Gallardo-Guti\'errez,Siskakis and Yakubovich that there are no non-trivial norm-strongly continuousweighted composition semigroups on Banach spaces $\mathcal{F}(\mathbb{D})$ ofholomorphic 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 thereare non-trivial weighted composition semigroups on such spaces which arestrongly continuous w.r.t. the mixed topology between the norm-topology and thecompact-open topology. We study such weighted composition semigroups in thegeneral setting of Banach spaces of continuous functions and derive necessaryand sufficient conditions on the spaces involved, the semiflows andsemicocycles for strong continuity w.r.t. the mixed topology and as a byproductfor norm-strong continuity as well. Moreover, we give several characterisationsof their generator and their space of norm-strong continuity.

Weighted composition semigroups on spaces of continuous functions and their subspaces