Weighted composition semigroups on Banach spaces of holomorphic functions

Abstract

We study, on certain Banach spaces X, families of weighted composition operators. Notably, we show that if this family forms a strongly continuous semigroup, then its infinitesimal generator (\(\Gamma , D(\Gamma )\)) is given by \(\Gamma f = gf+Gf^\prime \) with \(D(\Gamma ) = \{ f\in X \ | \ gf+Gf^\prime \in X \}\) where g, G are holomorphic functions. Moreover, our second main result is to study the reciprocal implication. That is, if \((\Gamma , D(\Gamma ))\), defined as above, generates a strongly continuous semigroup, then this one is a family of weighted composition operators.