Weighted composition semigroups on Banach spaces of holomorphic functions

Abstract

We study, to certain Banach spaces $X$, families of weighted composition operators. Notably, we show that if this family form 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 maim result is to study the reciprocal implication. That is if $(\Gamma , D(\Gamma))$, define like above, generate a strongly continuous semigroup, then this one is a family of weighted composition operators.