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 $\Gamma f=gf+G{f}^{\prime }$ with $D(\Gamma )=\{f\in X|gf+G{f}^{\prime }\in X\}$ 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.