Weighted composition semigroups on Banach spaces of holomorphic functions

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 ($\mathrm{\Gamma },D(\mathrm{\Gamma })$) is given by $\mathrm{\Gamma }f=gf+G{f}^{\mathrm{\prime }}$ 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 $(\mathrm{\Gamma },D(\mathrm{\Gamma }))$, define like above, generate a strongly continuous semigroup, then this one is a family of weighted composition operators.