Weighted composition semigroups on Banach spaces of holomorphic functions

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' with D(Gamma) = {f is an element of X vertical bar gf +Gf' is an element of 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.