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We prove that in a large class of Banach spaces of analytic functions in theunit disc $\mathbb{D}$ an (unbounded) operator $Af=G\cdot f'+g\cdot f$ with$G,\, g$ analytic in $\mathbb{D}$ generates a $C_0$-semigroup of weightedcomposition operators if and only if it generates a $C_0$-semigroup. Particularinstances of such spaces are the classical Hardy spaces. Our result generalizesprevious results in this context and it is related to cocycles of flows ofanalytic functions on Banach spaces. Likewise, for a large class ofnon-separable Banach spaces $X $ of analytic functions in $\mathbb{D}$contained in the Bloch space, we prove that no non-trivial holomorphic flowinduces a $C_0$-semigroup of weighted composition operators on $X$. Thisgeneralizes previous results regarding $C_0$-semigroup of (unweighted)composition operators.

Generators of $C_0$-semigroups of weighted composition operators