  Let $A$ be the generator of a $C_0$-semigroup $T$ on a Banach space of
analytic functions on the open unit disc. If $T$ consists of composition
operators, then there exists a holomorphic function $G:{\mathbb D}\to{\mathbb
C}$ such that $Af=Gf'$ with maximal domain. The aim of the paper is the study
of the reciprocal implication.
