On generators of $C_0$-semigroups of composition operators

Avicou, Chalendar and Partington proved that an (unbounded) operator $(Af)=G\cdot f^{\mathrm{\prime }}$ on the classical Hardy space generates a $C_0$ semigroup of composition operators if and only if it generates a quasicontractive semigroup. Here we prove that if such an operator $A$ generates a $C_0$ semigroup, then it is automatically a semigroup of composition operators, so that the condition of quasicontractivity of the semigroup in the cited result is not necessary. Our result applies to a rather general class of Banach spaces of analytic functions in the unit disc.