On generators of C0-semigroups of composition operators

Abstract

Avicou, Chalendar and Partington proved in [4] that an (unbounded) operator Af = G.f′ on the classical Hardy space generates a C0 semigroup of composition operators if and only if it generates a quasicontractive semigroup. Here we prove that if such an operator A generates a C0 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. 1.