On generators of $C_0$-semigroups of composition operators

TL;DR

This paper proves that on certain Banach spaces of analytic functions, operators generating a $C_0$-semigroup automatically form a semigroup of composition operators, removing the need for quasicontractivity conditions.

Contribution

It shows that if an operator on these spaces generates a $C_0$-semigroup, then this semigroup is necessarily composed of composition operators, simplifying previous characterizations.

Findings

Operators generating $C_0$-semigroups are automatically composition operators.

Quasicontractivity condition is unnecessary for such generators.

Results apply to broad classes of Banach spaces of analytic functions.

Abstract

Avicou, Chalendar and Partington proved that an (unbounded) operator $(Af)=G\cdot f'$ 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.