A class of quasicontractive semigroups acting on Hardy and Dirichlet space

This paper provides a complete characterization of quasicontractive $C_0$-semigroups on Hardy and Dirichlet space with a prescribed generator of the form $Af=G{f}^{\mathrm{\prime }}$. We show that such semigroups are semigroups of composition operators and we give simple sufficient and necessary condition on $G$. Our techniques are based on ideas from semigroup theory, such as the use of numerical ranges.