A class of quasicontractive semigroups acting on Hardy and Dirichlet space

This paper provides a complete characterization of quasicontractive C0-semigroups on Hardy and Dirichlet space with a prescribed generator of the form Af = Gf ′. 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.