A class of quasicontractive semigroups acting on Hardy and Dirichlet space

Abstract

This paper provides a complete characterization of quasicontractive C 0-semigroups on Hardy and Dirichlet space with a prescribed generator of the form $${Af=Gf^\prime}$$ . 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.