A class of quasicontractive semigroups acting on Hardy and Dirichlet space

TL;DR

This paper characterizes quasicontractive $C_0$-semigroups on Hardy and Dirichlet spaces generated by $Af=Gf'$, showing they are composition operator semigroups with conditions on $G$.

Contribution

It provides a complete characterization of such semigroups, linking their generators to composition operators with explicit conditions on $G$.

Findings

Semigroups are composition operators on Hardy and Dirichlet spaces.

Necessary and sufficient conditions on $G$ for quasicontractivity.

Use of numerical ranges to analyze semigroup properties.

Abstract

This paper provides a complete characterization of quasicontractive $C_0$-semigroups on Hardy and Dirichlet space with a prescribed generator of the form $Af=Gf'$. 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.