Analyticity and compactness of semigroups of composition operators

TL;DR

This paper characterizes quasicontractive groups and analytic semigroups of composition operators on Hardy and Dirichlet spaces, providing conditions for analyticity, compactness, and trace class membership, with a comparative analysis for the half-plane case.

Contribution

It offers a complete characterization of semigroups generated by specific operators on Hardy and Dirichlet spaces, including conditions for analyticity and compactness.

Findings

Complete characterization of quasicontractive groups and analytic semigroups

Conditions for compactness and trace class membership

Different behavior observed in the half-plane case

Abstract

This paper provides a complete characterization of quasicontractive groups and analytic $C_0$-semigroups on Hardy and Dirichlet space on the unit disc with a prescribed generator of the form $Af=Gf'$. In the analytic case we also give a complete characterization of immediately compact semigroups. When the analyticity fails, we obtain sufficient conditions for compactness and membership in the trace class. Finally, we analyse the case where the unit disc is replaced by the right-half plane, where the results are drastically different.