Semigroups of composition operators and Integral operators in BMOA-type spaces

TL;DR

This paper investigates semigroups of composition operators on BMOA-type spaces, characterizing their strong continuity, and explores the boundedness and compactness of associated Volterra-type operators, extending known results to new parameter ranges.

Contribution

It provides new characterizations of semigroup properties on BMOA_p spaces and extends the understanding of Volterra operators' boundedness and compactness.

Findings

Characterization of strong continuity subspaces for semigroups.

Necessary and sufficient conditions for semigroup properties.

Extension of boundedness and compactness results for Volterra operators.

Abstract

The aim of this article is to study semigroups of composition operators on the BMOA-type spaces $BMOA_p$, and on their "little oh" analogues $VMOA_p$. The spaces $BMOA_p$ were introduced by R. Zhao as part of the large family of F(p,q,s) spaces, and are the M\"{o}bius invariant subspaces of the Dirichlet spaces $D^p_{p-1}$. We study the maximal subspace of strong continuity, providing a sufficient condition on the infinitesimal generator of ${\phi}$, under which $[{\phi}_t,BMOA_p]=VMOA_p$, and a related necessary condition in the case where the Denjoy - Wolff point of the semigroup is in $\mathbb{D}$. Further, we characterize those semigroups, for which $[{\phi}_t, BMOA_p]=VMOA_p$, in terms of the resolvent operator of the infinitesimal generator of $T_t$. In addition we provide a connection between the maximal subspace of strong continuity and the Volterra-type operators $T_g$. We characterize the symbols g for which $T_g$ acting from $BMOA$ to $BMOA_1$ is bounded or compact, thus extending a related result to the case $p=1$. We also prove that for $1<p<2$ compactness of $T_g$ on $BMOA_p$ is equivalent to weak compactness.