Semigroups of Composition Operators and Integral Operators in BMOA-Type Spaces

Abstract

The aim of this article is to study semigroups of composition operators \(T_t=f\circ \phi _t\) on the BMOA-type spaces \(\textit{BMOA}_p\), and on their “little oh” analogues \(\textit{VMOA}_p\). The spaces \(\textit{BMOA}_p\) were introduced by R. Zhao as part of the large family of F(p, q, s) spaces, and are the Möbius invariant subspaces of the Dirichlet spaces \(D^p_{p-1}\). We study the maximal subspace \([\phi _t, \textit{BMOA}_p]\) of strong continuity, providing a sufficient condition on the infinitesimal generator of \(\{\phi _t\}\), under which \([\phi _t, \textit{BMOA}_p]=\textit{VMOA}_p\), 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 \([\phi _t, \textit{BMOA}_p]=\textit{VMOA}_p\), in terms of the resolvent operator of the infinitesimal generator of \((T_t|_{\textit{VMOA}_p})\). 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 \(T_g:\,\textit{BMOA}\rightarrow \textit{BMOA}_1\) 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 \(\textit{BMOA}_p\) is equivalent to weak compactness.