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

Abstract

The aim of this article is to study semigroups of composition operators on the BMOA-type spaces $BMO{A}_p$, and on their "little oh" analogues $VMO{A}_p$. The spaces $BMO{A}_p$ 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 $D_{p-1}^p$. We study the maximal subspace of strong continuity, providing a sufficient condition on the infinitesimal generator of $\varphi$, under which $[{\varphi }_t,BMO{A}_p]=VMO{A}_p$, 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 $[{\varphi }_t,BMO{A}_p]=VMO{A}_p$, 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 $BMO{A}_1$ 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 $BMO{A}_p$ is equivalent to weak compactness.

24 pages

References in corpus (1)

• Composition Semigroups on BMOA and $H^{\infty }$

Cited by in corpus (2)

• Holomorphic semigroups and Sarason's characterization of vanishing mean oscillation
• Weighted composition semigroups on spaces of continuous functions and their subspaces