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 $BMOA_p$, 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 Möbius 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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2021
- DOI:
- 10.48550/arXiv.2103.04371
- arXiv:
- arXiv:2103.04371
- Bibcode:
- 2021arXiv210304371D
- Keywords:
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- Mathematics - Functional Analysis;
- Mathematics - Complex Variables
- E-Print:
- 24 pages