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

Abstract

We study $[{\varphi }_t,X]$, the maximal space of strong continuity for a semigroup of composition operators induced by a semigroup $\{{\varphi }_t{\}}_{t\ge 0}$ of analytic self-maps of the unit disk, when $X$ is BMOA, $H^{\infty }$ or the disk algebra. In particular, we show that $[{\varphi }_t,\text{BMOA}]\ne \text{BMOA}$ for all nontrivial semigroups. We also prove, for every semigroup $\{{\varphi }_t{\}}_{t\ge 0}$, that ${\lim }_{t\to 0^{+}}{\varphi }_t(z)=z$ not just pointwise, but in $H^{\infty }$ norm. This provides a unified proof of known results about $[{\varphi }_t,X]$ when $X\in \{H^p,A^p,{\mathcal{B}}_0,\text{VMOA}\}$.

References in corpus (1)

• Some integral operators acting on $H^{\infty }$

Cited by in corpus (5)

• On generators of $C_0$-semigroups of composition operators
• Semigroups of composition operators and Integral operators in BMOA-type spaces
• Dirichlet-to-Robin Operators via Composition Semigroups
• Weighted composition semigroups on spaces of continuous functions and their subspaces
• Composition Semigroups on the Besov Spaces