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

Abstract

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

Cited by in corpus (1)

• Composition Semigroups on the Besov Spaces