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

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