Composition semigroups on BMOA and H∞

We study [φt,X], the maximal space of strong continuity for a semigroup of composition operators induced by a semigroup {φt}t≥0 of analytic self-maps of the unit disk, when X is BMOA, H∞ or the disk algebra. In particular, we show that [φt,BMOA]≠BMOA for all nontrivial semigroups. We also prove, for every semigroup {φt}t≥0, that limt→0+⁡φt(z)=z not just pointwise, but in H∞ norm. This provides a unified proof of known results about [φt,X] when X∈{Hp,Ap,B0,VMOA}.