A weighted composition semigroup on BMOA (Q371835)

Let \(\{\varphi_t: t\geq 0\}\) be a semigroup of analytic self-maps of the unit disk \(\mathbb{D}\). The author obtains several results related to the strong continuity of the weighted composition semigroup \[ T_t f(z) = \varphi_t^\prime(z) f(\varphi(z)) \] on the space \(\mathrm{BMOA}(\mathbb{D})\). In particular, there exists a maximal subspace of \(\mathrm{BMOA}\), denoted by \([\varphi_t^\prime, \mathrm{BMOA}]\), on which \(\{T_t\}\) is strongly continuos. An analytic decription of \([\varphi_t^\prime, \mathrm{BMOA}]\) is given in terms of the generator of \(\{\varphi_t\}\). Also, the author finds conditions under which \([\varphi_t^\prime, \mathrm{BMOA}]=\mathrm{VMOA}\).