Weak infinite powers of Blaschke products

Abstract

Letb be a Blaschke product with zeros {z _n } in the open unit disk Δ. Let$\mathcal{P}(b)$ be the set of sequences of non-negative integersp=(p ₁,p ₂,…) such that ∑ ^∞ _n=1 p _n (1 − |z _n |) < ∞ andp _n →∞ asn→∞. We study the class of weak infinite powers ofb,$b^p (z) = \mathop \Pi \limits_{n = 1}^\infty \left( {\frac{{ - \bar z_n }}{{\left| {z_n } \right|}}{\mathbf{ }}\frac{{z - z_n }}{{1 - \bar z_n z}}} \right)^{pn} ,{\mathbf{ }}p \in \mathcal{P}(b).$ Properties of these classes depend on the setS(b) of the cluster points in ∂Δ of {z _n }. It is proved thatS(b)=∂Δ if and only if$L^\infty = H^\infty [\overline {b^p } :p \in \mathcal{P}(b)]$, the Douglas algebra generated by$\overline {b^p } ,p \in \mathcal{P}(b)$. Also, it is proved thatdθ(S(b))=0 if and only if there exists an interpolating Blaschke productB such that$H^\infty [\overline {b^p } :p \in \mathcal{P}(b)] \subset H^\infty [\bar B$.