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Weak infinite powers of Blaschke products

Authors:	Keiji Izuchi

Institution:	(1) Department of Mathematics, Niigata University, 950-2181 Niigata, Japan

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 $H^\infty \overline {b^p } :p \in \mathcal{P}(b)] \subset H^\infty \bar B$ .

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