Weak infinite powers of Blaschke products
Let \(b(z)= \prod^\infty_{n= 1}{| a_n|\over a_n} {a_n- z\over 1-\overline a_nz}\) be a Blaschke product with zeros \((a_n)\). Note that \(\sum^\infty_{n=1}| a_n|< \infty\). Let \({\mathcal P}(b)\) be the set of sequences of positive integers \(p= (p_1,p_2,\dots)\) such that \[ \sum^\infty_{n=1} p_n(1-| a_n|)< \infty \] and \(p_n\to\infty\). The author studies the class of weak infinite powers of \(b\), defined as \[ b^p(z)= \prod^\infty_{n= 1} \Biggl({| a_n|\over a_n} {a_n- z\over 1-\overline a_nz}\Biggr)^{p_n},\quad p\in{\mathcal P}(b). \] First this class appeared in a paper of \textit{D. J. Newman} [Ann. Math., II. Ser. 70,438-445 (1959; Zbl 0092.11802)] on the characterization of the Shilov boundary of \(H^\infty\) by inner functions. An important result of Izuchi's paper tells us that whenever \(b\) is a Blaschke product and \((K_n)\) a sequence of compact sets in the spectrum of \(H^\infty+ C\) such that \(K_n\cap\overline{\{| b|< 1\}}= \emptyset\), then there exists \(p\in{\mathcal P}(b)\) such that \(\overline{\{| b|< 1\}}\subseteq Z(b^p)\) and \(| b^p|= 1\) on \(\bigcup^\infty_{n=1} K_n\). This theorem has also been proved, independently, by P. Gorkin. Another interesting result of the paper tells us that the Douglas algebra \(D(b)= H^\infty[\overline{b^p}: p\in{\mathcal P}(b)]\) generated by the weak infinite powers of a single Blaschke product \(b\) equals \(L^\infty\) if and only if the set \(S(b)\) of cluster points in \(\mathbb{C}\) of the zeros of \(b\) in \(\mathbb{D}\) is the whole unit circle. As a corollary one obtains that whenever \(b\) is a Blaschke product with \(S(b)= \partial\mathbb{D}\) and \(g\in L^\infty\), then there exists a weak infinite power of \(b\) multiplying \(g\) into \(H^\infty+ C\); this generalizes a result of \textit{S. Axler} [Ann. Math., II. Ser. 106, 567-572 (1977; Zbl 0373.46063)]. The author also investigates when the Douglas algebra \(D(b)\) above is contained in a Douglas algebra of the form \(H^\infty[\overline B]\), where \(B\) is an interpolating Blaschke product such that \(S(B)= S(b)\). Necessary and sufficient conditions for this are given. One of those, for example, says that the set of singularities \(S(b)\) of \(b\) must have Lebesgue measure zero. The paper ends with some local properties of the sets \({\mathcal R}(b)= \bigcup_{p\in{\mathcal P}(b)}\{| b^p|< 1\}\).
scientific article; zbMATH DE number 1221953
scientific article