Weak infinite powers of Blaschke products
type: Item
instance of: scholarly article
title: Weak infinite powers of Blaschke products
zbMATH Open document ID: 0914.30022
DOI: 10.1007/BF02788696
published in: Journal d'Analyse Mathématique
publication date: 1999-06-23T00:00:00Z
review text: 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 \(H^\infty[\overline B]\),…
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 1221953
zbMATH Keywords: Douglas algebra
zbMATH Keywords: bounded analytic functions
zbMATH Keywords: Blaschke product
zbMATH Keywords: Shilov boundary
author: Kei-Ji Izuchi
reviewed by: Raymond Mortini
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1221953