Inner functions and division in Douglas algebras
type: Item
instance of: scholarly article
title: Inner functions and division in Douglas algebras
zbMATH Open document ID: 0614.46045
author: Yuko Izuchi
author: Kei-Ji Izuchi
published in: Michigan Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: In this interesting paper the authors continue the study of division in Douglas algebras begun by \textit{C. Guillory} and \textit{D. Sarason} [Mich. Math. J. 28, 173-181 (1981; Zbl 0417.30024)]. In that paper the problem was posed of finding the best constant \(N\in {\mathbb{N}}\) such that the following holds. If I is an inner function and if \(f\in H^{\infty}+C\) such that \(| f| \leq | I|\) on the maximal ideal space of \(H^{\infty}+C\), then \(f^ N\bar I\in H^{\infty}+C\). It was known that N cannot be one. The paper under review now settles this problem; it tells us that one may take \(N=2\). Quite unexpectedly, the techniques of the proof are essentially the same as those of Guillory and Sarason, who had shown the existence of such an N. Several related questions are also answered. The paper concludes with a nice factorization theorem for Blaschke products. Remark. Meanwhile, the reviewer has obtained from the authors a manuscript where the result mentioned above is generalized to arbitrary Douglas algebras.
reviewed by: Raymond Mortini
Mathematics Subject Classification ID: 46J30
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 46J10
zbMATH DE Number: 3994303
zbMATH Keywords: Douglas algebras
zbMATH Keywords: inner function
zbMATH Keywords: maximal ideal space
zbMATH Keywords: factorization theorem for Blaschke products
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003423
OpenAlex ID: W2018971061
DOI: 10.1307/MMJ/1029003423
description: scientific article; zbMATH DE number 3994303