On a generalization of the Corona problem
type: Item
instance of: scholarly article
title: On a generalization of the Corona problem
zbMATH Open document ID: 0613.46046
DOI: 10.1155/S0161171286000959
author: Graziano Gentili
author: D. C. Struppa
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46150
review text: Let \(H^{\infty}\) denote the algebra of bounded analytic functions in the open unit disk and let \(f,f_ 1,...,f_ N\) be functions in \(H^{\infty}\) satisfying \[ (1)\quad | f| \leq \sum^{N}_{i=1}| f_ i|. \] As a generalization of the Corona problem, T. H. Wolff proved that (1) implies the existence of functions \(g_ 1,...,g_ N\in H^{\infty}\) such that \[ (2)\quad f^ 3=\sum^{N}_{i=1}g_ if_ i. \] As was noted earlier by R. Rao, the conclusion does not hold (in general) if the exponent 3 is replaced by 1. The question of whether (2) holds for \(f^ 2\) is still a challenging open problem. The authors of the present paper now give conditions on the functions \(f_ 1,...,f_ N\) in order that (1) implies that f resp. \(f^ 2\) belongs to the ideal generated by \(f_ 1,...,f_ N.\) Reviewers remark. Apparently the authors are not aware of the work of M. v. Renteln and V. A. Tolokonnikov. For instance, Theorem 6 has been proved a long time ago by \textit{M. v. Renteln} [Collect. Math. 26, 115-126 (1975; Zbl 0315.46044)]. It is today a standard fact. Even far more reaching results have been obtained by \textit{V. A. Tolokonnikov} [Sov. J. Math. 27, 2549-2553 (1984; Zbl 0518.46046)]. On the whole the paper under review does not yield any substantial progress on the generalized corona problem. Finally let us point out an error on p. 786. The ideal \(I_ H\) should be replaced by \(I_ H\cap H^{\infty}\).
reviewed by: Raymond Mortini
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 30D50
zbMATH DE Number: 3992190
zbMATH Keywords: Carleson measure
zbMATH Keywords: algebra of bounded analytic functions in the open unit disk
zbMATH Keywords: Corona problem
MaRDI profile type: Publication
OpenAlex ID: W2104712769
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description: scientific article; zbMATH DE number 3992190