The generalized corona theorem
type: Item
instance of: scholarly article
title: The generalized corona theorem
zbMATH Open document ID: 0834.46034
DOI: 10.1007/BF01446623
author: Pamela Gorkin
author: Raymond Mortini
author: Artur Nicolau
published in: Mathematische Annalen
publication date: 1996-04-01T00:00:00Z
full work available at URL: https://eudml.org/doc/165285
review text: This is another beautiful paper on the ideal theory of \(H^\infty\) by one of the leading experts (R.M.) and colleagues. \(H^\infty\) is the ring of bounded holomorphic functions in the open unit disk \(\mathbb{D}\). For \(f_1,\dots, f_N\in H^\infty\), \(J:= \{f\in H^\infty: |f|\leq C_f \sum^N_{j= 1} |f_j|\), for some finite constant \(C_f\}\) is an ideal that contains the ideal \(I\) generated by \(f_1,\dots, f_N\), and simple examples show that proper containment can occur. L. Carleson's famous Corona Theorem asserts that equality occurs if \(J= H^\infty\). It is proved that if \(N= 2\) and \(f_1\), \(f_2\) have no common non-invertible \(H^\infty\)-factor, then \(I= J\) if and only if either \(I\) or \(J\) contains an interpolating Blaschke product. [Earlier work of \textit{V. Tolokonnikov} is relevant, especially J. Soc. Math. 27, 2549-2553 (1984; Zbl 0546.46046).] But for \(N= 3\) an (easy) example is constructed of \(I= J\) without this ideal containing an interpolating Blaschke product. Denote by \(M\) the maximal ideal space of \(H^\infty\), and by \(\widehat f: M\to \mathbb{C}\) the Gelfand transform of \(f\in H^\infty\). The order of \(m\in M\) as a zero of \(\widehat f\) is defined in a natural way and the minimum of these over \(f\in I\), denoted \(\text{ord}(I, m)\), plays a significant role. Use is made of an earlier result of the second author's [Analysis 14, No. 1, 67-73 (1984; Zbl 0810.46055)]that an ideal \(I\subsetneqq H^\infty\) is generated by interpolating Blaschke products if and only if \(\text{ord}(I, m)\in \{0, 1\}\) for every \(m\in M\). Thomas Wolff showed some time ago that \(f\in J\) implies \(f^3\in I\) always holds, and he posed the problem [cf. Lect. Notes Math. 1043 (1984; Zbl 0545.30038)] whether the stronger conclusion \(f^2\in I\) always holds as well. In the second part of this paper, the authors provide an affirmative answer under the additional assumption that at every \(m\in M\), \(\text{ord}(\widehat f_j, m)\) is finite for …
Mathematics Subject Classification ID: 46J20
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 30D50
zbMATH DE Number: 734583
zbMATH Keywords: finitely generated ideals
zbMATH Keywords: zero sets
zbMATH Keywords: ideal theory of \(H^ \infty\)
zbMATH Keywords: ring of bounded holomorphic functions
zbMATH Keywords: Corona Theorem
zbMATH Keywords: interpolating Blaschke product
zbMATH Keywords: maximal ideal space
zbMATH Keywords: Gelfand transform
zbMATH Keywords: one-point Gleason part
zbMATH Keywords: Carleson measures
reviewed by: Robert Burckel
MaRDI profile type: Publication
OpenAlex ID: W2160686343
description: scientific article; zbMATH DE number 734583