On an example of J. Bourgain on closures of finitely generated ideals
type: Item
instance of: scholarly article
title: On an example of J. Bourgain on closures of finitely generated ideals
zbMATH Open document ID: 0870.30035
DOI: 10.1007/PL00004299
published in: Mathematische Zeitschrift
publication date: 1997-07-03T00:00:00Z
review text: Let \(H^\infty\) be the algebra of bounded analytic functions in the unit disk \(\mathbb{D}\). For two functions \(f\) and \(g\) in \(H^\infty\) let \(I= I(f,g)\) be the ideal generated by \(f\) and \(g\) and let \[ J= J(f,g)= \{h\in H^\infty: |h|\leq C(|f|+|g|)\text{ for some }C>0\}. \] It is obvious that \(J\) is an ideal containing \(I\). We assume that the functions \(f\) and \(g\) have no common factor. In [Math. Ann. 301, No. 1, 135-154 (1995; Zbl 0564.46046)], \textit{P. Gorkin}, \textit{A. Nicolau} and the author proved that \(I(f,g)= J(f,g)\) if and only if \[ |f|+|g|+(1-|z|^2)(|f'|^2+|g'|^2)\geq\delta>0\quad\text{on }\mathbb{D}. \] On the other hand, as was shown by \textit{J. Bourgain} [Ann. Inst. Fourier 35, No. 4, 163-174 (1985; Zbl 0564.46044)], there exists functions \(f\) and \(g\) in \(H^\infty\) for which \(J(f,g)\) is not even contained in the closure \(\overline{I(f,g)}\) of \(I\). He proved the existence of two Blaschke products \(B\) and \(C\) such that \(BC\not\in \overline{I(B^2,C^2)}\). In the paper under review, it is shown that, in contrast to Bourgain's example, the ideal \(J(f,g)\) is contained in the closure of \(I(f,g)\) whenever \(f\) and \(g\) are powers of interpolating Blaschke products. It is also shown that the ideal \(J(B^N,C^N)\) is finitely generated whenever \(B\) and \(C\) are interpolating Blaschke products.
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 1028170
zbMATH Keywords: Blaschke products
author: Raymond Mortini
reviewed by: Raymond Mortini
MaRDI profile type: Publication
full work available at URL: pl00004299
OpenAlex ID: W1984825359
Recommended article: Generating sets for ideals of finite type in \(H^\infty\)
Recommended article: IDEALS GENERATED BY INTERPOLATING BLASCHKE PRODUCTS
Recommended article: Q3752641
Recommended article: Factorization of Blaschke products
Recommended article: Zero sets of interpolating Blaschke products
Recommended article: Q4869349
Recommended article: Q5484158
Recommended article: Q4889441
Recommended article: On ideals in \(H^\infty\) whose closures are intersections of maximal ideals
Recommended article: Q4300429
description: scientific article; zbMATH DE number 1028170