A new estimate for the vector valued corona problem
type: Item
instance of: scholarly article
title: A new estimate for the vector valued corona problem
zbMATH Open document ID: 1004.30026
author: Tavan T. Trent
published in: Journal of Functional Analysis
publication date: 2003-02-03T00:00:00Z
review text: In this paper the author gives an improved estimate in the vector valued corona theorem for the unit disk. His proof splits into two parts: first, he resurrects the operator corona theorem from the 1970s, and then he uses a Hilbert space argument, based on a version of Wolff's ideas. Since he uses Hilbert space methods, the Riesz representation theorem replaces the usual \(\bar\partial\)-techniques. Then the author considers the operator version of the corona problem on the unit disk \(D\) and gives an improvement of Vasyunin's estimate. He gets estimates for analytic solutions, \(G(z)\), of \(F(z)G(z)=I\) for all \(z\in D\) of the form \[ \sup\limits_{z\in D}||G(z)||_{B(\mathbb{C}^m,\mathbb{C}^n)}\leq 2[2\sqrt{e}+ 2\sqrt{2}e]\frac 1{\varepsilon^{n+1}}\ln\left(\frac 1{\varepsilon^{2n}}\right) . \] Since estimates independent of \(m\), the case \(m=\infty\) follows from a routine argument.
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 32A38
Mathematics Subject Classification ID: 47C10
zbMATH DE Number: 1747022
zbMATH Keywords: Hilbert space
zbMATH Keywords: vector valued corona theorem
zbMATH Keywords: ideals of operators
zbMATH Keywords: Carleson
zbMATH Keywords: Tolokonnikov
zbMATH Keywords: Vasyunin
MaRDI profile type: Publication
full work available at URL: jfan.2001.3842
OpenAlex ID: W1998071038
cites work: Nevanlinna-Pick interpolation on the bidisk
cites work: Interpolation problems in nest algebras
cites work: A commutant lifting theorem on the polydisc
cites work: Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables
cites work: Interpolations by bounded analytic functions and the corona problem
cites work: Q5511041
cites work: On the Corona Theorem and its Application to Spectral Problems in Hilbert Space
cites work: Q3922907
cites work: Q3743859
cites work: The H p -Corona Theorem for the Polydisc
cites work: Q4301741
cites work: A corona theorem for countably many functions
cites work: The Corona Theorem as an Operator Theorem
cites work: On contractions similar to isometries and Toeplitz operators
cites work: Estimates in Carleson's corona theorem and finitely generated ideals in an \(H^\infty\) algebra
cites work: Q4205954
cites work: Linear and complex analysis. Problem book. 199 research problems
DOI: 10.1006/JFAN.2001.3842
description: scientific article; zbMATH DE number 1747022