An \(H^{2}\)-corona theorem on the bidisk for infinitely many functions.
type: Item
instance of: scholarly article
title: An \(H^{2}\)-corona theorem on the bidisk for infinitely many functions.
zbMATH Open document ID: 1069.30060
DOI: 10.1016/S0024-3795(03)00505-6
author: Tavan T. Trent
published in: Linear Algebra and its Applications
publication date: 2004-03-14T00:00:00Z
review text: The author establishes the \(H^2\)-corona theorem on the bidisk for infinitely many functions. For the general polidisk, an extension of this method applies. Assuming the data are smooth across the boundary, the author explicitly writes a solution for the \(H^2\) problem, using the representation of the kernel of a matrix valued analytic function. Estimates to show that this solution has the appropriate size are based on iterating the one-variable Littlewood-Paley identity. Let \(D\) be the open unit disk. \(F({\mathbf z})\) denotes the \(n\times m\) matrix, giving a linear transformation in \(B(l^2_m,l^2_n)\) in the natural way. Assume that \(\varepsilon\leq F({\mathbf z})F({\mathbf z})^*\leq 1\) for all \({\mathbf z}\in D^2\). Then \[ T_FT_F^*\geq\left(731e\left[ \frac{1}{\varepsilon^5}\ln \frac{1}{\varepsilon^2} \right]\right)^{-2}\text{I}, \] where \(T_F\) is the analytic Toeplitz operator received by restricting \(F\) to \(H^2(D^2)^{(m)}\).
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 32A38
zbMATH DE Number: 2056685
zbMATH Keywords: corona theorem
zbMATH Keywords: bidisk
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: matrix valued analytic function
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2056685