Invertible factorization over multiplier algebras
type: Item
instance of: scholarly article
title: Invertible factorization over multiplier algebras
zbMATH Open document ID: 1278.46025
author: Tavan T. Trent
published in: Integral Equations and Operator Theory
publication date: 2013-04-15T00:00:00Z
review text: A consequence of the classical Fejer-Riesz theorem says that, if \(T\) is a positive, invertible Toeplitz operator on the Hardy space \(H^2(\mathbb{D})\), then \(T = M^{*}_{a}M_{a}\), where \(a\) and \(\frac{1}{a}\) are bounded analytic functions on the unit disk. A vector-valued version of this result was established by \textit{D. B. Lowdenslager} [Ann. Math. (2) 78, 450--454 (1963; Zbl 0117.14701)]. In the paper under review, the author is concerned with such factorizations for more general positive, invertible operators acting on certain nice vector-valued reproducing kernel Hilbert spaces. A Hilbert space \(\mathcal{H}\) of analytic functions on a domain \(\Omega\subset\mathbb{C}^n\) is a reproducing kernel Hilbert space if point evaluation is bounded at every point in \(\Omega\). Such a space possesses a kernel function \(k\) and one usually denotes \(\mathcal{H}\) by \(H^2(k)\). If \(E\) is another Hilbert space, then \(H^2_{E}(k)\) is the tensor product \(H^2(k)\otimes E\), which consists of \(E\)-valued analytic functions. The author first proves, under the assumption that \(H^2(k)\) is nice (see the paper), a general abstract result which characterizes when a positive, invertible, bounded operator \(T\) on \(H^2_{E}(k)\) can be factored as \(T = A^{*}A\) with \(A\) and \(A^{-1}\) in \(\mathcal{M}(H^2_{E}(k))\), the multiplier algebra of \(H^2_{E}(k)\). The author then applies his theorem to derive interesting results on classical spaces such as the Bergman space over the unit disk and the Hardy space over the unit disk, polydisk, and the unit ball in higher dimensions. In each of these cases, a list of more concrete conditions is given. In the case of the Hardy space over the unit disk, the author recovers Lowdenslager's theorem which, as he states in the paper, provided the motivation for the main result.
reviewed by: Trieu Le
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47A68
Mathematics Subject Classification ID: 46E20
Mathematics Subject Classification ID: 32A35
zbMATH DE Number: 6153931
zbMATH Keywords: reproducing kernels
zbMATH Keywords: Fejer-Riesz theorem
zbMATH Keywords: factorizations
zbMATH Keywords: Hardy space
zbMATH Keywords: vector-valued reproducing kernel Hilbert spaces
MaRDI profile type: Publication
full work available at URL: s00020-012-2017-1
OpenAlex ID: W1987857077
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DOI: 10.1007/S00020-012-2017-1
description: scientific article; zbMATH DE number 6153931