Sarason interpolation and Toeplitz corona theorem for almost periodic matrix functions
type: Item
instance of: scholarly article
title: Sarason interpolation and Toeplitz corona theorem for almost periodic matrix functions
zbMATH Open document ID: 0933.42003
DOI: 10.1007/BF01203771
author: Joseph A. Ball
author: Leiba Rodman
author: Yuri I. Karlovich
author: Ilya M. Spitkovsky
published in: Integral Equations and Operator Theory
publication date: 2000-03-21T00:00:00Z
review text: The authors investigate matrices whose entries are almost periodic functions in various senses (Bohr, Besicovitch or with absolutely convergent Fourier series). By \((AP)^{n\times n}\) one denotes the algebra of square matrices with (complex) almost periodic entries, of type \(n\) by \(n\). A matrix function \(G(t)\) in \((AP)^{n\times n}\) is said to possess a factorization of \(G(t)= G_+(t) G_-(t)\), \(t\in\mathbb{R}\), with \(G_+\) and \(G_-\) invertible for each \(t\in\mathbb{R}\), both almost periodic with their inverses, and such that the spectra of \(G_+\) and \((G_+)^{-1}\) are in \(\mathbb{R}_+\), while the spectra of \(G_-\) and \((G_-)^{-1}\) are in \(\mathbb{R}_-= (-\infty,0]\). A marathon type theorem is given, providing 16 equivalent conditions for the factorization of a matrix function \(G(t)\). Some of these conditions are, in fact, invertibility conditions for \(G(t)\) or some attached operators. It is shown, among other properties of factorizable matrices with almost periodic entries, that once the factorization holds with almost periodic matrices \(G_+\) and \(G_-\), it also holds when requiring the absolute convergence of the Fourier series of the factor matrices. The proof of the theorem requires several nontrivial auxiliary results related to function algebras. Besides various particular cases, the basic theorem is used in connection with the Sarason interpolation concept and the Toeplitz corona theorem.
Mathematics Subject Classification ID: 42A75
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 42A15
zbMATH DE Number: 1267973
zbMATH Keywords: matrix functions
zbMATH Keywords: almost periodic functions
zbMATH Keywords: factorization
zbMATH Keywords: almost periodic matrices
zbMATH Keywords: Sarason interpolation
zbMATH Keywords: Toeplitz corona theorem
reviewed by: Constantin Corduneanu
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01203771
OpenAlex ID: W2046819343
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description: scientific article; zbMATH DE number 1267973