Banach algebras of structured matrix sequences
type: Item
instance of: scholarly article
title: Banach algebras of structured matrix sequences
zbMATH Open document ID: 1161.47010
author: M. Seidel
author: B. Silbermann
published in: Linear Algebra and its Applications
publication date: 2009-02-10T00:00:00Z
review text: In the first part of this paper, the authors present a general theory about the asymptotic behavior of operator sequences. Given a Banach space \(X\) and an approximate projection \({\mathcal P}\) (a bounded sequence of operators in the algebra of all bounded and linear operators on \(X\) with certain properties), the authors introduce the notions of operator \({\mathcal P}\)-compact, operator \({\mathcal P}\)-Fredholm and \({\mathcal P}\)-strong convergence of a sequence of operators. Then they establish a Fredholm property for structured sequences in Banach algebras of the type \({\mathcal F}^T\) (certain \(C^*\)-algebras of bounded sequences of operators acting on finite-dimensional spaces) and prove a stability result. In the second part, the authors apply the general theory to the space \(\ell^p (\mathbb{Z}^K,\mathbb{C}^N)\), where \(K,N \in \mathbb{N}\).
reviewed by: Antonio Martinón
Mathematics Subject Classification ID: 47A58
Mathematics Subject Classification ID: 47N40
Mathematics Subject Classification ID: 47B36
Mathematics Subject Classification ID: 65J10
zbMATH DE Number: 5505626
zbMATH Keywords: matrix sequences
zbMATH Keywords: approximation numbers
zbMATH Keywords: band-dominated operators
zbMATH Keywords: finite sections
zbMATH Keywords: \(k\)-splitting-property
MaRDI profile type: Publication
full work available at URL: j.laa.2008.10.021
OpenAlex ID: W2051772019
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DOI: 10.1016/J.LAA.2008.10.021
description: scientific article; zbMATH DE number 5505626