Index calculus for approximation methods and singular value decomposition
type: Item
instance of: scholarly article
title: Index calculus for approximation methods and singular value decomposition
zbMATH Open document ID: 0923.65026
author: Steffen Roch
author: B. Silbermann
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-11-29T00:00:00Z
review text: Some approximation methods for the equation \(Ax=y\) with a linear bounded operator \(A\) on a Hilbert space \(H\) are considered. An approximate method for \(A\) is treated as a sequence \(\{A_n\}\), \(A_n\in L(H)\). Two classes of sequences are investigated: Moore-Penrose and Fredholm sequences which correspond to normally solvable and Fredholm operators, respectively. Approximation methods are investigated in the frame of \(C^*\)-algebra theory of bounded sequences of linear bounded operators on \(H\). The authors give a characterization of Moore-Penrose invertibility in a \(C^*\)-algebra which yields to the one of the main results of the paper on the behaviour of singular values of elements of Moore-Penrose sequences \(\{A_n\}\). For a Fredholm sequence the index and \(\alpha\)-number are introduced. The authors prove relations between the index and the \(\alpha\)-number of a Fredholm sequence and some other properties of this sequence, in particular, the asymptotic distribution of the singular values of \(A_n\).
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 46N40
zbMATH DE Number: 1218229
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: Moore-Penrose sequences
zbMATH Keywords: Fredholm sequences
zbMATH Keywords: singular value decomposition
zbMATH Keywords: regularization
zbMATH Keywords: index calculus
zbMATH Keywords: approximation methods
reviewed by: Teresa Regińska
MaRDI profile type: Publication
full work available at URL: jmaa.1998.6022
OpenAlex ID: W2012088908
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DOI: 10.1006/JMAA.1998.6022
description: scientific article; zbMATH DE number 1218229