Finite sections of band operators with slowly oscillating coefficients
type: Item
instance of: scholarly article
title: Finite sections of band operators with slowly oscillating coefficients
zbMATH Open document ID: 1056.65054
author: Marko Lindner
author: Steffen Roch
author: Vladimir Rabinovich
published in: Linear Algebra and its Applications
publication date: 2004-11-18T00:00:00Z
review text: The aim of this note is to prove that the finite section method for a band operator \(A\in L(l^2(\mathbb Z))\) with slowly oscillating coefficients \(a_{i,j}\) with respect to the standard basis of \(l^2(\mathbb Z)\) is stable if and only if the operator \(A\) is invertible and if the plus-index of \(A\) is zero. This result generalizes the classical stability criterion for the finite section method for band Toeplitz operators. The finite section method consists in replacing the operator equation \(Au=f\) by the sequences of the linear systems \(R_n AR_n u_n=R_n f,\;n\in \mathbb N\), where \[ R_n :\;l^2(\mathbb Z)\to l^2(\mathbb Z),\;(R_n x)_m= \begin{cases} x_m & \text{if}\;| m| >n, \\ 0 & \text{if}\;| m| >n \end{cases} \] and this method is called stable if the matrices \(R_n AR_n \) are invertible for sufficiently large \(n\) and if the norm of their inverses are uniformly bounded.
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 2116521
zbMATH Keywords: finite sections method
zbMATH Keywords: band operators
zbMATH Keywords: slowly oscillating functions
zbMATH Keywords: sequence space
zbMATH Keywords: linear operator
zbMATH Keywords: stability
reviewed by: Karel Najzar
MaRDI profile type: Publication
full work available at URL: j.laa.2004.05.005
OpenAlex ID: W2057181637
cites work: Toeplitz operators with symbols from \(C+H^{\infty}\) in the space \(\ell ^ p\)
cites work: Introduction to large truncated Toeplitz matrices
cites work: Q4505311
cites work: Fredholm indices of band-dominated operators
cites work: Fredholm theory and finite section method for band-dominated operators
cites work: Algebras of approximation sequences: Finite sections of band-dominated operators
DOI: 10.1016/J.LAA.2004.05.005
description: scientific article; zbMATH DE number 2116521