Limit operators and their applications in operator theory
type: Item
instance of: scholarly article
title: Limit operators and their applications in operator theory
zbMATH Open document ID: 1077.47002
author: Steffen Roch
author: Vladimir Rabinovich
author: B. Silbermann
published in: Operator Theory: Advances and Applications
publication date: 2004-11-23T00:00:00Z
review text: The purpose of the present book is two-fold: to develop a method of limit operators and to apply limit operators to the study of Fredholm properties of band and band dominated operators. In the space of two-sided infinite sequences \(\ell^2(\mathbb Z)\), a standard basis \(e^k=\{\delta_{kj}\}_{j=-\infty}^\infty\), \(k=0,\pm1,\dots\), is chosen, where \(\delta_{kj}\) is the Kroneker's symbol. Any linear bounded operator \(A\in{\mathcal L}(\ell^2(\mathbb Z))\) is represented in this standard basis as a two-sided infinite matrix \(A=[a_{jk}]_{\mathbb Z\times\mathbb Z}\), where \(a_{jk}=(Ae^j,e^k)\). The band operators are those which possess only a finite number of non-trivial diagonals in their representation \(a_{jk}=0\) whenever \(| j-k| \geq N\) for some \(N\). The band dominated operators are limits (in the operator norm) of convergent sequences of band operators and belong to the Banach subalgebra of \({\mathcal L}(\ell^2(\mathbb Z))\) generated by the band operators. We encounter such operators rather often among partial differential (more generally, among pseudodifferential) equations; their solution is a dominating problem in numerical methods for differential and pseudodifferential equations. The authors concentrate mainly on the Fredholm properties of band and band dominated operators. These properties play a crucial role not only in solving the equations explicitly (which turns out usually to be a rather theoretical result), but also in elaborating an efficient numerical approximation with strongly motivated convergence and stability. The study is based on the fact that Fredholm properties of an operator are stable under compact perturbations. Therefore, no finite parts of the matrix representation \(A=[a_{jk}]_{\mathbb Z\times\mathbb Z}\) of the operator influences the Fredholm properties, and only the asymptotic behaviour of \(a_{jk}\) as \(j,k\to\pm\infty\) matters. To extract information about the behaviour of the operator at ``infinity'', one can consi…
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 47L80
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 65R20
zbMATH DE Number: 2117720
zbMATH Keywords: limit operators
zbMATH Keywords: band operators
zbMATH Keywords: band dominated operators
zbMATH Keywords: Fredholm properties
zbMATH Keywords: convolution type operator
zbMATH Keywords: pseudodifferential operator
zbMATH Keywords: slowly oscillating symbol
zbMATH Keywords: almost periodic symbol
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: finite sections
zbMATH Keywords: spectral approximation
zbMATH Keywords: fractality of approximation methods
zbMATH Keywords: homogeneous groups
reviewed by: Roland Duduchava
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2117720