Toeplitz matrices, asymptotic linear algebra and functional analysis
type: Item
instance of: scholarly article
title: Toeplitz matrices, asymptotic linear algebra and functional analysis
zbMATH Open document ID: 0952.47025
published in: Texts and Readings in Mathematics
publication date: 2000-05-18T00:00:00Z
review text: This small book is a very nice introduction to some problems concerning the approximation of a Toeplitz matrix by its principal submatrices. Recall that for an essentially bounded measurable function \(a\) on the unit circle, the corresponding Toeplitz matrix \(T(a)\) is \([\widehat a_{i-j}] _{i,j=0}^\infty\), where \(\widehat a_j\) are the Fourier coefficients of~\(a\). Let \(T_n(a)=[\widehat a_{i-j}]_{i,j=0}^{n-1}\) be the \(n\times n\) principal submatrix of~\(T(a)\). The authors study the convergence of the norms, the norms of the inverses, the inverses themselves, the spectra, the pseudospectra, and the singular values of \(T_n(a)\) to those of \(T(a)\) as \(n\to\infty\). The basic facts concerning these are established in Chapter~2, after presenting some introductory material (basic properties of Toeplitz operators, basics of \(C^*\)-algebras and Fredholm theory) in Chapter~1. Chapter~3 is concerned with the speed of divergence of \(\|T_n(a)^{-1}\|\) to infinity for non-invertible~\(T(a)\). Chapter~4 discusses the rate of convergence of the norms \(\|T_n(a)\|\), the norms of the inverses \(\|T_n(a)^{-1}\|\), and the condition numbers \(\kappa(T_n(a))=\|T_n(a)\|\|T_n(a)^{-1}\|\) to \(\|T(a)\|\), \(\|T(a)^{-1}\|\) and \(\kappa(T(a))\), respectively, for invertible~\(T(a)\). Finally, in Chapter~5 the limiting behaviour of the singular values is investigated, both the individual limits of \(s_k(T_n(a))\) and \(s_{n-k}(T_n(a))\) as \(k\) is fixed while \(n\to\infty\), and the asymptotic distribution of the singular values (i.e. the limits of the spectra \(\sigma(|T_n(a)|)\) as \(n\) tends to infinity -- the Szegö limit theorem, the Avram-Parter theorem, etc.). The authors usually confine themselves to symbols \(a\) which are rational functions; some information on the general case can be found in \textit{A.~Böttcher} and \textit{B.~Silbermann}'s book ``Introduction to large truncated Toeplitz matrices'', New York (1999; Zbl 0916.15012). The book is fairly self-cont…
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15-01
Mathematics Subject Classification ID: 47-01
Mathematics Subject Classification ID: 47A56
zbMATH DE Number: 1448985
zbMATH Keywords: Toeplitz matrices
zbMATH Keywords: finite section method
zbMATH Keywords: stability
zbMATH Keywords: asymptotic linear algebra
zbMATH Keywords: spectra
zbMATH Keywords: pseudospectra
zbMATH Keywords: singular values
zbMATH Keywords: Toeplitz operators
zbMATH Keywords: \(C^*\)-algebras
zbMATH Keywords: Fredholm theory
zbMATH Keywords: rate of convergence
author: A. Böttcher
author: S. M. Grudsky
reviewed by: Miroslav Engliš
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1448985