Toeplitz operators on weighted \(\ell_ p\) spaces and associated asymptotics
type: Item
instance of: scholarly article
title: Toeplitz operators on weighted \(\ell_ p\) spaces and associated asymptotics
zbMATH Open document ID: 0713.47026
DOI: 10.1007/BF01199888
published in: Integral Equations and Operator Theory
publication date: 1990-01-01T00:00:00Z
review text: The author in his paper, is concerned with finite Toeplitz matrices generated, in some way, by a piecewise continuous function. More specifically we can find in it asymptotic information about the eigenvalues of such finite matrices. The first instance is Theorem 3.3 which shows that the finite Toeplitz matrices \[ T_ n[\Phi]=(\Phi_{j-k})_{j,k=0,...,n}, \] where \(\Phi\) is a piecewise continuous complex valued map on \(S^ 1\), and \(\lambda_ k\) are the eigenvalues of the image of \(\Phi\). Therefore \(f(T_ n[\Phi])\) makes sense for a large class of suitably defined analytic functions f and an asymptotic formula for \[ tr f(T_ n[\Phi])=\sum f(\lambda_ i) \] is given, where the \(\lambda_ i's\) are the eigenvalues of \(T_ n[\Phi]\). This formula is a generalization of a Szegö limit theorem [J. Math. Analysis Appl. 120, No.-1, 25-38 (1986; Zbl 0609.47036) or \textit{H. Widom}, Adv. Math. 21, 1-29 (1976; Zbl 0344.47016)]. The paper has some kind of bright so that it is easy to read it.
reviewed by: Florian-Horia Vasilescu
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 4173717
zbMATH Keywords: finite Toeplitz matrices generated by a piecewise continuous function
zbMATH Keywords: asymptotic information about the eigenvalues
zbMATH Keywords: Szegö limit theorem
author: Estelle Basor
MaRDI profile type: Publication
cites work: Trace formulas for Toeplitz matrices with piecewise continuous symbols
cites work: Q3700325
cites work: Toeplitz Operators and Determinants Generated by Symbols with One Fisher-Hartwig Singularity
cites work: Q5665763
cites work: Q5571194
cites work: Asymptotic behavior of block Toeplitz matrices and determinants. II
full work available at URL: https://doi.org/10.1007/bf01199888
OpenAlex ID: W2057538922
description: scientific article; zbMATH DE number 4173717