Chebyshev polynomials for operators
type: Item
instance of: scholarly article
title: Chebyshev polynomials for operators
zbMATH Open document ID: 1189.46042
author: Ernst Albrecht
published in: Archiv der Mathematik
publication date: 2009-06-16T00:00:00Z
review text: Let \(P^1_n\) denote the set of all monic polynomials of degree \(n\), where \(n\in\mathbb{N}_0\). Let \(({\mathcal R},\|\cdot \|)\) be a unital Banach algebra. If \(A\in{\mathcal R}\) and \(n\in\mathbb{N}_0\) then \(p_n\in{\mathcal P}^1_n\) is a Chebyshev polynomial for \(A\) if \[ \| p_n(A)\|= \underset{p\in{\mathcal P}^1_n}{}{\text{inf}}\,\| p(A)\|. \] If \(X\) is a normed space, let \({\mathcal L}(X)\) denote the algebra of bounded linear operators on \(X\). The paper is concerned with the question of uniqueness of Chebyshev polynomials and proves Theorem 2. If \((X,\|\cdot \|)\) is a uniformly convex normed space then for each \(A\in{\mathcal L}(X)\) and each \(n\in\mathbb{N}_0\) such that \(n< m\) if \(p\in{\mathcal P}^1_m\) and \(p(A)= 0\), then there exists a unique Chebyshev polynomial of degree \(n\) for \(A\). The result is placed in context by a racy discussion of the history of Chebyshev polynomials, by the suggestion that the supremum norm in Chebyshev's original problem (\({\mathcal R}= C([-1,1])\), \(A(t)= t\) for all \(t\in[-1, 1]\)) might be replaced by other `reasonable' norms, and by the exhibition of one `reasonable' and one `unreasonable' norm.
reviewed by: Aldric L. Brown
Mathematics Subject Classification ID: 46H99
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 41A99
zbMATH DE Number: 5565270
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: uniformly convex space
zbMATH Keywords: linear operator
zbMATH Keywords: uniqueness
zbMATH Keywords: Banach algebra
zbMATH Keywords: uniqueness of Chebyshev polynomials
MaRDI profile type: Publication
full work available at URL: s00013-009-3160-z
OpenAlex ID: W1982628741
DOI: 10.1007/S00013-009-3160-Z
description: scientific article; zbMATH DE number 5565270