Spectrum of the Cesàro operator in \(\ell ^{p }\)
type: Item
instance of: scholarly article
title: Spectrum of the Cesàro operator in \(\ell ^{p }\)
zbMATH Open document ID: 1275.47008
published in: Archiv der Mathematik
publication date: 2013-04-09T00:00:00Z
review text: For \(1<p<\infty\), the Cesàro operator \(C^{(p)}\) on \(\ell^{p}\) assigns to a sequence \((a_n)_{n=1}^{\infty}\) its sequence of averages \((1/n \sum_{k=1}^{n}a_k)_{n=1}^{\infty}\). By Hardy's inequality, \(C^{(p)}\) is bounded. In this paper, a novel proof of the fact that the spectrum of \(C^{(p)}\) is the closed disc \(D[p]\), which is centered at \(q/2\) and has radius \(q/2\), where \(q\) is the conjugate index of \(p\). The difficult part in proving the equality \(D[p]= \sigma(C^{(p)})\) is the inclusion \(D[p]\supseteq \sigma(C^{(p)})\). For this, one has to see that \((C^{(p)}-\lambda I)^{-1}\) is a bounded operator whenever \(\lambda \notin D[p]\). This is managed in a clever way by dominating the matrix \(E_{\lambda}:=-\lambda^2[(C-\lambda I)^{-1}-D_{\lambda}]\) by a more tractable positive matrix \(G_{\lambda}\). Here, \(D_{\lambda}\) is the diagonal part of \((C-\lambda I)^{-1}\), for which it is not hard to prove the boundedness. The problem is thereby reduced to show that \(G_{\lambda}\) is a bounded operator.
reviewed by: Janko Bračič
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 6152314
zbMATH Keywords: spectrum
zbMATH Keywords: Cesàro operator
zbMATH Keywords: \(\ell^p\) spaces
author: Guillermo P. Curbera
author: Werner J. Ricker
MaRDI profile type: Publication
full work available at URL: s00013-013-0498-z
OpenAlex ID: W2053064710
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DOI: 10.1007/S00013-013-0498-Z
description: scientific article; zbMATH DE number 6152314