On the approximate point spectrum of the Bergman space Cesàro operator
type: Item
instance of: scholarly article
zbMATH Open document ID: 1011.47025
publication date: 2002-07-04T00:00:00Z
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 1652732
zbMATH Keywords: Cesàro operator
zbMATH Keywords: Bergman space
zbMATH Keywords: Bishop's property (\(\beta\))
zbMATH Keywords: semigroup analysis
MaRDI profile type: Publication
author: V. Glass Miller
author: T. L. Miller
title: On the approximate point spectrum of the Bergman space Cesàro operator
published in: Houston Journal of Mathematics
review text: \textit{A. G. Siskakis} [J. Lond. Math. Soc., II. Ser. 36, 153-164 (1987; Zbl 0634.47038)] obtained the spectra and norm bounds for the operators \(C|_{H^p}\) on the Hardy space \(H^p\). The authors (1) identify the spectrum of the Cesàro operator \(\sigma ( C|_{L^p_a})\) on the Bergman spaces \(L^p_a(D)\) for \(p>1\); NEWLINENEWLINENEWLINE(2) prove that for \(p>2\) the approximate point spectrum of \(C|_{L^p_a}\) is the boundary of the spectrum; i.e., \(\sigma_{ap}(C|_{L^p_a}) = \partial B(\frac p4, \frac p4)\); NEWLINENEWLINENEWLINE(3) for \(p \geq 4\), give a growth condition on the resolvent and obtain as a consequence that \(C|_{L^p_a}\) has Bishop's property \((\beta)\). The basic tool used to investigate the Cesàro operator is the semigroup analysis of Siskakis.
reviewed by: Jinkee Lee
description: scientific article; zbMATH DE number 1652732