Size of the peripheral point spectrum under power or resolvent growth conditions
type: Item
instance of: scholarly article
title: Size of the peripheral point spectrum under power or resolvent growth conditions
zbMATH Open document ID: 1126.47001
author: Cătălin Badea
author: Sophie Grivaux
published in: Journal of Functional Analysis
publication date: 2007-06-26T00:00:00Z
review text: Let \(X\) be a complex infinite-dimensional Banach space, and let \({\mathcal L}(X)\) be the algebra of all bounded linear operators on \(X\). An operator \(T\in{\mathcal L}(X)\) is said to be power-bounded if \(\sup_{n\geq0}\| T^n\| <+\infty\), and is said be to partially power-bounded with respect to an increasing sequence \((n_k)_k\) of positive integers if \(\sup_{k\geq0}\| T^{n_k}\| <+\infty\). In [Proc.\ Am.\ Math.\ Soc.\ 16, 375--377 (1965; Zbl 0133.07202)], \textit{B.\,Jamison} proved that if \(T\in{\mathcal L}(X)\) is a power-bounded operator, then the peripheral point spectrum \(\sigma_p(T)\cap\mathbb{T}\) is at most countable, where \(\sigma_{p}(T)\) denotes the point spectrum of \(T\) and \(\mathbb{T}\) stands as usual for the unit circle. This result does not always hold when replacing the power-boundedness by partial power-boundedness, as shown in [\textit{T.\,Ransford} and \textit{M.\,Roginskaya}, J.~Funct.\ Anal.\ 230, No.\,2, 432--445 (2006; Zbl 1087.47004); \textit{T.\,Ransford}, Isr.\ J.\ Math.\ 146, 93--110 (2005; Zbl 1103.47002)]. Thus the choice of the following definition, introduced by the authors in [Adv.\ Math.\ 211, No.\,2, 766--793 (2007; Zbl 1123.47006)], is justified: A sequence \((n_k)_k\) of positive integers is said to be a \textit{Jamison sequence} if every bounded linear operator on a separable Banach space which is partially power-bounded with respect to \((n_k)_k\) has a countable peripheral point spectrum. The authors first characterize Jamison sequences and show that an increasing sequence \((n_k)_k\) is a Jamison sequence if and only if \[ \inf\big\{\{\sup| \lambda^{n_k}-1| :k\geq0\}:\lambda\in\mathbb{T}\backslash\{1\}\big\}>0. \] They next provide and discuss several examples of Jamison sequences including the ones which appeared in the above cited papers. For any sequence \((n_k)_k\) of positive integers for which \(\lim_{k\to+\infty}\frac{n_{k+1}}{n_k}=+\infty\), they construct a separable complex Banach space \(X\) and a p…
reviewed by: Abdellatif Bourhim
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 5167475
zbMATH Keywords: power-bounded operators
zbMATH Keywords: partially power-bounded operators
zbMATH Keywords: peripheral point spectrum
zbMATH Keywords: Jamison sequences
zbMATH Keywords: Faber polynomials
zbMATH Keywords: Faber-bounded operators
zbMATH Keywords: partially Faber-bounded operators
zbMATH Keywords: Kreiss condition
zbMATH Keywords: minimality of systems of eigenvectors
MaRDI profile type: Publication
full work available at URL: j.jfa.2007.02.009
OpenAlex ID: W2073527339
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DOI: 10.1016/J.JFA.2007.02.009
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description: scientific article; zbMATH DE number 5167475