Property \((\omega)\) and the single-valued extension property
type: Item
instance of: scholarly article
title: Property \((\omega)\) and the single-valued extension property
zbMATH Open document ID: 1494.47017
author: Lei Dai
author: Xiaohong Cao
author: Qi Guo
publication date: 2021-09-24T00:00:00Z
review text: Let \(X\) be a complex Banach space and \(T: X\to X\) be a continuous linear operator. Following \textit{V. Rakočević} [Mat. Vesn. 37, 423--426 (1985; Zbl 0596.47001)], the operator \(T\) is said to satisfy the property \((\omega)\) if \(\sigma_a(T)\setminus \sigma_{ea}(T) =\pi_{00} (T)\), where \[\sigma_a(T)=\{\lambda\in\mathbb{C}: T-\lambda I \textrm{ is not a bounded below operator}\},\] \[\sigma_{ae}=\{\lambda\in\mathbb{C}: T -\lambda I \text{ is not an upper semi-Weyl operator}\}\] and \[\pi_{00}(T) =\{\lambda\in\sigma(T): \lambda \text{ is an isolated point of } \sigma(T) \text{ and } 0<{\dim}\ker(T-\lambda I)<\infty\}.\] In the paper under review, the authors first introduce a new notion of spectrum inspired by the single-valued extension property. Using this new spectrum, they then give necessary and sufficient conditions for a continuous linear operator defined on a complex Banach space to satisfy property \((\omega)\). Moreover, the authors give examples which illustrate that there is no relationship between property \((\omega)\) for operators and hypercyclic (or supercyclic) property. They also present some results on the relationship between hypercyclic property (or supercyclic property) and property \((\omega)\).
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A16
zbMATH DE Number: 7400535
zbMATH Keywords: property \((\omega)\)
zbMATH Keywords: single-valued extension property
zbMATH Keywords: hypercyclic property
zbMATH Keywords: supercyclic property
MaRDI profile type: Publication
full work available at URL: s10114-021-0436-0
OpenAlex ID: W217921755
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DOI: 10.1007/S10114-021-0436-0
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published in: Acta Mathematica Sinica, English Series
description: scientific article