Cyclic properties of Volterra operator.
type: Item
instance of: scholarly article
title: Cyclic properties of Volterra operator.
zbMATH Open document ID: 1069.47008
author: Fernando León-Saavedra
author: Antonio Piqueras-Lerena
published in: Pacific Journal of Mathematics
publication date: 2004-09-16T00:00:00Z
review text: An operator \(T\) on a Hilbert space \(H\) is called supercyclic if there exists a vector \(x \in H\) such that the projective orbit \(\{ {\lambda} T^{n}x: {\lambda} \in \mathbb C\), \(n \in \mathbb N \}\) is dense in \(H\). An operator \(T\) is called hypercyclic if there exists a vector whose orbit under \(T\) is dense. An operator \(T\) is called cyclic if the linear span of some orbit is dense. In this paper, the authors deal with a question on supercyclicity of the Volterra operator on \(L^{2}[0,1]\) in \textit{N. H. Salas} [Stud. Math. 135, 55--74 (1999; Zbl 0940.47005)]. The paper consists of two main sections. In section \(2\), they introduce the elementary facts on hypercyclicity and cyclicity of the Volterra operator. In section \(3\), they prove that Volterra and identity plus Volterra operators are not supercyclic on \(L^{2}[0,1]\).
reviewed by: Ömer Gök
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 2101104
zbMATH Keywords: supercyclic operator
zbMATH Keywords: hypercyclic operator
zbMATH Keywords: cyclic operator
zbMATH Keywords: Volterra operator
MaRDI profile type: Publication
full work available at URL: pjm.2003.211.157
OpenAlex ID: W1965125756
DOI: 10.2140/PJM.2003.211.157
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description: scientific article; zbMATH DE number 2101104