Hypercyclic and cyclic vectors
type: Item
instance of: scholarly article
title: Hypercyclic and cyclic vectors
zbMATH Open document ID: 0853.47013
author: Shamim I. Ansari
published in: Journal of Functional Analysis
publication date: 1997-01-05T00:00:00Z
review text: Let \(\mathcal X\) denote a separable complex Banach space. A vector \(x\in {\mathcal X}\) is said to be hypercyclic for an operator \(T\) on \(\mathcal X\) if the set \(\{T^n x: n\in \mathbb{N}\}\) is norm dense in \(\mathcal X\). We say that \(x\) is supercyclic if the set \(\{aT^n x: n\in \mathbb{N}, a\in \mathbb{C}\}\) is norm dense. An operator is called hypercyclic (supercyclic) if it has a hypercyclic (supercyclic) vector. \textit{B. Beauzamy} [On the orbits of a linear operator, preprint] and \textit{C. Kitai} [Invariant closed sets for linear operators, Thesis, University of Toronto (1982)] have shown that the set of all hypercyclic (supercyclic) vectors for a hypercyclic (supercyclic) operator on a Banach space is norm dense. \textit{B. Sz-Nagy} and \textit{C. Foiaş} [Stud. Math. 31, 35-42 (1968; Zbl 0174.18203)] have shown that the set of cyclic vectors is norm dense when the underlying space is Hilbert. In the present article it is shown that if \(T\) is a cyclic operator on \(\mathcal X\) such that the point spectrum of \(T^*\) contains no nonempty open set then the set of all cyclic vectors is dense. A result of \textit{K. F. Clancey} and \textit{D. D. Rogers} [Indiana Univ. Math. J. 27, 689-696 (1978; Zbl 0396.47016)] on the density of cyclic vectors is also generalized. It is also shown that if a vector \(x\) in \(\mathcal X\) is hypercyclic for \(T\) then \(x\) is hypercyclic for all \(T^n\) for all \(n\geq 1\).
Mathematics Subject Classification ID: 47A65
zbMATH DE Number: 764013
zbMATH Keywords: hypercyclic
zbMATH Keywords: supercyclic
zbMATH Keywords: density of cyclic vectors
reviewed by: K. Seddighi
MaRDI profile type: Publication
full work available at URL: jfan.1995.1036
OpenAlex ID: W2005864902
DOI: 10.1006/JFAN.1995.1036
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description: scientific article; zbMATH DE number 764013