Frequently hypercyclic subspaces
type: Item
instance of: scholarly article
title: Frequently hypercyclic subspaces
zbMATH Open document ID: 1290.47007
published in: Monatshefte für Mathematik
publication date: 2012-11-29T00:00:00Z
review text: In [\textit{L. B. González} and \textit{A. Montes Rodríguez}, J. Approx. Theory 82, No. 3, 375--391 (1995; Zbl 0831.30024)], the authors started a new line of investigation by asking if an operator can possess an infinite-dimensional closed subspace all of whose non-zero vectors are hypercyclic, that is, vectors with a dense orbit. In [\textit{A. Montes-Rodríguez}, Mich. Math. J. 43, No. 3, 419--436 (1996; Zbl 0907.47023)], the author obtained a sufficient condition for an operator to have such a subspace, while in the Banach space setting, \textit{M. González} et al. [Proc. Lond. Math. Soc. (3) 81, No. 1, 169--189 (2000; Zbl 1028.47007)] even obtained a full characterization. In the present paper, the authors study the analogous problem for frequently hypercyclic operators, that is, they study the existence of operators with an infinite-dimensional closed subspace all of whose non-zero vectors are frequently hypercyclic. Such a subspace is called a frequently hypercyclic subspace. An operator \(T\) on a separable topological vector space \(X\) is said to be frequently hypercyclic if there is some vector \(x\in X\) such that, for any non-empty open subset \(U\) of \(X,\) the set \(\{n\in \mathbb{N}: \;T^nx\in U\}\) has positive lower density; the lower density of a subset \(A\subseteq \mathbb{N}\) is defined as \(\liminf_{N\rightarrow \infty}\frac{1}{N}\text{card}\{n\leq N: n\in A\}\). The authors give sufficient conditions by attacking the problem with the three methods that have been used for hypercyclicity: a constructive approach, an approach via left-multiplication operators, and an approach via tensor products. More concretely, for \(T\) an operator on a separable \(F\)-space \(X\) with a continuous norm, they add a strong demand on the Frequently Hypercyclicity Criterion (see [\textit{F. Bayart} and \textit{S. Grivaux}, C. R., Math., Acad. Sci. Paris 338, No. 9, 703--708 (2004; Zbl 1059.47006); Trans. Am. Math. Soc. 358, No. 11, 5083--5117 (2006; Zbl 1115.470…
reviewed by: María J. Beltrán
Mathematics Subject Classification ID: 47A16
zbMATH DE Number: 6111281
zbMATH Keywords: frequently hypercyclic operator
zbMATH Keywords: frequently hypercyclic subspace
zbMATH Keywords: left-multiplication operator
zbMATH Keywords: frequently hypercyclic tensor product
author: A. Bonilla
author: Karl-Goswin Grosse-Erdmann
MaRDI profile type: Publication
full work available at URL: s00605-011-0369-2
OpenAlex ID: W2049045140
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DOI: 10.1007/S00605-011-0369-2
description: scientific article; zbMATH DE number 6111281