Weakly supercyclic operators
type: Item
instance of: scholarly article
title: Weakly supercyclic operators
zbMATH Open document ID: 1073.47013
author: Rebecca Sanders
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-08-06T00:00:00Z
review text: An operator \(T\) on a Banach space \({\mathcal B}\) is said to be supercyclic if there is \(x\in{\mathcal B}\) such that \(\{\lambda T^n x: \lambda \in \mathbb C \text{ and } n=0,1,\ldots \}\) is dense in \(\mathcal B\). If norm density is replaced by density with respect to the weak topology, then one obtains the concept of weakly supercyclic operator. If there is no need of scalar multiples to have the required density, one obtains the special concepts of (norm) hypercyclicity and weakly hypercyclicity. The weak versions are, of course, not mere generalizations. Indeed, they are really interesting notions, since a theorem of Mazur, acording to which the norm closure and the weak closure of a convex set coincide, implies that weak supercyclic operators are cyclic. Thus the class of operators that are weakly supercyclic but not supercyclic is a new class of cyclic operators. On the other hand, although some results from supercyclicity can be generalized easily to weak supercyclicity, this is not always the case and, as is well-known, weak topologies are much more difficult to handle than norm topologies. For instance, \textit{F.~Bayart} and \textit{E.~Matheron} [``Hyponormal operators, weighted shifts, and weak forms of supercyclicity'', Proc. Edinb. Math. Soc. (to appear)] have recently proved that some unitary operators are weakly supercyclic in Hilbert space, which is impossible for norm supercyclicity [\textit{H. M. Hilden} and \textit{L. J. Wallen}, Indiana Univ. Math. J. 23, 557--565 (1974; Zbl 0274.47004)]. Indeed, Bayart's and Matheron's result answers the question posed in the paper under review: Is there a weakly supercyclic hyponormal operator? In the paper under review, there are several very nice results. First of all, the author gives a proof that if \(T: \mathcal B\to \mathcal B\) is a bounded linear operator, then the vector \(x\oplus \mu\) in \(\mathcal B\oplus \mathbb C\) is weakly supercyclic for \(T \oplus \alpha I_\mathbb C\), \(\alpha\not=0\),…
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 2081788
zbMATH Keywords: weakly supercyclic operator
zbMATH Keywords: weakly hypercyclicity
zbMATH Keywords: weakly supercyclic hyponormal operator
reviewed by: Alfonso Montes-Rodriguez
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.11.049
OpenAlex ID: W1980494725
cites work: Q4382087
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cites work: Q5182704
cites work: Multi-hypercyclic operators are hypercyclic
cites work: Supercyclicity and weighted shifts
DOI: 10.1016/J.JMAA.2003.11.049
description: scientific article; zbMATH DE number 2081788