Common hypercyclic vectors for the unitary orbit of a hypercyclic operator
type: Item
instance of: scholarly article
title: Common hypercyclic vectors for the unitary orbit of a hypercyclic operator
zbMATH Open document ID: 1230.47020
author: Kit C. Chan
author: Rebecca Sanders
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-11-15T00:00:00Z
review text: The authors continue their recent work on common hypercyclic vectors for paths and similarity orbits of operators \(T\) on a separable Hilbert space \(H\) [J. Oper. Theory 61, 191--233 (2009); ibid. 66, No.~1, 107--124 (2011; Zbl 1230.47021); J. Math. Anal. Appl. 375, No.~1, 139--148 (2011; Zbl 1208.47013); Integral Equations Oper. Theory 65, No.~1, 131--149 (2009; Zbl 1196.47008)]. In the present paper, they investigate the unitary orbit of an operator \(T\), that is, the set of all operators of the form \(U^* T U\), with \(U\) a unitary operator on \(H\). Since all the elements of the unitary orbit have the same norm of \(T\), the orbit cannot be norm dense in the space of all operators. The unitary orbit is path connected. The authors prove that, if \(T\) is a hypercyclic operator on \(H\), then its unitary orbit contains a path of operators whose closure in the strong operator topology contains the orbit and the whole path has a \(G_\delta\) set of common hypercyclic vectors. Several consequences and improvements of this result are also included.
reviewed by: José Bonet
Mathematics Subject Classification ID: 47A16
zbMATH DE Number: 5971752
zbMATH Keywords: hypercyclic operators
zbMATH Keywords: path of operators
zbMATH Keywords: unitary orbit
zbMATH Keywords: common hypercyclic vectors
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.08.056
OpenAlex ID: W1977589117
cites work: Common hypercyclic vectors for multiples of backward shift.
cites work: Q3689716
cites work: Q5492480
cites work: Q5708184
cites work: How to get common universal vectors
cites work: Hypercyclicity and unimodular point spectrum
cites work: Approximation by chaotic operators and by conjugate classes.
cites work: Hereditarily hypercylic operators
cites work: Invariant Manifolds of Hypercyclic Vectors
cites work: Q4421998
cites work: Q3621830
cites work: Q5705641
cites work: Q3097298
cites work: Common hypercyclic vectors for the conjugate class of a hypercyclic operator
cites work: Hypercyclic behaviour of operators in a hypercyclic \(C_{0}\)-semigroup
cites work: Genericity of wild holomorphic functions and common hypercyclic vectors.
cites work: Q5660202
cites work: Universal Vectors for Operators on Spaces of Holomorphic Functions
cites work: Q5601136
cites work: Rotations of hypercyclic and supercyclic operators
cites work: Common hypercyclic vectors and the hypercyclicity criterion
DOI: 10.1016/J.JMAA.2011.08.056
description: scientific article; zbMATH DE number 5971752