Common hypercyclic vectors for the conjugate class of a hypercyclic operator
type: Item
instance of: scholarly article
title: Common hypercyclic vectors for the conjugate class of a hypercyclic operator
zbMATH Open document ID: 1208.47013
author: Rebecca Sanders
author: Kit C. Chan
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-01-12T00:00:00Z
full work available at URL: https://epublications.marquette.edu/cgi/viewcontent.cgi
review text: Let \(X\) be an infinite-dimensional, separable Banach space and \(B(X)\) the algebra of all bounded linear operators on \(X\). The authors show that, if \(T\) is a continuous hypercyclic operator on \(X\), then the conjugate set \(S(T):=\{L^{-1}TL: L\in B(X)\) invertible\} contains a path \(\{F_t\in B(X)\), \(t\in [1,\infty[\}\) of operators which is SOT dense in \(B(X)\), and for which the set \(\bigcap_{t\geq 1}{\mathcal{HC}}(F_t)\) of common cyclic vectors for the whole path is a dense \(G_\delta\)-set. The following interesting corollary is obtained. Let \(g\) be a nonzero vector in \(X\). Then the set \(\mathcal A=\{T\in B(X): g\in{\mathcal{HC}}(T)\} \) is SOT-dense and SOT-connected in \(B(X)\). Furthermore, the set \(\bigcap_{T\in {\mathcal A}}{\mathcal{HC}}(T)\) of common hypercyclic vectors for \(A\) is \(\{\lambda g:\lambda\not=0\}\). It is also shown that the set of all hypercyclic operators on \(X\) is SOT-connected in \(B(X)\).
reviewed by: Raymond Mortini
Mathematics Subject Classification ID: 47A16
zbMATH DE Number: 5835285
zbMATH Keywords: hypercyclic operator
zbMATH Keywords: common hypercyclic vectors
zbMATH Keywords: paths of operators
zbMATH Keywords: strong operator topology
zbMATH Keywords: conjugate set
MaRDI profile type: Publication
OpenAlex ID: W1979964767
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DOI: 10.1016/J.JMAA.2010.08.018
description: scientific article; zbMATH DE number 5835285