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Epsilon-hypercyclic operators

Published online by Cambridge University Press:  04 November 2009

CATALIN BADEA ,
SOPHIE GRIVAUX  and
VLADIMIR MÜLLER
CATALIN BADEA
Affiliation:
Laboratoire Paul Painlevé, UMR CNRS 8524, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France (email: badea@math.univ-lille1.fr, grivaux@math.univ-lille1.fr)
SOPHIE GRIVAUX
Affiliation:
Laboratoire Paul Painlevé, UMR CNRS 8524, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France (email: badea@math.univ-lille1.fr, grivaux@math.univ-lille1.fr)
VLADIMIR MÜLLER
Affiliation:
Institute of Mathematics AV CR, Zitna 25, 115 67 Prague 1, Czech Republic (email: muller@math.cas.cz)
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Abstract

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Let X be a separable infinite-dimensional Banach space, and T a bounded linear operator on X; T is hypercyclic if there is a vector x in X with dense orbit under the action of T. For a fixed ε∈(0,1), we say that T is ε-hypercyclic if there exists a vector x in X such that for every non-zero vector y∈X there exists an integer n with $\|T^nx-y\|\leq \varepsilon \|y\|$. The main result of this paper is a construction of a bounded linear operator T on the Banach space ℓ1 which is ε-hypercyclic without being hypercyclic. This answers a question from V. Müller [Three problems, Mini-Workshop: Hypercyclicity and linear chaos, organized by T. Bermudez, G. Godefroy, K.-G. Grosse-Erdmann and A. Peris. Oberwolfach Rep.3 (2006), 2227–2276].

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 6 , December 2010 , pp. 1597 - 1606
Copyright
Copyright © Cambridge University Press 2009

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