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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Multiples of hypercyclic operators


Authors: Catalin Badea, Sophie Grivaux and Vladimir Müller
Journal: Proc. Amer. Math. Soc. 137 (2009), 1397-1403
MSC (2000): Primary 47A16, 47B37
Published electronically: October 27, 2008
MathSciNet review: 2465665
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Abstract: We give a negative answer to a question of Prăjitură by showing that there exists an invertible bilateral weighted shift $ T$ on $ \ell_2(\mathbb{Z})$ such that $ T$ and $ 3T$ are hypercyclic but $ 2T$ is not.

Moreover, any $ G_\delta$ set $ M \subseteq (0,\infty)$ which is bounded and bounded away from zero can be realized as $ M=\{t>0 \mid tT \textrm{ is hypercyclic}\}$ for some invertible operator $ T$ acting on a Hilbert space.


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Additional Information

Catalin Badea
Affiliation: Laboratoire Paul Painlevé, UMR CNRS 8524, Université des Sciences et Technologies de Lille, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France
Email: badea@math.univ-lille1.fr

Sophie Grivaux
Affiliation: Laboratoire Paul Painlevé, UMR CNRS 8524, Université des Sciences et Technologies de Lille, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France
Email: 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

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09696-2
PII: S 0002-9939(08)09696-2
Keywords: Hypercyclic operators, bilateral weighted shifts
Received by editor(s): May 7, 2008
Published electronically: October 27, 2008
Additional Notes: The first two authors were partially supported by ANR-Projet Blanc DYNOP
The third author was partially supported by grant No. 201/06/0128 of GA CR. The main part of the paper was written during the stay of the authors in Oberwolfach, Germany, under the MFO-RiP (“Research in Pairs”) programme. We would like to thank the Mathematisches Forschungsinstitut Oberwolfach for excellent working conditions.
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.