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Multiples of hypercyclic operators
Author(s):
Catalin
Badea;
Sophie
Grivaux;
Vladimir
Müller
Journal:
Proc. Amer. Math. Soc.
137
(2009),
1397-1403.
MSC (2000):
Primary 47A16, 47B37
Posted:
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  on  such that  and  are hypercyclic but  is not. Moreover, any  set  which is bounded and bounded away from zero can be realized as  for some invertible operator  acting on a Hilbert space.
References:
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- 1.
- F. BAYART, É. MATHERON,
Dynamics of Linear Operators, Cambridge University Press,
to appear. - 2.
- J. BèS, A. PERIS,
Hereditarily hypercyclic operators,
J. Funct. Anal. 167 (1999), pp. 94-112. MR 1710637 (2000f:47012) - 3.
- M. DE LA ROSA, C. READ,
A hypercyclic operator whose direct sum  is not hypercyclic,
to appear in J. Operator Theory. - 4.
- F. LEóN-SAAVEDRA, V. MüLLER,
Rotations of hypercyclic and supercyclic operators,
Integral Equations Operator Theory 50 (2004), pp. 385-391. MR 2104261 (2005g:47009) - 5.
- V. MüLLER,
Three problems,
Oberwolfach Reports 37 (2006), p. 2272. - 6.
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personal communication. - 7.
- H. SALAS,
Hypercyclic weighted shifts,
Trans. Amer. Math. Soc. 347 (1995), pp. 993-1004. MR 1249890 (95e:47042)
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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:
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
Posted:
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
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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