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

 

Power bounded operators and supercyclic vectors


Author: V. Müller
Journal: Proc. Amer. Math. Soc. 131 (2003), 3807-3812
MSC (1991): Primary 47A16, 47A15
Published electronically: March 25, 2003
MathSciNet review: 1999927
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Abstract: By the well-known result of Brown, Chevreau and Pearcy, every Hilbert space contraction with spectrum containing the unit circle has a nontrivial closed invariant subspace. Equivalently, there is a nonzero vector which is not cyclic.

We show that each power bounded operator on a Hilbert space with spectral radius equal to one has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant positive cone.


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Additional Information

V. Müller
Affiliation: Mathematical Institute, Czech Academy of Sciences, Zitna 25, 115 67 Prague 1, Czech Republic
Email: muller@math.cas.cz

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06962-4
PII: S 0002-9939(03)06962-4
Keywords: Supercyclic vector, invariant subspace problem, power bounded operator
Received by editor(s): June 19, 2002
Received by editor(s) in revised form: July 10, 2002
Published electronically: March 25, 2003
Additional Notes: This research was supported by grant No. 201/03/0041 of GA ČR
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2003 American Mathematical Society