Power bounded operators and supercyclic vectors
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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.References
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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
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 3807-3812
- MSC (1991): Primary 47A16, 47A15
- DOI: https://doi.org/10.1090/S0002-9939-03-06962-4
- MathSciNet review: 1999927