Power bounded operators and supercyclic vectors II
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Abstract:
We show that each power bounded operator with spectral radius equal to one on a reflexive Banach space 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 cone if $1$ belongs to its spectrum. This generalizes the corresponding results for Hilbert space operators. For non-reflexive Banach spaces these results remain true; however, the non-supercyclic vector (invariant cone, respectively) relates to the adjoint of the operator.References
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Additional Information
- V. Müller
- Affiliation: Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic
- Email: muller@math.cas.cz
- Received by editor(s): April 15, 2004
- Received by editor(s) in revised form: May 17, 2004
- Published electronically: March 22, 2005
- Additional Notes: This research was supported by grant No. 201/03/0041 of GA ČR
- Communicated by: Joseph A. Ball
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 2997-3004
- MSC (2000): Primary 47A16, 47A15
- DOI: https://doi.org/10.1090/S0002-9939-05-07829-9
- MathSciNet review: 2159778