Power bounded operators and supercyclic vectors II

Author:
V. Müller

Journal:
Proc. Amer. Math. Soc. **133** (2005), 2997-3004

MSC (2000):
Primary 47A16, 47A15

Published electronically:
March 22, 2005

MathSciNet review:
2159778

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Abstract | References | Similar Articles | Additional Information

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 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.

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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

DOI:
https://doi.org/10.1090/S0002-9939-05-07829-9

Keywords:
Supercyclic vectors,
invariant subspace problem,
positive operators,
power bounded operators

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

Article copyright:
© Copyright 2005
American Mathematical Society