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Power bounded operators and supercyclic vectors II
Author(s):
V.
Müller
Journal:
Proc. Amer. Math. Soc.
133
(2005),
2997-3004.
MSC (2000):
Primary 47A16, 47A15
Posted:
March 22, 2005
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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  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, Zitná 25, 115 67 Prague 1, Czech Republic
Email:
muller@math.cas.cz
DOI:
10.1090/S0002-9939-05-07829-9
PII:
S 0002-9939(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
Posted:
March 22, 2005
Additional Notes:
This research was supported by grant No.~201/03/0041 of GA CR
Communicated by:
Joseph A. Ball
Copyright of article:
Copyright
2005,
American Mathematical Society
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