Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
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 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
Full-text PDF Free Access

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 $1$ 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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47A16, 47A15

Retrieve articles in all journals with MSC (2000): 47A16, 47A15


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: http://dx.doi.org/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
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