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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Topological mixing and Hypercyclicity Criterion for sequences of operators


Authors: Jeng-Chung Chen and Sen-Yen Shaw
Journal: Proc. Amer. Math. Soc. 134 (2006), 3171-3179
MSC (2000): Primary 47A16, 47B37; Secondary 46A16, 47B33
Published electronically: June 5, 2006
MathSciNet review: 2231900
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Abstract: For a sequence $ \{T_n\}$ of continuous linear operators on a separable Fréchet space $ X$, we discuss necessary conditions and sufficient conditions for $ \{T_n\}$ to be topologically mixing, and the relations between topological mixing and the Hypercyclicity Criterion. Among them are: 1) topological mixing is equivalent to being hereditarily densely hypercyclic; 2) the Hypercyclicity Criterion with respect to the full sequence $ \mathbb{N}$ implies topological mixing; 3) topological mixing implies the Hypercyclicity Criterion with respect to some sequence $ \{n_k\}\subset \mathbb{N}$ that cannot be syndetic in general, and also implies condition (b) of the Hypercyclicity Criterion with respect to the full sequence. Applications to two examples of operators on the Fréchet space $ H(\mathbb{C})$ of entire functions are also discussed.


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Additional Information

Jeng-Chung Chen
Affiliation: Department of Mathematics, National Central University, Chung-Li, 320 Taiwan

Sen-Yen Shaw
Affiliation: Graduate School of Engineering, Lunghwa University of Science and Technology, Gueishan, Taoyuan, 333 Taiwan
Email: shaw@math.ncu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08308-0
PII: S 0002-9939(06)08308-0
Keywords: Hypercyclic sequence of operators, densely hypercyclic, topologically transitive, hereditarily densely hypercyclic, Hypercyclicity Criterion, topologically mixing
Received by editor(s): November 1, 2004
Received by editor(s) in revised form: December 18, 2004, March 29, 2005, and April 25, 2005
Published electronically: June 5, 2006
Additional Notes: This research was partially supported by the National Science Council of Taiwan
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.