Topological mixing and Hypercyclicity Criterion for sequences of operators
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- by Jeng-Chung Chen and Sen-Yen Shaw PDF
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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.References
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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
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 3171-3179
- MSC (2000): Primary 47A16, 47B37; Secondary 46A16, 47B33
- DOI: https://doi.org/10.1090/S0002-9939-06-08308-0
- MathSciNet review: 2231900