Bulletin of the Belgian Mathematical Society - Simon Stevin

Dynamics of linear operators on non-Archimedean vector spaces

Farrukh Mukhamedov and Otabek Khakimov

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Abstract

In the present paper we study dynamics of linear operators defined on topological vector space over non-Archimedean valued fields. We give sufficient and necessary conditions of hypercyclicity (resp. supercyclicity) of linear operators on separable $F$-spaces. It is proven that a linear operator $T$ on topological vector space $X$ is hypercyclic (supercyclic) if it satisfies Hypercyclicity (resp. Supercyclicity) Criterion. We consider backward shifts on $c_0(\bz)$ and $c_0(\bn)$, respectively, and characterize hypercyclicity and supercyclicity of such kinds of shifts. Finally, we study hypercyclicity, supercyclicity of operators $\lambda I+\mu B$, where $I$ is identity and $B$ is backward shift. We note that there are essential differences between the non-Archimedean and real cases.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 25, Number 1 (2018), 85-105.

Dates
First available in Project Euclid: 11 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1523412055

Digital Object Identifier
doi:10.36045/bbms/1523412055

Mathematical Reviews number (MathSciNet)
MR3784508

Zentralblatt MATH identifier
06882544

Subjects
Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators 37A25: Ergodicity, mixing, rates of mixing 47S10: Operator theory over fields other than $R$, $C$ or the quaternions; non- Archimedean operator theory

Keywords
non-Archimedean valuation hypercylic operator supercyclic operator backward shift operator

Citation

Mukhamedov, Farrukh; Khakimov, Otabek. Dynamics of linear operators on non-Archimedean vector spaces. Bull. Belg. Math. Soc. Simon Stevin 25 (2018), no. 1, 85--105. doi:10.36045/bbms/1523412055. https://projecteuclid.org/euclid.bbms/1523412055


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