Dynamics of linear operators on non-Archimedean vector spaces

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.