A hypercyclicity criterion for non-metrizable topological vector spaces

TL;DR

This paper establishes a new criterion for hypercyclicity of operators on non-metrizable topological vector spaces and applies it to specific examples, solving existing problems and highlighting limitations of previous methods.

Contribution

It introduces a sufficient condition for sequential hypercyclicity in non-metrizable spaces and demonstrates its application to composition and snake shift operators.

Findings

The criterion applies to certain non-metrizable spaces.

It solves two open problems related to hypercyclic operators.

Examples show hypercyclicity cannot always be deduced from F-space restrictions.

Abstract

We provide a sufficient condition for an operator $T$ on a non-metrizable and sequentially separable topological vector space $X$ to be sequentially hypercyclic. This condition is applied to some particular examples, namely, a composition operator on the space of real analytic functions on $]0,1[$, which solves two problems of Bonet and Doma\'nski \cite{bd12}, and the "snake shift" constructed in \cite{bfpw} on direct sums of sequence spaces. The two examples have in common that they do not admit a densely embedded F-space $Y$ for which the operator restricted to $Y$ is continuous and hypercyclic, i.e., the hypercyclicity of these operators cannot be a consequence of the comparison principle with hypercyclic operators on F-spaces.