Hypercyclic operators on topological vector spaces

TL;DR

This paper extends the construction of hypercyclic operators to broader classes of locally convex spaces, characterizing when such operators exist on sums of Fréchet spaces and inductive limits.

Contribution

It provides necessary and sufficient conditions for the existence of hypercyclic operators on sums of Fréchet spaces and characterizes inductive limits supporting hypercyclicity.

Findings

Hypercyclic operators exist on sums of Fréchet spaces if each is separable and infinitely many are infinite dimensional.

Characterization of inductive limits of separable Banach spaces supporting hypercyclic operators.

Extension of previous constructions to more general locally convex spaces.

Abstract

Bonet, Frerick, Peris and Wengenroth constructed a hypercyclic operator on the locally convex direct sum of countably many copies of the Banach space $\ell_1$. We extend this result. In particular, we show that there is a hypercyclic operator on the locally convex direct sum of a sequence $\{X_n\}_{n\in\N}$ of Fr\'echet spaces if and only if each $X_n$ is separable and there are infinitely many $n\in\N$ for which $X_n$ is infinite dimensional. Moreover, we characterize inductive limits of sequences of separable Banach spaces which support a hypercyclic operator.