Hypercyclic operators on countably dimensional spaces

TL;DR

This paper explores the conditions under which countably dimensional spaces support hypercyclic operators, showing that in certain spaces, the invertible operators act transitively on dense sets, and all such spaces support hypercyclic operators.

Contribution

It establishes that for separable infinite dimensional Fréchet spaces, the transitivity of the invertible group on dense sets is equivalent to having a continuous norm, and confirms all countably dimensional metrizable locally convex spaces support hypercyclic operators.

Findings

Transitivity of $GL(X)$ on $A(X)$ iff $X$ has a continuous norm.

Every countably dimensional metrizable locally convex space supports a hypercyclic operator.

Supports for hypercyclic operators exist in all countably dimensional normed spaces.

Abstract

According to Grivaux, the group $GL(X)$ of invertible linear operators on a separable infinite dimensional Banach space $X$ acts transitively on the set $\Sigma(X)$ of countable dense linearly independent subsets of $X$. As a consequence, each $A\in \Sigma(X)$ is an orbit of a hypercyclic operator on $X$. Furthermore, every countably dimensional normed space supports a hypercyclic operator. We show that for a separable infinite dimensional Fr\'echet space $X$, $GL(X)$ acts transitively on $\Sigma(X)$ if and only if $X$ possesses a continuous norm. We also prove that every countably dimensional metrizable locally convex space supports a hypercyclic operator.