Year: 2012
Type: preprint
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 $Σ(X)$ of countable dense linearly independent subsets of $X$. As a consequence, each $A\in Σ(X)$ 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 Fréchet space $X$, $GL(X)$ acts transitively on $Σ(X)$ if and only if $X$ possesses a continuous norm. We also prove that every countably dimensional metrizable locally convex space supports a hypercyclic operator.
Source: arXiv (Cornell University)
Authors: Schenke, Andre, Shkarin, Stanislav
Institutions: Queen's University Belfast, Bielefeld University
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2949575344