A short proof of the existence of disjoint hypercyclic operators

TL;DR

This paper presents a concise proof demonstrating the existence of disjoint hypercyclic operator tuples of arbitrary length on separable infinite-dimensional spaces, extending to dual hypercyclic tuples on Banach spaces with separable duals.

Contribution

The paper introduces a simplified proof for the existence of disjoint hypercyclic and dual hypercyclic tuples on broad classes of infinite-dimensional spaces, generalizing previous results.

Findings

Disjoint hypercyclic tuples exist for any length on separable infinite-dimensional Frechet spaces.

Disjoint dual hypercyclic tuples exist for any length on infinite-dimensional Banach spaces with separable duals.

The proof simplifies previous approaches to establishing hypercyclicity in operator theory.

Abstract

We give a short proof of the existence of disjoint hypercyclic tuples of operators of any given length on any separable infinite dimensional Frechet space. A similar argument provides disjoint dual hypercyclic tuples of operators of any length on any infinite dimensional Banach space with separable dual.