Disjoint frequently hypercyclic pseudo-shifts

TL;DR

This paper establishes a criterion for disjoint frequent hypercyclicity and characterizes it for pseudo-shifts and weighted shifts on classical sequence spaces, revealing differences between disjoint and non-disjoint hypercyclicity.

Contribution

It introduces a Disjoint Frequent Hypercyclicity Criterion and characterizes disjoint frequent hypercyclicity for pseudo-shifts and weighted shifts, including counterexamples.

Findings

Disjoint frequent hypercyclicity criterion characterizes pseudo-shifts.

Characterization of disjoint frequently hypercyclic weighted shifts.

Counterexamples show differences between disjoint and non-disjoint hypercyclicity.

Abstract

We obtain a Disjoint Frequent Hypercyclicity Criterion and show that it characterizes disjoint frequent hypercyclicity for a family of unilateral pseudo-shifts on $c_0(\mathbb{N})$ and $\ell^p(\mathbb{N})$, $1\le p <\infty$. As an application, we characterize disjoint frequently hypercyclic weighted shifts. We give analogous results for the weaker notions of disjoint upper frequent and reiterative hypercyclicity. Finally, we provide counterexamples showing that, although the frequent hypercyclicity, upper frequent hypercyclicity, and reiterative hypercyclicity coincide for weighted shifts on $\ell^p(\mathbb{N})$, this equivalence fails for disjoint versions of these notions.