A note on disjoint hypercyclicity for invertible bilateral pseudo-shifts on $\ell^{p}(\mathbb{Z})$

TL;DR

This paper explores disjoint hypercyclicity of invertible bilateral pseudo-shifts on ell^{p}(\u007fZ) and shows that such shifts can be disjoint hypercyclic along with their inverses, challenging previous assumptions and addressing an open problem.

Contribution

It demonstrates that invertible bilateral pseudo-shifts can be disjoint hypercyclic along with their inverses, and partially resolves an open problem on disjoint reiterative hypercyclicity.

Findings

Invertible bilateral pseudo-shifts can be disjoint hypercyclic with their inverses.

Disjoint hypercyclicity does not imply non-invertibility for pseudo-shifts.

Reiterative hypercyclicity and disjoint reiterative hypercyclicity are equivalent for operators on reflexive Banach spaces.

Abstract

We first give a note on disjoint hypercyclicity for invertible bilateral pseudo-shifts on $\ell^{p}(\mathbb{Z})$, $1\leq p <\infty$. It is already known that if a tuple of bilateral weighted shifts on $\ell^{p}(\mathbb{Z})$, $1\leq p <\infty$, is disjoint hypercyclic, then non of the weighted shifts is invertible. We show that as for pseudo-shifts which is a generalization of weighted shifts, this fact is not true. We give an example of invertible bilateral pseudo-shifts on $\ell^{p}(\mathbb{Z})$, $1\leq p <\infty$, which are disjoint hypercyclic and whose inverses are also disjoint hypercyclic. Next we partially answer to the open problem posed by Martin, Menet and Puig (2022)\cite{MMP22} concerned with disjoint reiteratively hypercyclic, that is, we show that as for the operators on a reflexive Banach space, reiteratively hypercyclic ones are disjoint hypercyclic if and only if they are disjoint reiteratively hypercyclic.