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arXiv · 2412.19115

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

SongUng Ri·HyonHui Ju·JinMyong Kim
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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.

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SongUng Ri, HyonHui Ju, JinMyong Kim. 2025-12-23. A note on disjoint hypercyclicity for invertible bilateral pseudo-shifts on $\ell^{p}(\mathbb{Z})$. https://arxiv.org/abs/2412.19115

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