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

SongUng Ri, HyonHui Ju, JinMyong Kim

TL;DR

This paper investigates disjoint hypercyclicity for invertible bilateral pseudo-shifts on $\ell^{p}(\mathbb{Z})$, showing that invertibility does not preclude disjoint hypercyclicity and even allowing both an operator and its inverse to be jointly disjoint hypercyclic. It introduces a Disjoint Blow-up/Collapse Criterion to establish disjoint hypercyclicity for a family of bilateral pseudo-shifts and provides explicit invertible examples, including constructions not reducible to powers of bilateral weighted shifts. It also resolves a question about reiterative hypercyclicity on reflexive Banach spaces by proving the equivalence between disjoint hypercyclicity and disjoint reiterative hypercyclicity under reiterative hypercyclicity, and extends results to $\ell^{p}$-spaces with $p>1$ without the same inducing map condition. The results broaden the known dynamics of pseudo-shifts beyond weighted shifts and contribute to the understanding of joint dynamical properties in Banach spaces, with implications for open problems in disjoint (reiterative) hypercyclicity.

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.

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

TL;DR

This paper investigates disjoint hypercyclicity for invertible bilateral pseudo-shifts on , showing that invertibility does not preclude disjoint hypercyclicity and even allowing both an operator and its inverse to be jointly disjoint hypercyclic. It introduces a Disjoint Blow-up/Collapse Criterion to establish disjoint hypercyclicity for a family of bilateral pseudo-shifts and provides explicit invertible examples, including constructions not reducible to powers of bilateral weighted shifts. It also resolves a question about reiterative hypercyclicity on reflexive Banach spaces by proving the equivalence between disjoint hypercyclicity and disjoint reiterative hypercyclicity under reiterative hypercyclicity, and extends results to -spaces with without the same inducing map condition. The results broaden the known dynamics of pseudo-shifts beyond weighted shifts and contribute to the understanding of joint dynamical properties in Banach spaces, with implications for open problems in disjoint (reiterative) hypercyclicity.

Abstract

We first give a note on disjoint hypercyclicity for invertible bilateral pseudo-shifts on , . It is already known that if a tuple of bilateral weighted shifts on , , 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 , , 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.
Paper Structure (3 sections, 8 theorems, 29 equations)

This paper contains 3 sections, 8 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Disjoint hypercyclic bilateral pseudo-shifts
  3. Disjoint reiterative hypercyclicity

Key Result

Theorem 2.1

(Theorem 2.2 of CMS20) (Disjoint Blow-up/Collapse Criterion) For operators $T_{1}$,...,$T_{N}\in \mathcal{L}(X)$ with $N\geq 2$, suppose there exist a strictly increasing sequence $(n_{k})$ of positive integers, a dense subset $X_{0}$ of $X$ and maps $S_{k}:\oplus^{N}_{i=1}X_{0}\rightarrow X$ which

Theorems & Definitions (13)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • Theorem 2.4
  • proof
  • Example 2.5
  • Example 2.6
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • ...and 3 more