Recurrence and mixing recurrence of multiplication operators

• Published in: Mathematica bohemica Vol. 149; no. 1; pp. 1 - 11
• Main Authors: Mohamed Amouch, Hamza Lakrimi
• Format: Journal Article
• Language: English
• Published: Institute of Mathematics of the Czech Academy of Science 01.04.2024
• Subjects: banach ideal of operators; hypercyclicity; left multiplication operator; recurrent operator; right multiplication operator; tensor product
• ISSN: 0862-7959; 2464-7136
• DOI: 10.21136/MB.2023.0047-22

Let $X$ be a Banach space, $\mathcal{B}(X)$ the algebra of bounded linear operators on $X$ and $(J, \|{\cdot}\|_J)$ an admissible Banach ideal of $\mathcal{B}(X)$. For $T\in\mathcal{B}(X)$, let $L_{J, T}$ and $R_{J, T}\in\mathcal{B}(J)$ denote the left and right multiplication defined by $L_{J, T}(A)=TA$ and $R_{J, T}(A)=AT$, respectively. In this paper, we study the transmission of some concepts related to recurrent operators between $T\in\mathcal{B}(X)$, and their elementary operators $L_{J, T}$ and $R_{J, T}$. In particular, we give necessary and sufficient conditions for $L_{J, T}$ and $R_{J, T}$ to be sequentially recurrent. Furthermore, we prove that $L_{J, T}$ is recurrent if and only if $Tøplus T$ is recurrent on $Xøplus X$. Moreover, we introduce the notion of a mixing recurrent operator and we show that $L_{J, T}$ is mixing recurrent if and only if $T$ is mixing recurrent.