Recurrence and mixing recurrence of multiplication operators

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.