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Keywords:
hypercyclicity; recurrent operator; left multiplication operator; right multiplication operator; tensor product; Banach ideal of operators
hypercyclicity; recurrent operator; left multiplication operator; right multiplication operator; tensor product; Banach ideal of operators
Summary:
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 $T\oplus T$ is recurrent on $X\oplus X$. 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.
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