Supercyclicity of the left and right multiplication operators on Banach ideal of operators

Let $X$ be a Banach space with $\dim X>1$ such that $X^{\ast }$, its dual, is separable and $\mathcal{B}(X)$ the algebra of bounded linear operators on $X$. In this paper, we study the passage of property of being supercyclic from an operator $T\in \mathcal{B}(X)$ to the left and right multiplication induced by $T$ on separable admissible Banach ideal of $\mathcal{B}(X)$. We give a sufficient condition for the tensor product $T\widehat{\otimes }R$ of two operators to be supercyclic. As a consequence, we give another equivalent conditions for the Supercyclicity Criterion.