Property (gw) for tensor product and left-right multiplication operators

Given Banach spaces $X$ and $Y$ and operators $A\in B(X)$ and $B\in B(Y)$, property $(gw)$ does not in general transfer from $A$ and $B$ to the tensor product operator $A\otimes B\in B(X\bar{\otimes }Y)$ or to the elementary operator defined by $A$ and $B$, ${\tau }_{AB}=L_A{R}_B\in B(B(Y,X))$. In this article necessary and sufficient conditions ensuring that property $(gw)$ transfers from $A$ and $B$ to $A\otimes B$ and to ${\tau }_{AB}$ will be given.