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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 $A\otimes B\in B(X\overline{\otimes} Y)$ or to the elementary operator defined by $A$ and $B$, $\tau_{AB}=L_AR_B\in B(B(Y,X))$. 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., Comment: 14 pages, original research article