Abstract. Given Banach spaces ${\cal X}$ and ${\cal Y}$ and operators $A\in B({\cal X})$ and $B\in B({\cal Y})$, property $(gw)$ does not in general transfer from $A$ and $B$ to the tensor product operator $A\otimes B\in B({\cal X}\overline{\otimes } {\cal Y})$ or to the elementary operator defined by $A$ and $B$, $\tau_{AB}=L_AR_B\in B(B(Y,{\cal X}))$. 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.

AMS Subject Classification (1991): 47A80, 47A53, 47A10

Keyword(s): Banach space, property $(gw)$, tensor product operator, left-right multiplication operator

Received May 28, 2011, and in revised form March 2, 2012. (Registered under 26/2011.)