Property $ \bf{(wL)}$ and the reciprocal Dunford-Pettis property in projective tensor products

Abstract

A Banach space $X$ has the reciprocal Dunford-Pettis property ($RDPP$) if every completely continuous operator $T$ from $X$ to any Banach space $Y$ is weakly compact. A Banach space $X$ has the $RDPP$ (resp. property $(wL)$) if every $L$-subset of $X^*$ is relatively weakly compact (resp. weakly precompact). We prove that the projective tensor product $X \otimes{_\pi} Y$ has property $(wL)$ when $X$ has the $RDPP$, $Y$ has property $(wL)$, and $L(X,Y^*)=K(X,Y^*)$.