## Ioana Ghenciu

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

Comment.Math.Univ.Carolin. 56,3 (2015) 319-329.**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 $L(X,Y^*)=K(X,Y^*)$.

**Keywords:** the reciprocal Dunford-Pettis property; property $(wL)$; spaces of compact operators; weakly precompact sets

**DOI:** DOI 10.14712/1213-7243.2015.126

**AMS Subject Classification:** 46B20 46B28 28B05

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