Pełczyński’s property (V) on positive tensor products of Banach lattices

Let $E$ be an atomic reflexive Banach lattice and $X$ be any Banach lattice with Pełczyński’s property (V). We show that (i) the positive injective tensor product $E\mathbin {\check {\otimes }_{|\varepsilon |}}X$ has Pełczyński’s property (V); (ii) the positive projective tensor product $E\mathbin {\hat {\otimes }_{|\pi |}}X$ has Pełczyński’s property (V) if and only if every positive linear operator from $E$ to $X^*$ is compact. As an application, we provide new examples of non-reflexive Banach lattices with Pełczyński’s property (V).