Pelczyński’s property (V) and ( V ∗) on tensor products and spaces of compact operators

Let U be a reflexive Banach space with an unconditional basis and X be any Banach space. We show that if X has Pelczyński's property (V) then (U,X) has Pelczyński's property (V). We also show that if X has Pelczyński's property (V∗) then (i) XY has Pelczyński's property (V∗), and (ii) (U,X) has Pelczyński's property (V∗) if and only if every bounded linear operator from U to X is compact. As an application, we provide new examples of non-reflexive Banach spaces with Pelczyński's property (V) and Pelczyński's property (V∗), respectively.