Some properties of space of compact operators

Let $X$ be a separable Banach space, $Y$ be a Banach space and $Λ$ be a subset of the dual group of a given compact metrizable abelian group. We prove that if $X^*$ and $Y$ have the type I-$Λ$-RNP (resp. type II-$Λ$-RNP) then $K(X,Y)$ has the type I-$Λ$-RNP (resp. type II-$Λ$-RNP) provided $L(X,Y)=K(X,Y)$. Some corollaries are then presented as well as results conserning the separability assumption on $X$. Similar results for the NearRNP and the WeakRNP are also presented.

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