The Radon‐Nikodym Property for Spaces of Operators

We show that if X ∗ and Y have the Radon‐Nikodym property and every bounded linear operator from X to Y is compact, then the space of bounded linear operators from X to Y has the Radon‐Nikodym property provided that the w ∗ ‐closure of every bounded norm separable subset of X ∗ is w ∗ ‐metrizable. If Y is a dual space, then this additional condition on X ∗ may be dropped. We also show that if X ∗ and Y have the Radon‐Nikodym property, then the spaces of p ‐absolutely summing operators from X to Y have the Radon‐Nikodym property provided that 1 ⩽ p < + ∞. The same result holds for the spaces of p ‐nuclear operators from X to Y provided that X ∗ has the approximation property.