Radon-Nikodym property in L(X,Y)

TL;DR

This paper investigates conditions under which the space of bounded linear operators from a separable Banach space to a space with the Radon-Nikodym property also has this property, focusing on weak local uniform convexity and weak compact generation.

Contribution

It establishes that L(X,Y) inherits the Radon-Nikodym property when it is weakly locally uniformly convex or weakly compactly generated.

Findings

L(X,Y) has the Radon-Nikodym property under weak local uniform convexity.

L(X,Y) has the Radon-Nikodym property if it is weakly compactly generated.

The results extend the understanding of Radon-Nikodym property in operator spaces.

Abstract

Let X be a separable Banach space and Y a space which has the Radon-Nikodym property. In this work, we show that L(X, Y) has the Radon-Nikodym property, if L(X, Y) is weakly locally uniformly convex or if L(X, Y) is a weakly compactly gen- erated space.