On quasi norm attaining operators between Banach spaces

We provide a characterization of the Radon-Nikodym property for a Banach space Y in terms of the denseness of bounded linear operators into Y which attain their norm in a weak sense, which complement the one given by Bourgain and Huff in the 1970s for domain spaces. To this end, we introduce the following notion: an operator T : X -> Y between the Banach spaces X and Y is quasi norm attaining if there is a sequence (x(n)) of norm one elements in X such that (Tx(n)) converges to some u is an element of Y with parallel to u parallel to = parallel to T parallel to. We prove that strong Radon-Nikodym operators can be approximated by quasi norm attaining operators, a result which does not hold for norm attaining operators. It shows that this new notion of quasi norm attainment allows us to characterize the Radon-Nikodym property in terms of denseness of quasi norm attaining operators for both domain and range spaces, which in the case of norm attaining operators, was only valid for domain spaces due to the celebrated counterexample by Gowers in 1990. A number of other related results are also included in the paper: we give some positive results on the denseness of norm attaining nonlinear maps, characterize both finite dimensionality and reflexivity in terms of quasi norm attaining operators, discuss conditions such that quasi norm attaining operators are actually norm attaining, study the relation with the norm attainment of the adjoint operator and, finally, present some stability results.