Introduction : A bounded linear operator T between Banach spaces X and Y is said to be a Radon-Nikodym operator if it maps bounded X-valued martingales into converging Y-valued martingales. In [G-J], an example is given of an R.N.P operator that does not factor through an R.N.P space (i.e a space where the identity is an R.N.P operator). However, the question is still open in the case of a strong R.N.P operator i.e. when the closure of the image of the unit ball by the operator is an R.N.P set in the range space.

This note consists of two parts. In the first, we shall prove that the interpolation method of [D-F-J-P] gives a positive answer to the above question if one considers what we call a controllable R.N.P operator (See the definition below). Most of the commonly known strong R.N.P operators are of this type. However, in the second part of this note, we shall give examples which are not. Actually, we shall construct strong R.N.P operators for which the [D-F-J-P] factorization scheme gives Banach spaces that contain C0. On the other hand, these operators are not counterexamples to the general problem because we do not know whether they factor through an R.N.P space by another scheme.

It is well known that the above mentioned type of questions is equivalent to the following interpolation problem:

Suppose C is a closed circled convex bounded R.N.P subset of a Banach space X.