Factorization property in rearrangement invariant spaces

Let X be a Banach space with a basis (e(k))(k) and biorthogonals (e(k)*)(k). An operator on X is said to have a large diagonal if inf(k) |e(k)* (T(e(k)))| > 0. The basis (e(k))(k) is said to have the factorization property if the identity factors through any operator with a large diagonal. Under the assumption that the Rademacher sequence is weakly null, we study the factorization property of the Haar system in a Haar system space. A Haar system space is the completion of the span of characteristic functions of dyadic intervals with respect to a rearrangement invariant norm. We show that every bounded operator with a large diagonal on a Haar system space is approximatively a factor of some diagonal operator with a large diagonal. Moreover, when the Haar system is an unconditional basis for a Haar system space, it has the factorization property.