Factorization Property in Rearrangement Invariant Spaces

Let $X$ be a Banach space with a basis $(e_k)_k$ and biorthogonals $(e^\ast_k)_k$. An operator on $X$ is said to have a $\textit {large diagonal}$ if $\inf\limits_{k} |e_k^\ast(T(e_k))| > 0$. The basis $(e_k)_k$ is said to have the $\textit {factorization property}$ 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.