Factorization Property in Rearrangement Invariant Spaces

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