Factorization property in rearrangement invariant spaces (Q6086411)

Let \(X\) be a Banach space with a basis \((e_k)_k\) and biorthogonal functionals \((e_k^*)_k\). 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. It is shown 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.