In this paper we consider the following problem: let Xk, be a Banach space with a normalised basis (e(k, j))j, whose biorthogonals are denoted by
be their l∞-sum, and let
be a bounded linear operator with a large diagonal, i.e.,
Under which condition does the identity on Z factor through T? The purpose of this paper is to formulate general conditions for which the answer is positive.