On factorization of operators through the spaces \(l^p.\)

Abstract

We give conditions on a pair of Banach spaces \(X\) and \(Y,\) under which each operator from \(X\) to \(Y,\) whose second adjoint factors compactly through the space \(l^p,\) \(1\le p\le+\infty\), itself compactly factors through \(l^p.\) The conditions are as follows: either the space \(X^*,\) or the space \(Y^{***}\) possesses the Grothendieck approximation property. Leaving the corresponding question for parameters \(p>1, p\neq 2,\) still open, we show that for \(p=1\) the conditions are essential.