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 $X^{\ast },$ or the space $Y^{\ast \ast \ast }$ possesses the Grothendieck approximation property. Leaving the corresponding question for parameters $p>1,p\ne 2,$ still open, we show that for $p=1$ the conditions are essential.

8 pages, AMSTeX; for the next paper see math.FA/0107113