Containment of $\c_0$ and $\ell_1$ in $Π_1(E,F)$

Abstract

Suppose $Π_1(E, F)$ is the space of all absolutely 1-summing operators between two Banach spaces $E$ and $F$. We show that if $F$ has a copy of $c_0$, then $Π_1(E, F)$ will have a copy of $c_0$, and under some conditions if $E$ has a copy of $\ell_1$ then $Π_1(E, F)$ would have a complemented copy of $\ell_1$.