Possible Duplicate:
Weakly compact operators on $\ell_1$
There is another question I would like to ask, if you don't mind, which is not very far from my previous one.
Suppose we have a Banach space $V$ and a bounded linear operator $T\colon V\to c_0$ which is not weakly compact. By Gantmacher's theorem we known that $T^\ast\colon \ell_1\to V^*$ is not weakly compact either. Suppose moreover that there is some mysterious complemented isomorphic copy of $\ell_1$ in the dual space $V^\ast$, $W$ say. Can we infer that there is a complemented copy of $c_0$ in $V$?
A naive strategy is to consider the operator $S\colon V^\ast \oplus \ell_1 \to V^\ast$ given by $S(x,y)=T^\ast y$ (provided the answer to my previous question is affirmative) because $V^\ast \oplus \ell_1$ is isomorphic to $V^\ast$ as $\ell_1$ is isomorphic to its square and is complemented in $V^\ast$. Unfortunately, $V\oplus c_0$ need not be isomorphic to $V$ (take the JL space!). But I am not sure whether the assumption that $T\colon V\to c_0$ is not weakly compact can help us here...