It is well-known that non-weakly compact operators from $\ell\_\infty$ into any Banach space act as isomorphisms on some subspace of $\ell\_\infty$ isomorphic to $\ell\_\infty$. I have a question in this spirit.


Suppose that $T\colon \ell\_\infty \to c\_0$ is a bounded linear operator. Then $T$ is not weakly compact. Can it happen that $T|\_{c\_0}\colon c\_0\to c\_0$ is weakly compact?



Operators from $\ell_\infty$
Every bounded linear operator from $\ell\_\infty$ to a separable space is weakly compact. This follows, for example, from the fact that $\ell\_\infty$ is a Grothendieck space (use Google).


