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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?

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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).

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