Is every weakly compact operator from $\ell\_1$ into $c\_0$ extendible
to any larger space? Equivalently, is every weakly compact operator from $\ell\_1$ into $c\_0$ extendible to $\ell\_\infty$?



Extension of weakly compact operators from $\ell_1$ into $c_0$
@Joaquin: This one pushed me. It is, IMO, one of the nicest problems on Banach space theory asked on MO.


The answer is no. For a counterexample, take any weakly compact operator $T:\ell\_1 \to c\_0$ that preserves $\ell\_1^n$ uniformly for all $n$. In fact, since $\ell\_1^n$ embeds isometrically into $\ell\_\infty^{2^n}$, it is easy to construct one that has norm one so that for each $n$, there is a subspace of $\ell\_1$ isometric to $\ell\_1^n$ on which the operator acts isometrically.


Embed $\ell\_1$ isometrically into $\ell\_\infty$ and assume that $T$ extends to an operator $S$ from $\ell\_\infty$ into $c\_0$. The operator $S$ necessarily preserves (copies of) $\ell\_\infty^n$ uniformly for all $n$. A soft way of seeing this is to pass to an ultrapower, which will be an operator from some $C(K)$ space that is an isomorphism on a copy of $\ell\_1$, hence cannot be weakly compact, whence preserves a copy of $c\_0$ by Pelczynski's classical theorem. Taking adjoints, we see that $S^\*$ is an operator from $\ell\_1$ into an $L\_1$ space that preserves $\ell\_1^n$-s uniformly, hence preserves a copy of $\ell\_1$, whence is not weakly compact. But every operator from $\ell\_\infty$ into a separable space is weakly compact.


