# Weakly compact operators into $c_0$ and other separable spaces

A Banach space $$E$$ is called Grothendieck if every weak* convergent sequence in the dual space $$E^*$$ is weakly convergent. A typical example of a Grothendieck space is $$\ell_\infty$$. Diestel proved the following characterization of Grothendieck Banach spaces:

For every Banach space $$E$$, TFAE:

1. $$E$$ is Grothendieck,
2. for each separable Banach space $$F$$, every bounded operator $$T\colon E\to F$$ is weakly compact,
3. every bounded operator $$T\colon E\to c_0$$ is weakly compact.

My question: Can we replace the space $$c_0$$ in item 3 by some/any other fixed separable space? In particular, can we use here $$\ell_1$$?

(By "other" I of course mean "non-isomorphic to $$c_0$$".)

• Every bounded linear operator $c_0 \to \ell^1$ is compact (this is a version of Pitt's theorem), hence weakly compact. But $c_0$ is, of course, not Grothendieck. Jun 1 at 10:05
• @JochenGlueck, good point! Thanks! So, $\ell_1$ (and all $\ell_p$'s for $1<p<\infty$ as well) is eliminated. Jun 1 at 10:10
• Yes - but please note that $\ell^p$ for $1 < p < \infty$ aren't suitable candidates anyway since those spaces are reflexive and every bounded linear operator from a Banach space into a reflexive space is weakly compact. Jun 1 at 10:28
• Obviously you can replace $c_0$ with any separable space that has a subspace isomorphic to $c_0$. Your question should have been "if you can replace $c_0$ by $X$ in item 3, must $X$ have a subspace that is isomorphic to $c_0$". Jun 2 at 15:01

In the light of Professor Johnson's comment, we cannot replace $$c_0$$ with another $$X$$ unless $$X$$ contains a copy of $$c_0$$.
Indeed, let $$X$$ be a Banach space that contains no isomorphic copy of $$c_0$$. Consider a $$C(K)$$ space that is not a Grothendieck space (e.g., $$C([0,1])$$ ). Any bounded linear $$T:C(K)\to X$$ is unconditionally converging since it does not fix a copy of $$c_0$$, thus weakly compact since $$C(K)$$ has property $$(V)$$.
Generally, if $$E$$ (may not be a Grothendieck space) has property (V) & $$X$$ contains no copy of $$c_0$$, then every $$T\in B(E,X)$$ is weakly compact.
• Professor @JochenWengenroth, an operator is said to be unconditionally converging (u.c.) if it maps w.u.c. series to unconditionally convergent series. These are characterized by the fact that an operator $T$ is u.c. iff $T$ doesn't fix a copy of $c_0$. Jun 3 at 10:18