TITLE: Compactness of a bounded operator $T\colon c_0 \to \ell^1$
QUESTION [22 upvotes]: Pitt Theorem says that any bounded linear operator $T\colon \ell^r \to \ell^p$, $1 \leq p < r < \infty$, or $T\colon c_0 \to \ell^p$ is compact.
I know how to prove this in case $\ell^r \to \ell^p$, and $c_0 \to \ell^p$, where $p > 1$. Main idea in the first case is that $\ell^r$ is reflexive and hence closed ball $B_{\ell^r}$ is weakly compact. In the second case we could just use Schauder Theorem ($T$ is compact if and only if $T^*$ is compact).
The only case left is $T\colon c_0 \to \ell^1$. I have tried something like this:
By Schauder Theorem we need to prove that $T^*\colon \ell^\infty \to \ell^1$ is compact. By Banach-Alaoglu Theorem we know that $B_{\ell^\infty}$ is compact in the $weak^{*}$ topology on $\ell^\infty$. Moreover, we know, since $\ell^1$ is separable, that $B_{\ell^\infty}$ is metrizable. Hence, it is enough to prove that if $(x_n)$ is a $weak{}^{*}$ convergent (say, to $x$) sequence in $B_{\ell^\infty}$ then $(Tx_n)$ converges (to $Tx$, I think). Since Schur Theorem (weak and norm convergence is the same in $\ell^1$) we only need to show that $(Tx_n)$ converges weakly in $\ell^1$.
And here I stuck. Could you give me any ideas or references? In every book I have looked so far this particular case was omitted.

Edit (4.4.2011): I found in Diestel's Sequences and series in Banach spaces (chap. VII, Exercise 2(ii)) something like this:
A bounded operator $T: c_0 \to X$ is compact if and only if every subseries of $\sum_{n=1}^\infty Te_n$ is convergent, where $(e_n)$ is canonical basis for $c_0$.
I know how to prove this, but how we can show that operators $T: c_0 \to \ell^1$ possess the subseries property?

REPLY [2 votes]: Any operator $T:c_0\to X$ fails to be compact only if it fails to be strictly singular (see Albiac and Kalton, Theorem $2.4.10$).  This would mean there is an infinite-dimensional subspace $E\leq c_0$ so that $T|_E$ is an embedding.  But $c_0$ is self-saturated, so there is some $F\leq E$ isomorphic to $c_0$, and $T|_F$ is an embedding.  So if $T:c_0\to X$ fails to be compact, $X$ contains a copy of $c_0$.  A corollary is that all operators $T:c_0\to \ell_p$ are compact, $1\leq p<\infty$.  
In fact, this is a characterization of spaces which contain $c_0$.  $X$ does not contain a copy of $c_0$ if and only if all operators $T:c_0\to X$ are compact.