TITLE: Compact operators and completely continuous operators
QUESTION [8 upvotes]: A compact operator between Banach spaces is an operator that maps bounded sets into relatively compact sets, while a completely continuous operator maps all weakly convergent sequences into convergent sequences. 
Compact operators are always completely continuous, but completely continuous operators may be non-compact: the identity operator in the Schur space ${\rm l}_1$ is an example. In reflexive spaces, completely continuous operators are compact, so the two classes of operators are the same.
Here are my questions:

If the classes of compact operators and completely continuous operators are the same, what can we say about the space? Must it be reflexive?
What can we say about the spectrum of completely continuous operators?

REPLY [8 votes]: Sorry to drag up such an old thread, but concerning the first question:
For $E$ a Banach space, a theorem of Edward Odell states that the following assertions are equivalent:
$\ \ \ $1) $E$ contains no copy of $\ell_1$.
$\ \ \ $2) Every completely continuous operator on $E$ (to some Banach space) is compact.
This result is referenced in Corollary 5 of  this paper.  
Odell's Theorem above is proved on page 377 in the paper: H. P. Rosenthal, Point-wise compact subsets of the first Baire class, Amer. J. Math., $\bf 99$, (1977), pg. 362-378.  
A copy of this can be found here.