Spaces of Compact Operators

In this paper we study the structure of the Banach space K(E, F) of all compact linear operators between two Banach spaces E and F. We study three distinct problems: weak compactness in K(E, F), subspaces isomorphic to l~ and complementation of K(E, F) in L(E, F), the space of bounded linear operators. In § 2 we derive a simple characterization of the weakly compact subsets of K(E, F) using a criterion of Grothendieck. This enables us to study reflexivity and weak sequential convergence. In § 3 a rather different problem is investigated from the same angle. Recent results of Tong [20] indicate that we should consider when K(E, F) may have a subspace isomorphic to l~. Although L(E, F) often has this property (e.g. take E = F =/2) it turns out that K(E, F) can only contain a copy of l~o if it inherits one from either E* or F. In § 4 these results are applied to improve the results obtained by Tong and also to approach the problem investigated by Tong and Wilken [21] of whether K(E, F) can be non-trivially complemented in L(E,F) (see also Thorp [19] and Arterburn and Whitley [2]). It should be pointed out that the general trend of this paper is to indicate that K(E, F) accurately reflects the structure of E and F, in the sense that it has few properties which are not directly inherited from E and F. It is also worth stressing that in general the theorems of the paper do not depend on the approximation property, which is now known to fail in some Banach spaces; the paper is constructed independently of the theory of tensor products. These results were presented at the Gregynog Colloquium in May 1972.