Remarks on Banach spaces of compact operators

Abstract

Let E and F be Banach spaces. We generalize several known results concerning the nature of the compact operators K( E, F) as a subspace of the bounded linear operators L( E, F). The main results are: (1) If E is a c 0 or l p (1 < p < ∞) direct sum of a family of finite dimensional Banach spaces, then each bounded linear functional on K( E) admits a unique norm preserving extension to L( E). (2) If F has the bounded approximation property there is an isomorphism of L(E, F) into K(E, F) ∗∗ such that its restriction to K( E, F) is the canonical injection. (3) If E is infinite dimensional and if F contains a complemented copy of c 0, K( E, F) is not complemented in L( E, F).