Remarks on Banach spaces of compact operators

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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 c0 or lp (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 c0, K(E, F) is not complemented in L(E, F).

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Supported in part by an Oklahoma State University College of Arts and Sciences summer grant.