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For Banach spaces X, Z, let B(X, Z) denote the space of bounded linear operators from X into Z and K(X, Z) (resp. W(X, Z)) the subspace of compact (resp. weakly compact) operators. It is shown that (a) if X contains an isomorph of Co, then K(X, Γ) is not complemented in B(X, Γ), (b) if S is a compact Hausdorff space which is not scattered, then K(C(S), Z) is not complemented in W(C(S), Z) for Z = c0 or Γ. In particular, K(l°°, c0) is not complemented in B{Γt c0), which gives a negative answer to a question proposed by Arterburn and Whitley.

Projections in the spaces of bounded linear operations