Smooth noncompact operators from C(K), K scattered

Let X be a Banach space, K be a scattered compact and T : BC (K) -> X be a Frechet smooth operator whose derivative is uniformly continuous. We introduce the smooth biconjugate T** : B-C(K)** -> X** and prove that if T is noncompact, then the derivative of T** at some point is a noncompact linear operator. Using this we conclude, among other things, that either <(T(B-c0))over bar> is compact or that l(1) is a complemented subspace of X*. We also give some relevant examples of smooth functions and operators, in particular, a C-1,(u)-smooth noncompact operator from Boo which does not fix any (affine) basic sequence.