Let X be a Banach space, K be a scattered compact and T: B C(K) → X be a Fréchet 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 c0ℳ 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 B c O which does not fix any (affine) basic sequence.
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