Smooth noncompact operators from C(K), K scattered

Abstract

Let X be a Banach space, K be a scattered compact and T: B C(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 \(\overline {T(B_{c_0 } )} \) is compact or that l1 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.