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 Fréchet smooth operator whose derivative is uniformly continuous. We introduce the smooth biconjugate T**: BC(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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overline {T(B_{c_0 } )} $$ \end{document} 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 C1,u-smooth noncompact operator from BcO which does not fix any (affine) basic sequence.