Differentiable functions and nice operators (Q258954)

Let \(K_1,K_2\) be compact intervals of real numbers and let \(C^1(K_1),C^1(K_2)\) be the spaces of continuously differentiable functions (real or complex valued) equipped with the norm \(\|x\|_{\infty}+\|x'\|_{\infty}\). In this paper, the authors study the structure of linear operators between these spaces whose adjoints preserve the extreme points of the dual unit ball (such operators are called nice operators). Depending on the length of the intervals, they are of the form \(\alpha_0 x(t_0)+\beta_0 x'(s_0)\) where \(|\alpha_0|=1=|\beta_0|\), \(t_0,s_0 \in K_1\), or the non-constant form \(\alpha_0 x \circ \phi\) for an isometry \(\phi: K_2 \rightarrow K_1\).