Limited operators and differentiability

Abstract

We characterize the limited operators by differentiability of convex continuous functions. Given Banach spaces \(Y\) and \(X\) and a linear continuous continuous operator \(T: Y\longrightarrow X\), we prove that \(T\) is a limited operator if and only if, for every convex continuous function \(f: X \longrightarrow \mathbb{R}\) and every point \(y\in Y\), \(f\circ T\) is Fréchet differentiable at \(y \in Y\) whenever \(f\) is Gâteaux differentiable at \(T(y)\in X\).