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 operator $T:Y⟶X$, we prove that $T$ is a limited operator if and only if, for every convex continuous function $f:X⟶\mathbb{R}$ and every point $y\in Y$, $f\circ T$ is Fréchet differentiable at $y\in Y$ whenever $f$ is Gâteaux differentiable at $T(y)\in X$.