Limited operators and differentiability

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

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