Limited operators and differentiability

TL;DR

This paper characterizes limited operators between Banach spaces through the differentiability properties of convex continuous functions composed with these operators, linking operator theory with differentiability concepts.

Contribution

It provides a new characterization of limited operators using the differentiability of convex functions, connecting operator properties with functional analysis.

Findings

Limited operators are characterized by the differentiability of convex functions composed with them.

The paper establishes an equivalence between limited operators and the preservation of Fréchet differentiability.

It bridges the gap between operator theory and convex analysis through differentiability conditions.

Abstract

We characterize the limited operators by differentiability of convex continuous functions. Given Banach spaces $Y$ and $X$ and a linear continuous operator $T: Y \longrightarrow X$, 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 Fr\'echet differentiable at $y\in Y$ whenever $f$ is G\^ateaux differentiable at $T(y)\in X$.