Non-linear operators and differentiability of Lipschitz functions

TL;DR

This paper characterizes various linear and non-linear operators between Banach spaces through the differentiability of Lipschitz functions, providing new insights in an abstract framework and applications to classical operator classes.

Contribution

It introduces a unified abstract framework linking operator types to Lipschitz differentiability, extending known results and characterizations in Banach space theory.

Findings

Characterization of linear and non-linear maps via Lipschitz differentiability

Application to compact, weakly-compact, limited, and completely continuous operators

New insights into Gelfand-Phillips, Schur, and reflexive spaces

Abstract

In this work we provide a characterization of distinct type of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological and non-linear framework. Restricted to the linear case, we can apply our results to compact, weakly-compact, limited and completely continuous linear operators. Moreover, our results yield a characterization of Gelfand-Phillips spaces and recover some known result of Schur spaces and reflexive spaces concerning the differentiability of real-valued Lipschitz functions.