Non-Linear Operators and Differentiability of Lipschitz Functions

In this work we provide a characterization of distinct types 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 results about Schur spaces and reflexive spaces concerning the differentiability of real-valued Lipschitz functions.