Compact and Limited Operators

TL;DR

This paper characterizes compact and finite rank operators between normed spaces through differentiability properties of Lipschitz functions, and applies these results to establish a Banach-Stone-like theorem and extend related operator results.

Contribution

It introduces novel characterizations of compact and finite rank operators via Lipschitz differentiability, linking operator theory with nonlinear function properties.

Findings

Characterization of compact operators via Lipschitz differentiability.

Extension of finite rank operator characterization.

Application to a Banach-Stone-like theorem.

Abstract

Let $T:Y\to X$ be a bounded linear operator between two normed spaces. We characterize compactness of $T$ in terms of differentiability of the Lipschitz functions defined on $X$ with values in another normed space $Z$. Furthermore, using a similar technique we can also characterize finite rank operators in terms of differentiability of a wider class of functions but still with Lipschitz flavour. As an application we obtain a Banach-Stone-like theorem. On the other hand, we give an extension of a result of Bourgain and Diestel related to limited operators and cosingularity.