Compact and Limited operators

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