Compact and limited operators

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