Let $T:Y\to X$ be a bounded linear operator between two real 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.

中文翻译：

---

紧凑且有限的运算符

让 $吨：是\to X$是两个实范数空间之间的有界线性算子。我们根据定义在X上的 Lipschitz 函数的可微性来表征T 的紧致性，其值在另一个规范空间Z 中。此外，使用类似的技术，我们还可以根据更广泛的函数类别的可微性来表征有限秩算子，但仍然具有 Lipschitz 风格。作为应用，我们获得了类 Banach-Stone 定理。另一方面，我们给出了 Bourgain 和 Diestel 与有限算子和共奇异性相关的结果的扩展。