# 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.
Document type :
Journal articles

https://hal-paris1.archives-ouvertes.fr/hal-03039783
Contributor : Mohammed Bachir <>
Submitted on : Friday, December 4, 2020 - 8:48:49 AM
Last modification on : Tuesday, January 19, 2021 - 11:08:40 AM

### Identifiers

• HAL Id : hal-03039783, version 1

### Citation

Mohammed Bachir, Gonzalo Flores, Sebastian Tapia-Garcia. Compact and Limited operators. Mathematical News / Mathematische Nachrichten, Wiley-VCH Verlag, In press. ⟨hal-03039783⟩

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