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arXiv · 2407.05545

Extreme contractions on finite-dimensional Banach spaces

Debmalya Sain·Shamim Sohel·Kallol Paul
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Abstract

We study extreme contractions in the setting of finite-dimensional polyhedral Banach spaces. Motivated by the famous Krein-Milman Theorem, we prove that a \emph{rank one} norm one linear operator between such spaces can be expressed as a convex combination of \emph{rank one} extreme contractions, whenever the domain is two-dimensional. We establish that the same result holds true in the space of all linear operators from $\ell_{\infty}^n(\mathbb{C}) $ to $ \ell_1^n (\mathbb{C}). $ Furthermore, we present a geometric characterization of extreme contractions between finite-dimensional polyhedral Banach spaces.

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Debmalya Sain, Shamim Sohel, Kallol Paul. 2024-07-08. Extreme contractions on finite-dimensional Banach spaces. https://doi.org/10.4064/cm8803-7-2022

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