Extreme contractions on finite-dimensional Banach spaces
Volume 172 / 2023
Colloquium Mathematicum 172 (2023), 65-83
MSC: Primary 46B20; Secondary 47L05.
DOI: 10.4064/cm8803-7-2022
Published online: 24 October 2022
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 rank $1$ linear operator of unit norm between such spaces can be expressed as a convex combination of rank $1$ 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.
