Extreme contractions on finite-dimensional Banach spaces

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.