Extreme contractions on finite-dimensional Banach spaces

TL;DR

This paper investigates the structure of extreme contractions in finite-dimensional polyhedral Banach spaces, providing convex combination representations and geometric characterizations, especially for rank one operators and specific spaces like b^n and _1^n.

Contribution

It proves that rank one norm one operators in two-dimensional spaces can be expressed as convex combinations of extreme contractions, extending this to certain operator spaces and offering geometric characterizations.

Findings

Rank one operators in 2D can be decomposed into convex combinations of extreme contractions.

The result extends to operators from b^n(\u00a3) to _1^n().

Provides geometric characterizations of extreme contractions in finite-dimensional polyhedral 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.