On extreme contractions between real Banach spaces

TL;DR

This paper characterizes extreme contractions between two-dimensional strictly convex and smooth real Banach spaces, introduces new geometric notions, and applies these to describe specific rank one extreme contractions and characterize Hilbert spaces.

Contribution

It introduces the notions of compatible point pairs and μ-compatible point pairs, providing a complete characterization of extreme contractions in certain Banach space settings.

Findings

Complete characterization of extreme contractions in 2D strictly convex and smooth Banach spaces.

Identification of all rank one extreme contractions in specific operator spaces.

Characterization of real Hilbert spaces using compatible point pairs.

Abstract

We completely characterize extreme contractions between two-dimensional strictly convex and smooth real Banach spaces, perhaps for the very first time. In order to obtain the desired characterization, we introduce the notions of (weakly) compatible point pair (CPP) and $ \mu- $compatible point pair ($ \mu- $CPP) in the geometry of Banach spaces. As a concrete application of our abstract results, we describe all rank one extreme contractions in $ \mathbb{L}(\ell_4^2, \ell_4^2) $ and $ \mathbb{L}(\ell_4^2, \mathbb{H}) $, where $\mathbb{H}$ is any Hilbert space. We also prove that there does not exist any rank one extreme contractions in $ \mathbb{L}(\mathbb{H}, \ell_{p}^2), $ whenever $ p $ is even and $\mathbb{H}$ is any Hilbert space. We further study extreme contractions between infinite-dimensional Banach spaces and obtain some analogous results. Finally, we characterize real Hilbert spaces among real Banach spaces in terms of CPP, that substantiates our motivation behind introducing these new geometric notions.