On extreme contractions between real Banach spaces
Debmalya Sain, Kallol Paul, Arpita Mal

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 compatible point pair (CPP) in the geometry of Banach spaces. As a concrete application of our abstract results, we describe all rank one extreme contractions in and , where is any Hilbert space. We also prove that there does not exist any rank one extreme contractions in whenever is even and is any Hilbert space. We further study extreme contractions between infinite-dimensional Banach spaces and obtain some analogous results. Finally, we characterize real Hilbert…
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