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 μ-compatible point pair (μ-CPP) in the geometry of Banach spaces. As a concrete application of our abstract results, we describe all rank one extreme contractions in L(ℓ42,ℓ42) and L(ℓ42,H), where H is any Hilbert space. We also prove that there does not exist any rank one extreme contractions in L(H,ℓp2), whenever p is even and H is any Hilbert space. Finally, we characterize real Hilbert spaces among real Banach spaces in terms of CPP, that substantiates our motivation behind introducing these new geometric notions.
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