Nonlinear equivalence of Banach spaces based on Birkhoff-James orthogonality

Abstract

A Banach space X is said to be isomorphic to another Y with respect to the structure of Birkhoff-James orthogonality, denoted by X∼BJY, if there exists a (possibly nonlinear) bijection between X and Y that preserves Birkhoff-James orthogonality in both directions. It is shown that X≅Y if either X or Y is finite dimensional and X∼BJY, and that ℓp≁BJℓq if 1<p<q<∞. Moreover, if H is a Hilbert space with dim⁡H≥3 and H∼BJX, then H=X. In the two-dimensional case, it turns out that ℓp,q2∼BJℓ22, which indicates that nonlinear Birkhoff-James orthogonality preservers between Banach spaces are not necessarily scalar multiples of isometric isomorphisms.