Nonlinear Birkhoff–James orthogonality preservers in smooth normed spaces

Abstract

Let X and Y be smooth normed spaces which are either real and dim⁡X≥3, or infinite dimensional complex, and one of them is reflexive. Then a surjective mapping from X to Y preserves Birkhoff–James orthogonality in both directions if and only if it has the form x↦τ(x)Ux for some surjective linear or conjugate linear isometry U:X→Y and some scalar–valued mapping τ on X. In particular, there exists a surjective mapping from X to Y preserving Birkhoff–James orthogonality in both directions if and only if X and Y are isometrically isomorphic or conjugate isometrically isomorphic. Several illustrative examples and relations with Wigner's theorem are also given.