Di-orthographs of Banach spaces that are not invariant of nonlinear Birkhoff-James orthogonality preservers

Abstract

In this paper, we study the relationship between two known nonlinear equivalences of normed spaces induced by Birkhoff-James orthogonality. To be precise, we consider the nonlinear equivalences of normed spaces with respect to the structure of Birkhoff-James orthogonality and di-orthographs of them. It is shown that a graph isomorphism between di-orthographs of normed spaces gives rise to a Birkhoff-James orthogonality preserver, but the converse is false in general. A counterexample is constructed in the real two-dimensional setting. Radon planes and a reduced version of di-orthographs play important roles in this direction.