On Birkhoff-James orthogonality preservers between real non-isometric Banach spaces

TL;DR

This paper investigates the conditions under which Banach spaces preserve Birkhoff-James orthogonality, showing that smoothness implies isometric isomorphism, and constructing examples of non-isometric spaces with orthogonality-preserving maps.

Contribution

It establishes that smooth Banach spaces are isomorphic via Birkhoff-James orthogonality and provides explicit examples of non-isometric spaces with orthogonality-preserving maps.

Findings

Smooth Banach spaces are isomorphic if they preserve Birkhoff-James orthogonality.

Constructed examples of non-isometric Banach spaces with orthogonality-preserving maps.

Identified conditions for orthogonality preservation in various Banach space structures.

Abstract

Real smooth three-dimensional or higher Banach spaces are isomorphic with respect to the nonlinear structure of Birkhoff-James orthogonality if and only if they are isometrically isomorphic. Moreover, using smooth Radon planes and non-smooth direct sums, in arbitrary dimensions, we construct examples of non-isometric pairs of real Banach spaces that admit norm-preserving homogeneous bicontinuous Birkhoff-James orthogonality preservers among them.