On directional preservation of orthogonality and its application to isometries

TL;DR

This paper characterizes how linear operators preserve Birkhoff-James orthogonality locally in normed spaces, refining previous results and applying findings to identify isometries on polyhedral spaces like 0^n and 1^n.

Contribution

It provides a complete characterization of local orthogonality preservation and refines existing theorems, advancing understanding of isometries in polyhedral normed spaces.

Findings

Complete characterization of local orthogonality preservation.

Refinements of the Blanco-Koldobsky-Turn
61ek Theorem.

Identification of isometries on polyhedral spaces.

Abstract

We study the local preservation of Birkhoff-James orthogonality by linear operators between normed linear spaces, at a point and in a particular direction. We obtain a complete characterization of the same, which allows us to present refinements of the local preservation of orthogonality explored earlier. We also study the directional preservation of orthogonality with respect to certain special subspaces of the domain space, and apply the results towards identifying the isometries on a polyhedral normed linear space. In particular, we obtain refinements of the Blanco-Koldobsky-Turn\v{s}ek Theorem for polyhedral normed linear spaces, including $ \ell_{\infty}^{n}, \ell_{1}^{n}. $