On approximate preservation of orthogonality and its application to isometries

TL;DR

This paper explores how linear operators approximately preserve orthogonality in Banach spaces and refines existing characterizations of isometries, especially in finite-dimensional polyhedral spaces.

Contribution

It introduces new insights into approximate orthogonality preservation and enhances the understanding of isometries in specific Banach spaces.

Findings

Characterization of approximate orthogonality preservation in finite-dimensional polyhedral Banach spaces

Refinements of the Blanco-Koldobsky-Turnšek isometry characterization

Connections between geometric properties and operator behavior

Abstract

Motivated by the famous Blanco-Koldobsky-Turn\v{s}ek characterization of isometries, we study the \textit{approximate preservation of Birkhoff-James orthogonality by a linear operator between Banach spaces}. In particular, we investigate various geometric and analytic properties related to such preservation on finite-dimensional polyhedral Banach spaces. As an application of the results obtained here, we present refinements of the Blanco-Koldobsky-Turn\v{s}ek characterization of isometries on certain Banach spaces.