An improvement of the Blanco-Koldobsky-Turn\v{s}ek characterization of isometries

TL;DR

This paper enhances the characterization of isometries in normed spaces by introducing level vectors and their properties, providing a more detailed understanding of geometric and structural aspects, especially in spaces with the Krein-Milman property.

Contribution

It introduces the concept of level vectors via directional preservation of Birkhoff-James orthogonality and refines existing isometry characterizations in specific spaces.

Findings

Level vectors characterized through orthogonality preservation

Analysis of geometric phenomena induced by level vectors

Refinement of isometry characterization in Krein-Milman spaces

Abstract

We present an improvement of the Blanco-Koldobsky-Turn\v{s}ek characterization of isometries in normed linear spaces by using the concept of level vectors of an operator. In this context, we characterize level vectors entirely through directional preservation of Birkhoff-James orthogonality and analyze the associated geometric and structural phenomena that they induce. Furthermore, in spaces whose unit balls possess the \textit{Krein-Milman property}, we derive an additional refinement of the Blanco-Koldobsky-Turn\v{s}ek characterization of isometries.