An extension of orthogonality relations based on norm derivatives

TL;DR

This paper introduces a new orthogonality relation in normed spaces based on norm derivatives, characterizes inner product spaces through it, and studies linear mappings that preserve this relation, showing they are similarities.

Contribution

It extends orthogonality relations using norm derivatives, characterizes inner product spaces via a new functional, and describes linear maps preserving this orthogonality as similarities.

Findings

Characterizes inner product spaces via ${ ho}_{ ext{lambda}}$.

Shows linear maps preserving ${ ho}_{ ext{lambda}}$-orthogonality are similarities.

Establishes properties of the new orthogonality relation.

Abstract

We introduce the relation ${\rho}_{\lambda}$-orthogonality in the setting of normed spaces as an extension of some orthogonality relations based on norm derivatives, and present some of its essential properties. Among other things, we give a characterization of inner product spaces via the functional ${\rho}_{\lambda}$. Moreover, we consider a class of linear mappings preserving this new kind of orthogonality. In particular, we show that a linear mapping preserving ${\rho}_{\lambda}$-orthogonality has to be a similarity, that is, a scalar multiple of an isometry.