An orthogonality relation in complex normed spaces based on norm derivatives

TL;DR

This paper introduces a new orthogonality concept in complex normed spaces based on norm derivatives, characterizes inner product spaces via symmetry of this relation, and studies mappings preserving this orthogonality.

Contribution

It defines a novel orthogonality relation using norm derivatives, characterizes inner product spaces through symmetry, and analyzes structure-preserving maps in complex normed spaces.

Findings

Symmetry of the new orthogonality characterizes inner product spaces.

Mappings preserving the orthogonality are scalar multiples of isometries.

The paper discusses open problems in complex normed space geometry.

Abstract

Let $X$ be a complex normed space. Based on the right norm derivative $\rho_{_{+}}$, we define a mapping $\rho_{_{\infty}}$ by \begin{equation*} \rho_{_{\infty}}(x,y) = \frac1\pi\int_0^{2\pi}e^{i\theta}\rho_{_{+}}(x,e^{i\theta}y)d\theta \quad(x,y\in X). \end{equation*} The mapping $\rho_{_{\infty}}$ has a good response to some geometrical properties of $X$. For instance, we prove that $\rho_{_{\infty}}(x,y)=\rho_{_{\infty}}(y,x)$ for all $x, y \in X$ if and only if $X$ is an inner product space. In addition, we define a $\rho_{_{\infty}}$-orthogonality in $X$ and show that a linear mapping preserving $\rho_{_{\infty}}$-orthogonality has to be a scalar multiple of an isometry. A number of challenging problems in the geometry of complex normed spaces are also discussed.