An orthogonality relation in complex normed spaces based on norm derivatives

Abstract

Let $X$ be a complex normed space. Based on the right norm derivative $ρ_{_{+}}$, we define a mapping $ρ_{_{\infty}}$ by \begin{equation*} ρ_{_{\infty}}(x,y) = \frac1π\int_0^{2π}e^{iθ}ρ_{_{+}}(x,e^{iθ}y)dθ\quad(x,y\in X). \end{equation*} The mapping $ρ_{_{\infty}}$ has a good response to some geometrical properties of $X$. For instance, we prove that $ρ_{_{\infty}}(x,y)=ρ_{_{\infty}}(y,x)$ for all $x, y \in X$ if and only if $X$ is an inner product space. In addition, we define a $ρ_{_{\infty}}$-orthogonality in $X$ and show that a linear mapping preserving $ρ_{_{\infty}}$-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.