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 ${\rho }_{{}_{+}}$, we define a mapping ${\rho }_{{}_{\infty }}$ 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 ${\rho }_{{}_{\infty }}$ 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.