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 ${\rho }_{{}_{\mathrm{\infty }}}$ by ${\rho }_{{}_{\mathrm{\infty }}}(x,y)=\frac{1}{\pi }{\int }_0^{2\pi }{\mathrm{e}}^{i\theta }{\rho }_{{}_{+}}(x,{\mathrm{e}}^{i\theta }y)\mathrm{d}\theta (x,y\in X).$The mapping ${\rho }_{{}_{\mathrm{\infty }}}$ has a good response to some geometrical properties of X. For instance, we prove that ${\rho }_{{}_{\mathrm{\infty }}}(x,y)={\rho }_{{}_{\mathrm{\infty }}}(y,x)$ for all $x,y\in X$ if and only if X is an inner product space. In addition, we define a ${\rho }_{{}_{\mathrm{\infty }}}$-orthogonality in X and show that a linear mapping preserving ${\rho }_{{}_{\mathrm{\infty }}}$-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.

Keywords:

• Norm derivatives
• orthogonality
• orthogonality preserving mappings
• inner product spaces

Disclosure statement

No potential conflict of interest was reported by the author(s).