An extension of orthogonality relations based on norm derivatives

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.