Characterizations of smooth spaces by ${\rho }_{\ast }$-orthogonality

The aim of this paper is to present some results concerning the ${\rho }_{\ast }$-orthogonality in real normed spaces and its preservation by linear operators. Among other things, we prove that if $T:X⟶Y$ is a nonzero linear $(I,{\rho }_{\ast })$-orthogonality preserving mapping between real normed spaces, then $$\frac{1}{3}\mathrm{\parallel }T\mathrm{\parallel }\mathrm{\parallel }x\mathrm{\parallel }\le \mathrm{\parallel }Tx\mathrm{\parallel }\le 3[T]\mathrm{\parallel }x\mathrm{\parallel },(x\in X)$$ where $[T]:=\inf \{\mathrm{\parallel }Tx\mathrm{\parallel }:x\in X,\mathrm{\parallel }x\mathrm{\parallel }=1\}$. We also show that the pair $(X,{\perp }_{{\rho }_{\ast }})$ is an orthogonality space in the sense of R"{a}tz. Some characterizations of smooth spaces are given based on the ${\rho }_{\ast }$-orthogonality.