Characterizations of smooth spaces by $ρ_*$-orthogonality

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