  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.
