Characterizations of smooth spaces by $\rho_*$-orthogonality

TL;DR

This paper investigates the properties of $ ho_*$-orthogonality in real normed spaces, characterizes smooth spaces using this concept, and examines how linear operators preserve this orthogonality.

Contribution

It introduces new characterizations of smooth spaces via $ ho_*$-orthogonality and analyzes the preservation of this orthogonality by linear operators.

Findings

Linear $(I, ho_*)$-orthogonality preserving maps satisfy specific norm inequalities.

The pair $(X,ot_{ ho_*})$ forms an orthogonality space in R"{a}tz's sense.

Characterizations of smooth spaces are established based on $ ho_*$-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.