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

Abstract

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 $$\frac{1}{3}\Vert T\Vert \Vert x\Vert \le \Vert Tx\Vert \le 3[T]\Vert x\Vert ,(x\in X)$$ where $[T]:=\inf \{\Vert Tx\Vert :x\in X,\Vert x\Vert =1\}$. We also show that the pair $(X,{\perp }_{{\rho }_{\ast }})$ is an orthogonality space in the sense of Rätz. Some characterizations of smooth spaces are given based on the ${\rho }_{\ast }$-orthogonality.

arXiv comment19 pages, to appear in Houston J. Math