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arXiv · 1705.07032

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

Mohammad Sal Moslehian·Ali Zamani·Mahdi Dehghani
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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.

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Mohammad Sal Moslehian, Ali Zamani, Mahdi Dehghani. 2017-05-19. Characterizations of smooth spaces by $\rho_*$-orthogonality. https://arxiv.org/abs/1705.07032

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