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Approximate Roberts orthogonality

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Abstract

In a real normed space we introduce two notions of approximate Roberts orthogonality as follows:

$$x \perp_R^\varepsilon y \, {\rm if \, and \, only \, if} \left|\|x + ty\|^2 - \|x - ty\|^2\right| \leq 4\varepsilon\|x\|\|ty\| \, {\rm for \, all} \, t \in \mathbb{R}\,;$$

and

$$x^{\varepsilon} \perp_R y \, {\rm if \, and \, only \, if} \left|\|x + ty\|-\|x - ty\|\right| \leq \varepsilon(\|x + ty\| + \|x - ty\|) \, {\rm for \, all} \, t \in \mathbb{R}\,.$$

We investigate their properties and their relationship with the approximate Birkhoff orthogonality. Moreover, we study the class of linear mappings preserving approximately Roberts orthogonality of type  \({^{\varepsilon}\perp_R}\). A linear mapping \({U: \mathcal{X} \to \mathcal{Y}}\) between real normed spaces is called an \({\varepsilon}\)-isometry if \({(1 - \varphi_1 (\varepsilon))\|x\| \leq \|Ux\| \leq (1 + \varphi_2(\varepsilon))\|x\|\,\,(x \in \mathcal{X})}\), where \({\varphi_1 (\varepsilon)\rightarrow0}\) and \({\varphi_2 (\varepsilon)\rightarrow0}\) as \({\varepsilon\rightarrow 0}\). We show that a scalar multiple of an \({\varepsilon}\)-isometry is an approximately Roberts orthogonality preserving mapping.

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  1. Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, 91775, Mashhad, Iran

    Ali Zamani & Mohammad Sal Moslehian

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  1. Ali Zamani
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  2. Mohammad Sal Moslehian
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Correspondence to Mohammad Sal Moslehian.

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Zamani, A., Moslehian, M.S. Approximate Roberts orthogonality. Aequat. Math. 89, 529–541 (2015). https://doi.org/10.1007/s00010-013-0233-7

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  • DOI: https://doi.org/10.1007/s00010-013-0233-7

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