Aequationes mathematicae

, Volume 89, Issue 3, pp 529–541 | Cite as

Approximate Roberts orthogonality

Article

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.

Mathematics Subject Classification (1991)

Primary 46B20 Secondary 46C05 

Keywords

Roberts orthogonality approximate orthogonality \({\varepsilon}\)-isometry orthogonality preserving mapping 

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© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS)Ferdowsi University of MashhadMashhadIran

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