# Mappings approximately preserving orthogonality in normed spaces

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Blaž Mojškerc, Mar 30, 2014 Available from:-
##### Article: On a ρ-orthogonality

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**ABSTRACT:**In a normed space we introduce an exact and approximate orthogonality relation connected with “norm derivatives” r¢±{\rho^{\prime}_{\pm}} . We also consider classes of linear mappings preserving (exactly and approximately) this kind of orthogonality. Mathematics Subject Classification (2010)Primary 46B20-46C50-Secondary 39B82 KeywordsOrthogonality-approximate orthogonality-orthogonality preserving mappings-norm derivativeAequationes Mathematicae 09/2010; 80(1):45-55. DOI:10.1007/s00010-010-0042-1 · 0.55 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We survey mainly recent results on the two most important orthogonality types in normed linear spaces, namely on Birkhoff orthogonality and on isosceles (or James) orthogonality. We lay special emphasis on their fundamental properties, on their differences and connections, and on geometric results and problems inspired by the respective theoretical framework. At the beginning we also present other interesting types of orthogonality. This survey can also be taken as an update of existing related representations.Aequationes Mathematicae 02/2012; 83(1-2). DOI:10.1007/s00010-011-0092-z · 0.55 Impact Factor -
##### Article: 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.Aequationes Mathematicae 06/2013; 89(3). DOI:10.1007/s00010-013-0233-7 · 0.55 Impact Factor