caa a22 4500 605508631 CHVBK 20210128100637.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s00010-013-0233-7 doi (NATIONALLICENCE)springer-10.1007/s00010-013-0233-7 Approximate Roberts orthogonality [Elektronische Daten] [Ali Zamani, Mohammad Moslehian] In a real normed space we introduce two notions of approximate Roberts orthogonality as follows: $$x \perp_R^\varepsilon y \, {\rm if \, and \, only \, if} \left|\|x + ty\|^2 - \|x - ty\|^2\right| \leq 4\varepsilon\|x\|\|ty\| \, {\rm for \, all} \, t \in \mathbb{R}\,;$$ x ⊥ R ε y if and only if ‖ x + t y ‖ 2 - ‖ x - t y ‖ 2 ≤ 4 ε ‖ x ‖ ‖ t y ‖ for all t ∈ R ; and $$x^{\varepsilon} \perp_R y \, {\rm if \, and \, only \, if} \left|\|x + ty\|-\|x - ty\|\right| \leq \varepsilon(\|x + ty\| + \|x - ty\|) \, {\rm for \, all} \, t \in \mathbb{R}\,.$$ x ε ⊥ R y if and only if ‖ x + t y ‖ - ‖ x - t y ‖ ≤ ε ( ‖ x + t y ‖ + ‖ x - t y ‖ ) for all t ∈ 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 $${^{\varepsilon}\perp_R}$$ ε ⊥ R . A linear mapping $${U: \mathcal{X} \to \mathcal{Y}}$$ U : X → Y between real normed spaces is called an $${\varepsilon}$$ ε -isometry if $${(1 - \varphi_1 (\varepsilon))\|x\| \leq \|Ux\| \leq (1 + \varphi_2(\varepsilon))\|x\|\,\,(x \in \mathcal{X})}$$ ( 1 - φ 1 ( ε ) ) ‖ x ‖ ≤ ‖ U x ‖ ≤ ( 1 + φ 2 ( ε ) ) ‖ x ‖ ( x ∈ X ) , where $${\varphi_1 (\varepsilon)\rightarrow0}$$ φ 1 ( ε ) → 0 and $${\varphi_2 (\varepsilon)\rightarrow0}$$ φ 2 ( ε ) → 0 as $${\varepsilon\rightarrow 0}$$ ε → 0 . We show that a scalar multiple of an $${\varepsilon}$$ ε -isometry is an approximately Roberts orthogonality preserving mapping. Springer Basel, 2013 Roberts orthogonality nationallicence approximate orthogonality nationallicence $${\varepsilon}$$ ε -isometry nationallicence orthogonality preserving mapping nationallicence Zamani Ali Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, 91775, Mashhad, Iran aut Moslehian Mohammad Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, 91775, Mashhad, Iran aut Aequationes mathematicae Springer Basel 89/3(2015-06-01), 529-541 0001-9054 89:3<529 2015 89 10 https://doi.org/10.1007/s00010-013-0233-7 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00010-013-0233-7 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Zamani Ali Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, 91775, Mashhad, Iran aut NATIONALLICENCE 700 1- Moslehian Mohammad Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, 91775, Mashhad, Iran aut NATIONALLICENCE 773 0- Aequationes mathematicae Springer Basel 89/3(2015-06-01), 529-541 0001-9054 89:3<529 2015 89 10