Approximate Roberts orthogonality
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| 024 | 7 | 0 | |a 10.1007/s00010-013-0233-7 |2 doi |
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| 245 | 0 | 0 | |a Approximate Roberts orthogonality |h [Elektronische Daten] |c [Ali Zamani, Mohammad Moslehian] |
| 520 | 3 | |a 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. | |
| 540 | |a Springer Basel, 2013 | ||
| 690 | 7 | |a Roberts orthogonality |2 nationallicence | |
| 690 | 7 | |a approximate orthogonality |2 nationallicence | |
| 690 | 7 | |a $${\varepsilon}$$ ε -isometry |2 nationallicence | |
| 690 | 7 | |a orthogonality preserving mapping |2 nationallicence | |
| 700 | 1 | |a Zamani |D Ali |u Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, 91775, Mashhad, Iran |4 aut | |
| 700 | 1 | |a Moslehian |D Mohammad |u Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, 91775, Mashhad, Iran |4 aut | |
| 773 | 0 | |t Aequationes mathematicae |d Springer Basel |g 89/3(2015-06-01), 529-541 |x 0001-9054 |q 89:3<529 |1 2015 |2 89 |o 10 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s00010-013-0233-7 |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s00010-013-0233-7 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Zamani |D Ali |u Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, 91775, Mashhad, Iran |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Moslehian |D Mohammad |u Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, 91775, Mashhad, Iran |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Aequationes mathematicae |d Springer Basel |g 89/3(2015-06-01), 529-541 |x 0001-9054 |q 89:3<529 |1 2015 |2 89 |o 10 | ||