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Approximate Roberts orthogonality. (English) Zbl 1329.46019

Summary: In a real normed space we introduce two notions of approximate Roberts orthogonality as follows: \[ x\perp_R^\varepsilon y\text{ if and only if }\left|\| x+ty\|^2-\| x-ty\|^2\right|\leq 4\varepsilon\| x\|\| ty\|\text{ for all } t\in\mathbb R; \] and \[ x^\varepsilon\perp_R y\text{ if and only if }\left|\| x+ty\|-\| x-ty\|\right|\leq\varepsilon(\| x+ty\|+\| x-ty\|)\text{ for all } t\in\mathbb R. \] 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 \(\varphi_1(\varepsilon)\to 0\) and \(\varphi_2(\varepsilon)\to 0\) as \(\varepsilon\to 0\). We show that a scalar multiple of an \(\varepsilon\)-isometry is an approximately Roberts orthogonality preserving mapping.

MSC:

46B20 Geometry and structure of normed linear spaces
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
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