Mappings approximately preserving orthogonality in normed spaces

Nonlinear Analysis (Impact Factor: 1.33). 12/2010; 73(12):3821-3831. DOI: 10.1016/


We answer many open questions regarding approximately orthogonality preserving mappings (in Birkhoff–James sense) in normed spaces. In particular, we show that every approximately orthogonality preserving linear mapping (in Chmieliński sense) is necessarily a scalar multiple of an ε-isometry. Thus, whenever ε-isometries are close to isometries we obtain stability. An example is given showing that approximately orthogonality preserving mappings are in general far from scalar multiples of isometries, that is, stability does not hold.

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Available from: Blaž Mojškerc, Mar 30, 2014
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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 derivative
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