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.

Download full-text


Available from: Blaž Mojškerc, Mar 30, 2014
  • Source
    [Show abstract] [Hide abstract]
    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
    Full-text · Article · Sep 2010 · Aequationes Mathematicae
  • [Show abstract] [Hide abstract]
    ABSTRACT: We survey the results concerning the preservation (exact and approximate) of various types of orthogonality relations. We focus on the stability of the orthogonality preserving property. Our considerations are carried out in spaces with inner product structure as well as in normed spaces. Some related topics are also discussed. KeywordsOrthogonality–Birkhoff orthogonality–Isosceles-orthogonality–Approximate orthogonality–Orthogonality preserving property–Right-angle preserving property–Linear preservers–Stability–Orthogonality equation–Wigner equation–Inner product spaces–Hilbert modules–Normed spaces–Semi-innner product–Norm derivatives–Isometric mappings–Approximate isometry
    No preview · Chapter · Dec 2011
  • Source
    [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.
    Full-text · Article · Feb 2012 · Aequationes Mathematicae
Show more