Orthogonality Preserving Property and its Ulam Stability

Part of the Springer Optimization and Its Applications book series (SOIA, volume 52)


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.


Orthogonality 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 


  1. 1.
    Alonso, J., Benitez, C.: Orthogonality in normed linear spaces: A survey. Part I: Main properties. Extracta Math. 3, 1–15 (1988). Part II: Relations between main orthogonalities. Extracta Math. 4, 121–131 (1989)Google Scholar
  2. 2.
    Alonso, J.: Some properties of Birkhoff and isosceles orthogonality in normed linear space. In: Rassias Th.M. (ed.) Inner product spaces and applications, pp. 1–11. Pitman Res. Notes Math. Ser. 376, Addison Wesley Longman, Harlow (1997)Google Scholar
  3. 3.
    Alsina, C., Sikorska, J., Tomás, M.S.: Norm Derivatives and Characterizations of Inner Product Spaces. World Scientific, Hackensack, NJ (2009)CrossRefGoogle Scholar
  4. 4.
    Amir, D.: Characterization of Inner Product Spaces. Birkhäuser, Basel–Boston–Stuttgart (1986)Google Scholar
  5. 5.
    Birkhoff, G.: Orthogonality in linear metric spaces. Duke Math. J. 1, 169–172 (1935)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Blanco, A., Turnšek, A.: On maps that preserve orthogonality in normed spaces. Proc. Roy. Soc. Edinburgh Sect. A 136, 709–716 (2006)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Burgos, M., Fernández-Polo, F., Garcés, J.J., Martínez Moreno J., Peralta A.M.: Orthogonality preservers in C  ∗ -algebras, JB  ∗ -algebras and JB  ∗ -triples. J. Math. Anal. Appl. 348, 220–233 (2008)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Chmieliński, J.: On an ε-Birkhoff orthogonality. J. Inequal. Pure Appl. Math. 6(3), Art. 79 (2005)Google Scholar
  9. 9.
    Chmieliński, J.: Linear mappings approximately preserving orthogonality. J. Math. Anal. Appl. 304, 158–169 (2005)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Chmieliński, J., Stability of the orthogonality preserving property in finite-dimensional inner product spaces. J. Math. Anal. Appl. 318, 433–443 (2006)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Chmieliński, J.: On some approximate functional relations stemming from orthogonality preserving property. J. Inequal. Pure Appl. Math. 7, no. 3, Article 85, 1–14 (2006)Google Scholar
  12. 12.
    Chmieliński, J.: Orthogonality preserving property, Wigner equation and stability. J. Inequal. Appl. 2006, Article ID 76489, 1–9 (2006)Google Scholar
  13. 13.
    Chmieliński, J.: Stability of the Wigner equation and related topics. Nonlinear Funct. Anal. Appl. 11, 859–879 (2006)MathSciNetMATHGoogle Scholar
  14. 14.
    Chmieliński, J.: Remarks on orthogonality preserving mappings in normed spaces and some stability problems. Banach J. Math. Anal. 1, no. 1, 117–124 (2007)MathSciNetMATHGoogle Scholar
  15. 15.
    Chmieliński, J., Moslehian, M.S.: Approximately C  ∗ -inner product preserving mappings. Bull. Korean Math. Soc. 45, 157–167 (2008)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Chmieliński, J., Ilišević, D., Moslehian M.S., Sadeghi Gh.: Perturbation of the Wigner equation in inner product C  ∗ -modules. J. Math. Phys. 49, 033519 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Chmieliński, J., Wójcik, P.: Isosceles-orthogonality preserving property and its stability. Nonlinear Anal. 72, 1445-1453 (2010)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Chmieliński, J., Wójcik, P.: On a ρ-orthogonality. Aequationes Math. 80, 45–55 (2010)MATHCrossRefGoogle Scholar
  19. 19.
    Day, M.M.: Normed Linear Spaces. Springer-Verlag, Berlin – Heidelberg – New York (1973)Google Scholar
  20. 20.
    Ding, G.G.: The approximation problem of almost isometric operators by isometric operators. Acta Math. Sci. (English Ed.) 8, 361–372 (1988)Google Scholar
  21. 21.
    Dragomir, S.S.: On approximation of continuous linear functionals in normed linear spaces. An. Univ. Timişoara Ser. Ştiinţ. Mat. 29, 51–58 (1991)MATHGoogle Scholar
  22. 22.
    Dragomir, S.S.: Semi-Inner Products and Applications. Nova Science Publishers, Inc., Hauppauge, NY (2004)MATHGoogle Scholar
  23. 23.
    Forti, G.L.: Hyers–Ulam stability of functional equations in several variables. Aequationes Math. 50, 143–190 (1995)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Frank, M., Mishchenko, A.S., Pavlov, A.A.: Orthogonality-preserving, C  ∗ -conformal and conformal module mappings on Hilbert C  ∗ -modules. J. Funct. Anal. 260, 327–339 (2011)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Giles, J.R.: Classes of semi-inner-product spaces. Trans. Amer. Math. Soc. 129, 436–446 (1967)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Hyers, D,H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Progress in Nonlinear Differential Equations and Their Applications vol. 34, Birkhäuser, Boston-Basel-Berlin (1998)Google Scholar
  27. 27.
    Ilišević, D., Turnšek, A.: Approximately orthogonality preserving mappings on C  ∗ -modules. J. Math. Anal. Appl. 341, 298–308 (2008)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    James, R.C: Orthogonality in normed linear linear spaces. Duke Math. J. 12, 291–301 (1945)Google Scholar
  29. 29.
    James, R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Amer. Math. Soc. 61, 265–292 (1947)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Inc., Palm Harbor (2001)MATHGoogle Scholar
  31. 31.
    Koehler, D., Rosenthal, P.: On isometries of normed linear spaces. Studia Math. 36, 213–216 (1970)MathSciNetMATHGoogle Scholar
  32. 32.
    Koldobsky, A.: Operators preserving orthogonality are isometries. Proc. Roy. Soc. Edinburgh Sect. A 123, 835–837 (1993)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Kong, L., Cao, H.: Stability of orthogonality preserving mappings and the orthogonality equation (Chinese). J. Shaanxi Norm. Univ. Nat. Sci. Ed. 36, 10–14 (2008)MathSciNetMATHGoogle Scholar
  34. 34.
    Lance, E.C.: Hilbert C  ∗ -Modules. London Math. Soc. Lecture Note Series 210, Cambridge University Press, Cambridge (1995)Google Scholar
  35. 35.
    Leung, C.W., Ng, C.K., Wong, N.C.: Linear orthogonality preservers of Hilbert C  ∗ -modules over C  ∗ -algebras with real rank zero. Preprint, arXiv: 0910.2335Google Scholar
  36. 36.
    Leung, C.W., Ng, C.K., Wong, N.C.: Linear orthogonality preservers of Hilbert boundles. Preprint, arXiv:1005.4502Google Scholar
  37. 37.
    Leung, C.W., Ng, C.K., Wong, N.C.: Linear orthogonality preservers of Hilbert C  ∗ -modules over general C  ∗ -algebras. J. Aust. Math. Soc. 89, 245–254 (2010)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Lumer, G.: Semi–inner–product spaces. Trans. Amer. Math. Soc. 100, 29-43 (1961)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Miličić, P.M.: Sur la G-orthogonalité dans les espaces normés. Mat. Vesnik 39, 325–334 (1987)MathSciNetMATHGoogle Scholar
  40. 40.
    Mojškerc, B., Turnšek, A.: Mappings approximately preserving orthogonality in normed spaces. Nonlinear Anal. 73, 3821–3831 (2010)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Molnár, L.: Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces. Lecture Notes in Mathematics 1895, Springer, Berlin (2007)Google Scholar
  42. 42.
    Moszner, Z.: On the stability of functional equations. Aequationes Math. 77, 33–88 (2009)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Pedersen, G.K.: Analysis Now. Grad. Texts in Math. 118, Springer-Verlag, New-York (1989)Google Scholar
  44. 44.
    Protasov, V.Yu.: On stability of isometries in Banach spaces. In: Rassias Th.M., Brzdȩk J. (eds) Functional Equations in Mathematical Analysis, Springer (the paper is to be published in the same volume.)Google Scholar
  45. 45.
    Tissier, A.: A right-angle preserving mapping (a solution of a problem proposed in 1983 by H. Kestelman). Advanced Problem 6436, Amer. Math. Monthly 92, 291–292 (1985)Google Scholar
  46. 46.
    Turnšek, A.: On operators preserving James’ orthogonality. Lin. Algebra Appl. 407, 189–195 (2005)MATHCrossRefGoogle Scholar
  47. 47.
    Turnšek, A.: On mappings approximately preserving orthogonality. J. Math. Anal. Appl. 336, 625–631 (2007)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Ulam, S.M.: Problems in Modern Mathematics (Chapter VI, Some Questions in Analysis: §1, Stability). Science Editions, John Wiley & Sons, New York (1964)Google Scholar
  49. 49.
    Vestfrid, I.A.: Linear approximation of approximately biorthogonality preserving maps. J. Math. Anal. Appl. 336, 418–424 (2007)MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Wigner, E.P.: Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren. Friedr. Vieweg und Sohn Akt.-Ges. (1931)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Instytut MatematykiUniwersytet Pedagogiczny w KrakowieKrakówPoland

Personalised recommendations