Skip to main content
SpringerLink
Log in

Orthogonality Preserving Property and its Ulam Stability

  • Chapter
  • First Online:
Book cover Functional Equations in Mathematical Analysis

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

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.

Mathematics Subject Classification (2010): Primary 46B20, 47B49, 15A86, 39B52, 39B82

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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. 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. Alsina, C., Sikorska, J., Tomás, M.S.: Norm Derivatives and Characterizations of Inner Product Spaces. World Scientific, Hackensack, NJ (2009)

    Book  Google Scholar 

  4. Amir, D.: Characterization of Inner Product Spaces. Birkhäuser, Basel–Boston–Stuttgart (1986)

    Google Scholar 

  5. Birkhoff, G.: Orthogonality in linear metric spaces. Duke Math. J. 1, 169–172 (1935)

    Article  MathSciNet  Google Scholar 

  6. Blanco, A., Turnšek, A.: On maps that preserve orthogonality in normed spaces. Proc. Roy. Soc. Edinburgh Sect. A 136, 709–716 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chmieliński, J.: On an ε-Birkhoff orthogonality. J. Inequal. Pure Appl. Math. 6(3), Art. 79 (2005)

    Google Scholar 

  9. Chmieliński, J.: Linear mappings approximately preserving orthogonality. J. Math. Anal. Appl. 304, 158–169 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chmieliński, J., Stability of the orthogonality preserving property in finite-dimensional inner product spaces. J. Math. Anal. Appl. 318, 433–443 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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. Chmieliński, J.: Orthogonality preserving property, Wigner equation and stability. J. Inequal. Appl. 2006, Article ID 76489, 1–9 (2006)

    Google Scholar 

  13. Chmieliński, J.: Stability of the Wigner equation and related topics. Nonlinear Funct. Anal. Appl. 11, 859–879 (2006)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  15. Chmieliński, J., Moslehian, M.S.: Approximately C  ∗ -inner product preserving mappings. Bull. Korean Math. Soc. 45, 157–167 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  17. Chmieliński, J., Wójcik, P.: Isosceles-orthogonality preserving property and its stability. Nonlinear Anal. 72, 1445-1453 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chmieliński, J., Wójcik, P.: On a ρ-orthogonality. Aequationes Math. 80, 45–55 (2010)

    Article  MATH  Google Scholar 

  19. Day, M.M.: Normed Linear Spaces. Springer-Verlag, Berlin – Heidelberg – New York (1973)

    Google Scholar 

  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. Dragomir, S.S.: On approximation of continuous linear functionals in normed linear spaces. An. Univ. Timişoara Ser. Ştiinţ. Mat. 29, 51–58 (1991)

    MATH  Google Scholar 

  22. Dragomir, S.S.: Semi-Inner Products and Applications. Nova Science Publishers, Inc., Hauppauge, NY (2004)

    MATH  Google Scholar 

  23. Forti, G.L.: Hyers–Ulam stability of functional equations in several variables. Aequationes Math. 50, 143–190 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  25. Giles, J.R.: Classes of semi-inner-product spaces. Trans. Amer. Math. Soc. 129, 436–446 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  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. Ilišević, D., Turnšek, A.: Approximately orthogonality preserving mappings on C  ∗ -modules. J. Math. Anal. Appl. 341, 298–308 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. James, R.C: Orthogonality in normed linear linear spaces. Duke Math. J. 12, 291–301 (1945)

    Google Scholar 

  29. James, R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Amer. Math. Soc. 61, 265–292 (1947)

    Article  MathSciNet  Google Scholar 

  30. Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Inc., Palm Harbor (2001)

    MATH  Google Scholar 

  31. Koehler, D., Rosenthal, P.: On isometries of normed linear spaces. Studia Math. 36, 213–216 (1970)

    MathSciNet  MATH  Google Scholar 

  32. Koldobsky, A.: Operators preserving orthogonality are isometries. Proc. Roy. Soc. Edinburgh Sect. A 123, 835–837 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  34. Lance, E.C.: Hilbert C  ∗ -Modules. London Math. Soc. Lecture Note Series 210, Cambridge University Press, Cambridge (1995)

    Google Scholar 

  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: 0910.2335

    Google Scholar 

  36. Leung, C.W., Ng, C.K., Wong, N.C.: Linear orthogonality preservers of Hilbert boundles. Preprint, arXiv:1005.4502

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lumer, G.: Semi–inner–product spaces. Trans. Amer. Math. Soc. 100, 29-43 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  39. Miličić, P.M.: Sur la G-orthogonalité dans les espaces normés. Mat. Vesnik 39, 325–334 (1987)

    MathSciNet  MATH  Google Scholar 

  40. Mojškerc, B., Turnšek, A.: Mappings approximately preserving orthogonality in normed spaces. Nonlinear Anal. 73, 3821–3831 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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. Moszner, Z.: On the stability of functional equations. Aequationes Math. 77, 33–88 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Pedersen, G.K.: Analysis Now. Grad. Texts in Math. 118, Springer-Verlag, New-York (1989)

    Google Scholar 

  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. 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. Turnšek, A.: On operators preserving James’ orthogonality. Lin. Algebra Appl. 407, 189–195 (2005)

    Article  MATH  Google Scholar 

  47. Turnšek, A.: On mappings approximately preserving orthogonality. J. Math. Anal. Appl. 336, 625–631 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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. Vestfrid, I.A.: Linear approximation of approximately biorthogonality preserving maps. J. Math. Anal. Appl. 336, 418–424 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  50. Wigner, E.P.: Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren. Friedr. Vieweg und Sohn Akt.-Ges. (1931)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instytut Matematyki, Uniwersytet Pedagogiczny w Krakowie, Podchora̧żych 2, 30-084, Kraków, Poland

    Jacek Chmieliński

Authors
  1. Jacek Chmieliński
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jacek Chmieliński .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, National Technical University of Athens, Iroon Polytechneiou 9, Athens, 15780, Greece

    Themistocles M. Rassias

  2. Institute of Mathematics, Pedagogical University, ul. Podchorazych 2, Krakow, 30-084, Poland

    Janusz Brzdek

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Chmieliński, J. (2011). Orthogonality Preserving Property and its Ulam Stability. In: Rassias, T., Brzdek, J. (eds) Functional Equations in Mathematical Analysis. Springer Optimization and Its Applications(), vol 52. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0055-4_4

Download citation

Publish with us

Policies and ethics

Search

Navigation