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On Exact and Approximate Orthogonalities Based on Norm Derivatives

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Ulam Type Stability

Abstract

We survey mainly recent results on the orthogonality relations in normed linear spaces related to norm derivatives. We will focus on fundamental properties of norm derivatives orthogonality, differences and connections between these orthogonality types, and geometric results and problems closely related to them.

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References

  1. Abed, M.Y., Dehghani, M., Jahanipur, R.: Approximate \(\rho ^v_{\lambda }\)-orthogonality and its preservation. Ann. Funct. Anal. https://projecteuclid.org/accepted/euclid.afa

  2. Alonso, J., Martini, H., Wu, S.: On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces. Aequationes Math. 83, 153–189 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alsina, C., Sikorska, J., Tomás, M.S.: Norm Derivatives and Characterizations of Inner Product Spaces. World Scientific, Hackensack (2010)

    MATH  Google Scholar 

  4. Amir, D.: Characterizations of Inner Products Spaces. Birkauser, Basel (1986)

    Google Scholar 

  5. Ansari-piri, E., Sanati, R.G., Kardel, M.: A characterization of orthogonality preserving operators. Bull. Iran. Math. Soc. 43(7), 2495–2505 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Arambašić, L., Rajić, R.: Operators preserving the strong Birkhoff–James orthogonality on B(H). Linear Algebra Appl. 471, 394–404 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Balestro, V., Horváth, Á.G., Martini, H., Teixeira, R.: Angles in normed spaces. Aequationes Math. 91(2), 201–236 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Blanco, A., Turnšek, A.: On maps that preserve orthogonality in normed spaces. Proc. R. Soc. Edinb. Sect. A 136(4), 709–716 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brešar, M., Chebotar, M.A.: Linear preserver problems. In: Functional Identities, pp. 189–219. Birkhäuser, Basel (2007)

    Google Scholar 

  11. Chen, C., Lu, F.: Linear maps preserving orthogonality. Ann. Funct. Anal. 6(4), 70–76 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Chmieliński, J.: Stability of the orthogonality preserving property in finite-dimensional inner product spaces. J. Math. Anal. Appl. 318(2), 433–443 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chmieliński, J.: Remarks on orthogonality preserving mappings in normed spaces and some stability problems. Banach J. Math. Anal. 1(1), 117–124 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chmieliński, J.: Orthogonality preserving property and its Ulam stability. In: Rassias, T.M., Brzdȩk, J. (eds.) Functional Equations in Mathematical Analysis. Springer Optimization and Its Applications, vol. 52, pp. 33–58. Springer, Berlin (2011)

    Google Scholar 

  17. Chmieliński, J.: Normed spaces equivalent to inner product spaces and stability of functional equations. Aequationes Math. 87, 147–157 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chmieliński, J.: Orthogonality equation with two unknown functions. Aequationes Math. 90, 11–23 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chmieliński, J.: Operators reversing orthogonality in normed spaces. Adv. Oper. Theory 1(1), 8–14 (2016)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Chmieliński, J., Wójcik, P.: ρ-orthogonality and its preservation–revisited. Recent developments in functional equations and inequalities, 17–30, Banach Center Publications, vol. 99, Polish Academy of Sciences, Institute of Mathematic, Warsaw (2013)

    Google Scholar 

  23. Chmieliński, J., Łukasik, R., Wójcik, P.: On the stability of the orthogonality equation and the orthogonality-preserving property with two unknown functions. Banach J. Math. Anal. 10(4), 828–847 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dadipour, F., Sadeghi, F., Salemi, A.: Characterizations of inner product spaces involving homogeneity of isosceles orthogonality. Arch. Math. 104, 431–439 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dadipour, F., Sadeghi, F., Salemi, A.: An orthogonality in normed linear spaces based on angular distance inequality. Aequationes Math. 90, 281–297 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Day, M.M.: Some characterizations of inner-product spaces. Trans. Am. Math. Soc. 62, 320–337 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  27. Dehghani, M., Zamani, A.: Linear mappings approximately preserving ρ ∗-orthogonality. Indag. Math. 28, 992–1001 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Dehghani, M., Abed, M., Jahanipur, R.: A generalized orthogonality relation via norm derivatives in real normed linear spaces. Aequationes Math. 93(4), 651–667 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  30. Dragomir, S.S.: Continuous linear functionals and norm derivatives in real normed spaces. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 3, 5–12 (1992)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  32. 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 

  33. Frank, M., Moslehian, M.S., Zamani, A.: Orthogonality preserving property for pairs of operators on Hilbert C ∗-modules (2017). arXiv:1711.04724, Nov 10

    Google Scholar 

  34. Giles, J.R.: Classes of semi-inner product spaces. Trans. Am. Math. Soc. 116, 436–446 (1967)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Ionică, I.: On linear operators preserving orthogonality. Ann. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 58, 325–332 (2012)

    Google Scholar 

  37. Jahn, T.: Orthogonality in generalized Minkowski spaces. J. Convex Anal. 26(1), 49–76 (2019)

    MathSciNet  MATH  Google Scholar 

  38. James, R.C.: Orthogonality in normed linear spaces. Duke Math. J. 12(2), 291–302 (1945)

    Article  MathSciNet  MATH  Google Scholar 

  39. James, R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Am. Math. Soc. 61(2), 265–292 (1947)

    Article  MathSciNet  Google Scholar 

  40. James, R.C.: Inner product in normed linear spaces. Bull. Am. Math. Soc. 53, 559–566 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  41. Joichi, J.T.: Normed linear spaces equivalent to inner product spaces. Proc. Am. Math. Soc. 17, 423–426 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  42. Jordan, P., von Neumann, J.: On inner products in linear, metric spaces. Ann. Math. 36(3), 719–723 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  43. Karia, D.J., Parmar, Y.M.: Orthogonality preserving maps and pro-C ∗-modules. J. Anal. 26, 1–10 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  44. Kikianty, E., Sinnamon, G.: Angular equivalence of normed spaces. J. Math. Anal. Appl. 454(2), 942–960 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  46. Koldobsky, A.: Operators preserving orthogonality are isometries. Proc. R. Soc. Edinb. Sect. A. 123(5), 835–837 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kong, L., Cao, H.: Stability of orthogonality preserving mapping and the orthogonality equation. J. Shaanxi Normal Univ. Nat. Sci. Ed. 36(5), 10–14 (2008)

    MathSciNet  MATH  Google Scholar 

  48. Köthe, G.: Topological Vector Spaces I. Springer, Berlin (1969)

    MATH  Google Scholar 

  49. Leung, C.-W., Ng, C.-K., Wong, N.-C.: Linear orthogonality preservers of Hilbert bundles. J. Aust. Math. Soc. 89(2), 245–254 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  50. Leung, C.-W., Ng, C.-K., Wong, N.-C.: Linear orthogonality preservers of Hilbert C ∗-modules over C ∗-algebras with real rank zero. Proc. Am. Math. Soc. 140(9), 3151–3160 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  51. Leung, C.-W., Ng, C.-K., Wong, N.-C.: Linear orthogonality preservers of Hilbert C ∗-modules. J. Oper. Theory 71(2), 571–584 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  52. Li, C.K., Pierce, S.: Linear preserver problems. Am. Math. Mon. 108(7), 591–605 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  54. Martini, H., Wu, S.: On maps preserving isosceles orthogonality in normed linear spaces. Note Mat. 29(1), 55–59 (2009)

    MathSciNet  MATH  Google Scholar 

  55. Miličić, P.M.: Sur le semi-produit scalaire dans quelques espaces vectorial norḿes. Mat. Vesn. 8(55), 181–185 (1971)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  58. Moslehian, M.S.: On the stability of the orthogonal Pexiderized Cauchy equation. J. Math. Anal. Appl. 318, 211–223 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  59. Moslehian, M.S., Zamani, A.: Mappings preserving approximate orthogonality in Hilbert C ∗-modules. Math. Scand. 122, 257–276 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  60. Moslehian, M.S., Zamani, A., Dehghani, M.: Characterizations of smooth spaces by ρ ∗-orthogonality. Houst. J. Math. 43(4), 1187–1208 (2017)

    MathSciNet  MATH  Google Scholar 

  61. Moslehian, M.S., Zamani, A., Wójcik, P.: Approximately angle preserving mappings. Bull. Aust. Math. Soc. 99(3) 485–496 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  62. Nabavi Sales, S.M.S.: On mappings which approximately preserve angles. Aequationes Math. 92, 1079–1090 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  63. Pambuccian, V.: A logical look at characterizations of geometric transformations under mild hypotheses. Indag. Math. 11(3), 453–462 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  64. Papini, P.L.: Uńasservatione sui prodotti semi-scalari negli spasi di Banach. Boll. Un. Mat. Ital. 6, 684–689 (1969)

    Google Scholar 

  65. Precupanu, T.: Duality mapping and Birkhoff orthogonality. An. Stiint, Univ. Al. I. Cuza Iasi. Mat. (S.N.) 59(1), 103–112 (2013)

    Google Scholar 

  66. Rätz, J.: On orthogonally additive mappings. Aequationes Math. 28, 35–49 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  67. Roberts, B.D.: On the geometry of abstract vector spaces. Tôhoku Math. J. 39, 42–59 (1934)

    MATH  Google Scholar 

  68. Sikorska, J.: Orthogonalities and functional equations. Aequationes Math. 89, 215–277 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  69. Stypuła, T., Wójcik, P.: Characterizations of rotundity and smoothness by approximate orthogonalities. Ann. Math. Sil. 30, 193–201 (2016)

    MathSciNet  MATH  Google Scholar 

  70. Tapia, R.A.: A characterization of inner product spaces. Bull. Am. Math. Soc. 79, 530–531 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  71. Tapia, R.A.: A characterization of inner product spaces. Proc. Am. Math. Soc. 41, 569–574 (1973)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  73. Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1960)

    MATH  Google Scholar 

  74. Wang, S.-G., Ip, W.-C.: A matrix version of the Wielandt inequality and its applications to statistics. Linear Algebra Appl. 296, 171–181 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  75. Wójcik, P.: Linear mappings preserving ρ-orthogonality. J. Math. Anal. Appl. 386, 171–176 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  76. Wójcik, P.: Operators preserving and approximately preserving orthogonality and similar relations. Doctorial dissertation (in Polish), Katowice (2013)

    Google Scholar 

  77. Wójcik, P.: On mappings approximately transferring relations in finite-dimensional normed spaces. Linear Algebra Appl. 460, 125–135 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  78. Wójcik, P.: Characterizations of smooth spaces by approximate orthogonalities. Aequationes Math. 89, 1189–1194 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  79. Wójcik, P.: Operators preserving sesquilinear form. Linear Algebra Appl. 469, 531–538 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  80. Wójcik, P.: Linear mappings approximately preserving orthogonality in real normed spaces. Banach J. Math. Anal. 9(2), 134–141 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  81. Wójcik, P.: On certain basis connected with operator and its applications. J. Math. Anal. Appl. 423(2), 1320–1329 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  82. Wójcik, P.: Operators reversing orthogonality and characterization of inner product spaces. Khayyam J. Math. 3(1), 23–25 (2017)

    MathSciNet  MATH  Google Scholar 

  83. Wójcik, P.: Mappings preserving B-orthogonality. Indag. Math. 30, 197–200 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  84. Wu, S., He, C., Yang, G.: Orthogonalities, linear operators, and characterization of inner product spaces. Aequationes Math. 91(5), 969–978 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  85. Zamani, A.: Approximately bisectrix-orthogonality preserving mappings. Comment. Math. 54(2), 167–176 (2014)

    MathSciNet  MATH  Google Scholar 

  86. Zamani, A., Moslehian, M.S.: Approximate Roberts orthogonality. Aequationes Math. 89, 529–541 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  87. Zamani, A., Moslehian, M.S.: Approximate Roberts orthogonality sets and (δ, ε)-(a, b)-isosceles-orthogonality preserving mappings. Aequationes Math. 90, 647–659 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  88. Zamani, A., Moslehian, M.S.: An extension of orthogonality relations based on norm derivatives. Q. J. Math. 70(2), 379–393 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  89. Zamani, A., Moslehian, M.S., Frank, M.: Angle preserving mappings. Z. Anal. Anwend. 34, 485–500 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  90. Zhang, Y.: An identity for (δ, ε)-approximately orthogonality preserving mappings. Linear Algebra Appl. 554, 358–370 (2018)

    Article  MathSciNet  Google Scholar 

  91. Zhang, Y., Chen, Y., Hadwin, D., Kong, L.: AOP mappings and the distance to the scalar multiples of isometries. J. Math. Anal. Appl. 431(2), 1275–1284 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgement

The authors would like to thank the referee for her/his valuable suggestions and comments.

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Authors and Affiliations

  1. Department of Mathematics, Farhangian University, Tehran, Iran

    Ali Zamani

  2. Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran

    Mahdi Dehghani

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  1. Ali Zamani
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  2. Mahdi Dehghani
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Editors and Affiliations

  1. Department of Applied Mathematics, AGH University of Science and Technology, Krakow, Poland

    Janusz Brzdęk

  2. Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

    Dorian Popa

  3. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Zamani, A., Dehghani, M. (2019). On Exact and Approximate Orthogonalities Based on Norm Derivatives. In: Brzdęk, J., Popa, D., Rassias, T. (eds) Ulam Type Stability . Springer, Cham. https://doi.org/10.1007/978-3-030-28972-0_21

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