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APPROXIMATELY C*-INNER PRODUCT PRESERVING MAPPINGS

  • Chmielinski, Jacek ;
  • Moslehian, Mohammad Sal
  • Published : 2008.02.29

Abstract

A mapping f : $M{\rightarrow}N$ between Hilbert $C^*$-modules approximately preserves the inner product if $$\parallel<f(x),\;f(y)>-<x,y>\parallel\leq\varphi(x,y)$$ for an appropriate control function $\varphi(x,y)$ and all x, y $\in$ M. In this paper, we extend some results concerning the stability of the orthogonality equation to the framework of Hilbert $C^*$-modules on more general restricted domains. In particular, we investigate some asymptotic behavior and the Hyers-Ulam-Rassias stability of the orthogonality equation.

Keywords

Hilbert $C^*$-module;Hyers-Ulam-Rassias stability;superstability;orthogonality equation;asymptotic behavior

References

  1. M. Amyari, Stability of C*-inner products, J. Math. Anal. Appl. 322 (2006), 214-218 https://doi.org/10.1016/j.jmaa.2005.09.014
  2. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66 https://doi.org/10.2969/jmsj/00210064
  3. C. Baak, H. Chu, and M. S. Moslehian, On the Cauchy-Rassias inequality and linear n-inner product preserving mappings, Math. Inequal. Appl. 9 (2006), no. 3, 453-464
  4. R. Badora and J. Chmieli'nski, Decomposition of mappings approximately inner product preserving, Nonlinear Anal. 62 (2005), no. 6, 1015-1023 https://doi.org/10.1016/j.na.2005.04.009
  5. J. Chmielinski, On a singular case in the Hyers-Ulam-Rassias stability of the Wigner equation, J. Math. Anal. Appl. 289 (2004), no. 2, 571-583 https://doi.org/10.1016/j.jmaa.2003.08.042
  6. S. Czerwik, Functional equations and inequalities in several variables, World Scientific Publishing Co., Inc., River Edge, NJ, 2002
  7. G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), no. 1-2, 143-190 https://doi.org/10.1007/BF01831117
  8. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222-224
  9. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of functional equations in several variables, Progress in Nonlinear Differential Equations and their Applications, 34. Birkhauser Boston, Inc., Boston, MA, 1998
  10. D. H. Hyers, G. Isac, and Th. M. Rassias, On the asymptoticity aspect of Hyers-Ulam stability of mappings, Proc. Amer. Math. Soc. 126 (1998), no. 2, 425-430
  11. D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), no. 2-3, 125-153 https://doi.org/10.1007/BF01830975
  12. G. Isac and Th. M. Rassias, On the Hyers-Ulam stability of $\psi$-additive mappings, J. Approx. Theory 72 (1993), no. 2, 131-137 https://doi.org/10.1006/jath.1993.1010
  13. S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Hadronic Press, Inc., Palm Harbor, FL, 2001
  14. I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839-858 https://doi.org/10.2307/2372552
  15. E. C. Lance, Hilbert C*-Modules, LMS Lecture Note Series 210, Cambridge Univ. Press, 1995
  16. V. M. Manuilov and E. V. Troitsky, Hilbert C*-modules, Translations of Mathematical Monographs, 226. American Mathematical Society, Providence, RI, 2005
  17. M. S. Moslehian, Asymptotic behavior of the extended Jensen equation, Studia Sci. Math. Hungar (to appear)
  18. J. G. Murphy, C*-algebras and operator theory, Academic Press, Inc., Boston, MA, 1990
  19. W. L. Paschke, Inner product modules over B*-algebras, Trans. Amer. Math. Soc. 182 (1973), 443-468 https://doi.org/10.2307/1996542
  20. Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), no. 1, 23-130 https://doi.org/10.1023/A:1006499223572
  21. Th. M. Rassias, Stability of the Generalized Orthogonality Functional Equation, Inner product spaces and applications, 219-240, Pitman Res. Notes Math. Ser., 376, Longman, Harlow, 1997
  22. M. A. Rieffel, Induced representations of C*-algebras, Advances in Math. 13 (1974), 176-257 https://doi.org/10.1016/0001-8708(74)90068-1
  23. S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, Inc., New York 1964
  24. J. Chmielinski and S.-M. Jung, The stability of the Wigner equation on a restricted domain, J. Math. Anal. Appl. 254 (2001), no. 1, 309-320 https://doi.org/10.1006/jmaa.2000.7279
  25. Th. M. Rassias, A new generalization of a theorem of Jung for the orthogonality equation, Appl. Anal. 81 (2002), no. 1, 163-177 https://doi.org/10.1080/0003681021000021132
  26. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300

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