Browse > Article
http://dx.doi.org/10.4134/BKMS.2008.45.1.157

APPROXIMATELY C*-INNER PRODUCT PRESERVING MAPPINGS  

Chmielinski, Jacek (INSTITUTE OF MATHEMATICS PEDAGOGICAL UNIVERSITY OF CRACOW)
Moslehian, Mohammad Sal (DEPARTMENT OF MATHEMATICS FERDOWSI UNIVERSITY OF MASHHAD)
Publication Information
Bulletin of the Korean Mathematical Society / v.45, no.1, 2008 , pp. 157-167 More about this Journal
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;
Citations & Related Records

Times Cited By Web Of Science : 3  (Related Records In Web of Science)
Times Cited By SCOPUS : 2
연도 인용수 순위
1 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   DOI   ScienceOn
2 Th. M. Rassias, A new generalization of a theorem of Jung for the orthogonality equation, Appl. Anal. 81 (2002), no. 1, 163-177   DOI   ScienceOn
3 Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300
4 G. Isac and Th. M. Rassias, On the Hyers-Ulam stability of $\psi$-additive mappings, J. Approx. Theory 72 (1993), no. 2, 131-137   DOI   ScienceOn
5 M. Amyari, Stability of C*-inner products, J. Math. Anal. Appl. 322 (2006), 214-218   DOI   ScienceOn
6 D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222-224
7 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
8 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
9 D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), no. 2-3, 125-153   DOI
10 S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Hadronic Press, Inc., Palm Harbor, FL, 2001
11 M. A. Rieffel, Induced representations of C*-algebras, Advances in Math. 13 (1974), 176-257   DOI
12 S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, Inc., New York 1964
13 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   DOI   ScienceOn
14 T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66   DOI
15 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
16 R. Badora and J. Chmieli'nski, Decomposition of mappings approximately inner product preserving, Nonlinear Anal. 62 (2005), no. 6, 1015-1023   DOI   ScienceOn
17 S. Czerwik, Functional equations and inequalities in several variables, World Scientific Publishing Co., Inc., River Edge, NJ, 2002
18 G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), no. 1-2, 143-190   DOI
19 I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839-858   DOI   ScienceOn
20 E. C. Lance, Hilbert C*-Modules, LMS Lecture Note Series 210, Cambridge Univ. Press, 1995
21 V. M. Manuilov and E. V. Troitsky, Hilbert C*-modules, Translations of Mathematical Monographs, 226. American Mathematical Society, Providence, RI, 2005
22 M. S. Moslehian, Asymptotic behavior of the extended Jensen equation, Studia Sci. Math. Hungar (to appear)
23 J. G. Murphy, C*-algebras and operator theory, Academic Press, Inc., Boston, MA, 1990
24 W. L. Paschke, Inner product modules over B*-algebras, Trans. Amer. Math. Soc. 182 (1973), 443-468   DOI   ScienceOn
25 Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), no. 1, 23-130   DOI
26 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