## Annals of Functional Analysis

### A fixed point approach to the stability of $\varphi$-morphisms on Hilbert $C^*$-modules

#### Abstract

Let $E$, $F$ be two Hilbert $C^*$-modules over $C^*$-algebras $A$ and $B$ respectively. In this paper, by the alternative fixed point theorem, we give the Hyers-Ulam-Rassias stability of the equation $$\ip{U(x), U(y)}=\varphi( \ip{x,y})\qquad(x, y\in E),$$ where $U : E\to F$ is a mapping and $\varphi : A\to B$ is an additive map

#### Article information

Source
Ann. Funct. Anal., Volume 1, Number 1 (2010), 44- 50.

Dates
First available in Project Euclid: 12 May 2014

https://projecteuclid.org/euclid.afa/1399900992

Digital Object Identifier
doi:10.15352/afa/1399900992

Mathematical Reviews number (MathSciNet)
MR2755458

Zentralblatt MATH identifier
1221.39034

Subjects
Secondary: 46L08: $C^*$-modules

#### Citation

Abbaspour Tabadkan, Gh.; Ramezanpour, M. A fixed point approach to the stability of $\varphi$-morphisms on Hilbert $C^*$-modules. Ann. Funct. Anal. 1 (2010), no. 1, 44-- 50. doi:10.15352/afa/1399900992. https://projecteuclid.org/euclid.afa/1399900992

#### References

• Gh. Abbaspour Tabadkan, M.S. Moslehian and A. Niknam, Dynamical systems on Hilbert $C^*$-modules, Bull. Iranian Math. Soc. 31 (2005), 25–35.
• Gh. Abbaspour Tabadkan and M. Skeide,Genarators of dynamical systems on Hilbert modules, Commun. on Stoch. Anal. 1 (2007), no. 2, 193–207.
• D. Bakič and B. Guljaš, On class of module maps of Hilbert $C^*$-modules, Math. Commun. 7 (2002), 177–192.
• R. Badora and J. Chmieliński, Decomposition of mappings approximately inner product preserving, Nonlinear Anal. 62 (2005), 1015–1023.
• J. Chmieliński, D. Ilišević, M.S. Moslehian and Gh. Sadeghi, Perturbation of the Wigner equation in inner product C*-modules, J. Math. Phys. 49 (2008), no. 3, 033519, 8 pp.
• J. Chmieliński and M.S. Moslehian, Approximately $C^*$-inner product preserving mappings, Bull. Korean Math. Soc. 45 (2008), no. 1, 157–167.
• L. Cádariu and V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math. 4 (2003), no.1, Art. 4.
• J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309.
• Z. Gajda, On stability of additive mappings, Inter. J. Math. Sci. 14 (1991), 431–434.
• D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Soc. U.S.A. 27 (1941), 222–224.
• S.-M. Jung, A fixed point approach to the stability of an equation of the square spiral, Banach J. Math. Anal. 1 (2007), no. 2, 148–153.
• E.C. Lance, Hilbert $C^*$-Modules, LMS Lecture note Series 210, Cambridge Univ. Press, 1995.
• Y.-H. Lee, On the stability of the monomial functional equation, Bull. Korean Math. Soc. 45 (2008), no. 2, 397–403.
• A.K. Mirmostafaee, Approximate isometris in Hilbert $C^*$-modules, Math. Commun. 14 (2009), no.1, 167–176.
• C. Park and Th.M. Rassias, Fixed points and the stability of the Cauchy functional equation, Aust. J. Math. Anal. Appl. 6 (2009), no. 1, Art. 14.
• V. Radu, The fixed point alternative and stability of functional equations, Fixed point theorey 4 (2003), 190–199.
• Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.
• S.M. Ulam, A collection of the mathematical problems, Interscince Publ, New Yourk, 1960.