Authors

Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad 91735, Iran

Abstract

The aim of this paper is to show that under some mild conditions a functional equation of multiplicative $(\alpha,\beta )$-derivation is superstable on standard operator algebras. Furthermore, we prove that this generalized derivation can be a continuous and an inner $(\alpha,\beta)$-derivation.

Keywords

Main Subjects

[1] B. Aupetit, A primer on spectral theory, Springer- Verlag, New York, 1990.
 
[2] J. A. Baker, The stability of the cosine equation, Proc. Amer. Math. Soc. 80 (1980), 411-416.
 
[3] A. Bodaghi, Cubic derivations on Banach algebras, Acta Mathematica Vietnamica, 38, No.2 (2013),517-528.
 
[4] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke math, J. 16 (1949), 385-397.
 
[5] A. Hosseini, M. Hassani, A. Niknam, Generalized -derivation on Banach algebras, Bulletin of the Iranian Mathematical Society, 37 No. 4 (2011), 81-94.
 
[6] A. Hosseini, M. Hassani, A. Niknam, S. Hejazian, Some results on -derivations, Ann. Funct. Anal, No. 2 (2011), 75-84.
 
[7] Ch. Hou, W. Zhang, Q. Meng, A note on ( ; )-derivations, Linear Algebra and its Applications, 432 (2010), 2600-2607.
 
[8] Ch. Hou, Q. Meng, Continuity of ( ; )-derivation of operator algebras, J. Korean Math. Soc. 48(2011),823-835.
 
[9] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat'e. A cad. Sci. U. S. A. 27 (1941), 222-224.
 
[10] W. S. Martindale, when are multiplicative mappings additive, proceeding of the American Mathematical Soc. 21 No. 3 (1969), 695-698.
 
[11] L. Molanar, On isomorphisms on standard operators algebras,ar Xiv Preprint Math,2000.
 
[12] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.
 
[13] P. Semrl, Approximate homomorphisms, Proc 34th Internat. Symp. On Functional Equations,Wisa Jaronik, Poland, June 10-19 (1996).
 
[14] P. Semrl, The functional equation of multiplicative derivation is superstable on standard operator algebras, Integr Equat oper th, Vol. 18 (1994).
 
[15] S. M. Ulam, A Collection of Mathematical Problems, Inter Science, New York, 1960.
 
[16] S. Y. Yang, A. Bodaghi, K. A. M. Atan, Approximate cubic -derivations on Banach -algebras. Abstract and Applied Analysis, Volume 2012, Article ID 684179, 12 pages, doi:10.1155/2012/684179