On pexider type of Hilbert C$^* $-module higher derivations

Document Type : Research Paper

Author

Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran.

Abstract

In this paper, we introduce the concept of pexider Hilbert C$ ^* $-module higher $ \{A_n,B_n,D_n\} $-derivations. Specifically, we focus on a Hilbert C$ ^* $-module $ \mathcal{M} $ and provide a comprehensive characterization of these pexider Hilbert C$ ^* $-module higher $ \{A_n,B_n,D_n\} $-derivations $ \{\Phi_n\}_{n=0}^\infty $ on $ \mathcal{M} $ in relation to pexider Hilbert C$ ^* $-module $ \{\alpha_n, \beta_n, \delta_n\} $-derivations $ \{\varphi_n \}_{n=1}^\infty$ on $ \mathcal{M} $. We demonstrate that for every pexider Hilbert C$ ^* $-module higher $ \{A_n,B_n,D_n\} $-derivation $ \{\Phi_n\}_{n=0}^\infty $ on $ \mathcal{M} $, there exists a unique sequence of pexider Hilbert C$ ^* $-module $ \{\alpha_n, \beta_n, \delta_n\} $-derivations $ \{\varphi_n \}_{n=1}^\infty$ on $ \mathcal{M} $ such that
$$\left\{
\begin{array}{lr}
\varphi_n=\sum_{k=1}^n\Big(\sum_{\sum_{j=1}^k r_j=n}(-1)^{k-1}~r_1\Phi_{r_1}\Phi_{r_2}\ldots \Phi_{r_k}\Big),\\
\alpha_n=\sum_{k=1}^n\Big(\sum_{\sum_{j=1}^k r_j=n}(-1)^{k-1}~r_1A_{r_1}A_{r_2}\ldots A_{r_k}\Big),\\
\beta_n=\sum_{k=1}^n\Big(\sum_{\sum_{j=1}^k r_j=n}(-1)^{k-1}~r_1B_{r_1}B_{r_2}\ldots B_{r_k}\Big),\\
\delta_n=\sum_{k=1}^n\Big(\sum_{\sum_{j=1}^k r_j=n}(-1)^{k-1}~r_1D_{r_1}D_{r_2}\ldots D_{r_k}\Big),
\end{array}\right.$$
for all positive integers $ n$, where the inner summation is taken over all positive integers $r_j$ with $\sum_{j=1}^k r_j=n$.

Keywords


[1] H. M. Alnoghashi, F. A. Al-Qarni and N. U. Rehman, Results on generalized derivations in prime rings, Alg. Struc. Appl., 10 No. 2 (2023) 87-98.
[2] P. E. Bland, Higher derivations on rings and modules, Int. J. Math. Math. Sci., 15 (2005) 2373-2387.
[3] W. Cortes and C. Haetinger, On Jordan generalized higher derivations in rings, Turk. J. Math., 29 (2005) 1-10.
[4] S. Kh. Ekrami, Approximate orthogonally higher ring derivations, Control Optim. Appl. Math., 7 No. 1 (2022) 93-106.
[5] S. Kh. Ekrami, Characterization of Hilbert C-module higher derivations, Georgian Math. J., 31 No. 3 (2024) 397-403.
[6] S. Kh. Ekrami, Jordan higher derivations, a new approach, J. Algebr. Syst., 10 No. 1 (2022) 167-177.
[7] S. Kh. Ekrami, Jordan higher derivations on prime Hilbert C-modules, Kragujev. J. Math., 50 No. 5 (2026) 817-826.
[8] C. Haetinger, Higher derivations on Lie ideals, Tendencias em Matematica Aplicada e Computacional, 3 (2002) 141-145.
[9] A. R. Janfada, H. Saidi and M. Mirzavaziri, Characterization of Lie higher derivations on C-algebras, Bull. Iranian Math. Soc., 41 No. 4 (2015) 901-906.
[10] N. P. Jewell, Continuity of module and higher derivations, Pac. J. Math., 68 (1977) 91-98.
[11] I. Kaplansky, Modules Over Operator Algebras, Amer. J. Math., 75 (1953) 839-858.
[12] M. Mirzavaziri, Characterization of higher derivations on algebras, Comm. Alg., 38 (2010) 981-987.
[13] M. Mirzavaziri, Prime higher derivations on algebras, Bull. Iranian Math. Soc., 36 No. 1 (2010) 201-210.
[14] A. Nowicki, Inner derivations of higher orders, Tsukuba J. Math., 8 (1984) 219-225.
[15] N. U. Rehman, Sh. Huang and M. A. Raza, A note on derivations in rings and Banach algebras, Alg. Struc. Appl., 6 No. 1 (2019) 115-125.
[16] A. Roy and R. Sridharan, Higher derivations and central simple algebras, Nagoya Math. J., 32 (1968) 21-30.
[17] H. Saidi, A. R. Janfada and M. Mirzavaziri, Kinds of derivations on Hilbert C-modules and their operator algebras, Miskolc Math. Notes, 16 No. 1 (2015) 453-461.
[18] S. Xu and Z. Xiao, Jordan higher derivation revisited, Gulf J. Math., 2 No. 1 (2014) 11-21.