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

References

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Algebraic Structures and Their Applications
Volume 12, Issue 2
May 2025
Pages 155-167

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