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

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=۰}^\infty on \mathcal{M} in relation to pexider Hilbert C ^* -module \{\alpha_n, \beta_n, \delta_n\} -derivations \{\varphi_n \}_{n=۱}^\infty on \mathcal{M} . We demonstrate that for every pexider Hilbert C ^* -module higher \{A_n,B_n,D_n\} -derivation \{\Phi_n\}_{n=۰}^\infty on \mathcal{M} , there exists a unique sequence of pexider Hilbert C ^* -module \{\alpha_n, \beta_n, \delta_n\} -derivations \{\varphi_n \}_{n=۱}^\infty on \mathcal{M} such that\left\{\begin{array}{lr}\varphi_n=\sum_{k=۱}^n\Big(\sum_{\sum_{j=۱}^k r_j=n}(-۱)^{k-۱}~r_۱\Phi_{r_۱}\Phi_{r_۲}\ldots \Phi_{r_k}\Big),\\\alpha_n=\sum_{k=۱}^n\Big(\sum_{\sum_{j=۱}^k r_j=n}(-۱)^{k-۱}~r_۱A_{r_۱}A_{r_۲}\ldots A_{r_k}\Big),\\\beta_n=\sum_{k=۱}^n\Big(\sum_{\sum_{j=۱}^k r_j=n}(-۱)^{k-۱}~r_۱B_{r_۱}B_{r_۲}\ldots B_{r_k}\Big),\\\delta_n=\sum_{k=۱}^n\Big(\sum_{\sum_{j=۱}^k r_j=n}(-۱)^{k-۱}~r_۱D_{r_۱}D_{r_۲}\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=۱}^k r_j=n.

کلیدواژه ها:

derivation ، Higher derivation ، Hilbert C ^* -module