Automatic continuity of new generalized derivations

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two algebras and let $n$ be a positive integer. A linear mapping $D:\mathcal{A} \rightarrow \mathcal{B}$ is called a \emph{strongly generalized derivation of order $n$} if there exist families of linear mappings $\{E_k:\mathcal{A} \rightarrow \mathcal{B}\}_{k = 1}^{n}$, $\{F_k:\mathcal{A} \rightarrow \mathcal{B}\}_{k = 1}^{n}$, $\{G_k:\mathcal{A} \rightarrow \mathcal{B}\}_{k = 1}^{n}$ and $\{H_k:\mathcal{A} \rightarrow \mathcal{B}\}_{k = 1}^{n}$ which satisfy $D(ab) = \sum_{k = 1}^{n}\left[E_k(a) F_k(b) + G_k(a)H_k(b)\right]$ for all $a, b \in \mathcal{A}$. The purpose of this article is to study the automatic continuity of such derivations on Banach algebras and $C^{\ast}$-algebras.