On (m,n,l)-Jordan Centralizers of Some Algebras

Abstract

Let $\mathcal{A}$ be a unital algebra over the complex field $\mathbb{C}$. A linear mapping $\delta$ from $\mathcal{A}$ into itself is called a weak (\textit{m,n,l})-Jordan centralizer if $(m+n+l)\delta (A^2)-m\delta (A)A-nA\delta (A)-lA\delta (I)A\in \mathbb{C}I$ for every $A\in \mathcal{A}$, where $m\ge 0,n\ge 0,l\ge 0$ are fixed integers with $m+n+l\ne 0$. In this paper, we study weak (\textit{m,n,l})-Jordan centralizer on generalized matrix algebras and some reflexive algebras alg$\mathcal{L}$, where $\mathcal{L}$ is CSL or satisfies $\vee \{L:L\in \mathcal{J}(\mathcal{L})\}=X$ or $\wedge \{L_{-}:L\in \mathcal{J}(\mathcal{L})\}=(0)$, and prove that each weak (\textit{m,n,l})-Jordan centralizer of these algebras is a centralizer when $m+l\ge 1$ and $n+l\ge 1$.