On generalized ($m, n, l$)-Jordan centralizers of some algebras

Abstract

Let $\mathcal{A}$ be a unital algebra over a number field $\mathbb{K}$. A linear mapping $\delta$ from $\mathcal{A}$ into itself is called a generalized ($m, n, l$)-Jordan centralizer if it satisfies $(m+n+l)\delta(A^2)-m\delta(A)A-nA\delta(A)-lA\delta(I)A\in \mathbb{K}I$ for every $A\in \mathcal{A}$, where $m\geq0, n\geq0, l\geq0$ are fixed integers with $m+n+l\neq 0$. In this paper, we study generalized ($m, n, l$)-Jordan centralizers on generalized matrix algebras and some reflexive algebras alg$\mathcal{L}$, where $\mathcal{L}$ is a 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 generalized ($m, n, l$)-Jordan centralizer of these algebras is a centralizer when $m+l\geq1$ and $n+l\geq1$.