Source: Banach J. Math. Anal. Volume 6, Number 2
(2012), 19-37.
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
$n+l\geq1$.
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