Characterizations of $(m,n)$-Jordan derivations on some algebras

Abstract

Let $\mathcal{R}$ be a ring, $\mathcal{M}$ be a $\mathcal{R}$-bimodule and $m,n$ be two fixed nonnegative integers with $m+n\ne 0$. An additive mapping $\delta$ from $\mathcal{R}$ into $\mathcal{M}$ is called an \emph{$(m,n)$-Jordan derivation} if $(m+n)\delta (A^2)=2mA\delta (A)+2n\delta (A)A$ for every $A$ in $\mathcal{R}$. In this paper, we prove that every $(m,n)$-Jordan derivation from a $C^{\ast }$-algebra into its Banach bimodule is zero. An additive mapping $\delta$ from $\mathcal{R}$ into $\mathcal{M}$ is called a $(m,n)$-Jordan derivable mapping at $W$ in $\mathcal{R}$ if $(m+n)\delta (AB+BA)=2m\delta (A)B+2m\delta (B)A+2nA\delta (B)+2nB\delta (A)$ for each $A$ and $B$ in $\mathcal{R}$ with $AB=BA=W$. We prove that if $\mathcal{M}$ is a unital $\mathcal{A}$-bimodule with a left (right) separating set generated algebraically by all idempotents in $\mathcal{A}$, then every $(m,n)$-Jordan derivable mapping at zero from $\mathcal{A}$ into $\mathcal{M}$ is identical with zero. We also show that if $\mathcal{A}$ and $\mathcal{B}$ are two unital algebras, $\mathcal{M}$ is a faithful unital $(\mathcal{A},\mathcal{B})$-bimodule and $\mathcal{U}=\left[\begin{array}{cc}\mathcal{A}& \mathcal{M}\\ \mathcal{N}& \mathcal{B}\end{array}\right]$ is a generalized matrix algebra, then every $(m,n)$-Jordan derivable mapping at zero from $\mathcal{U}$ into itself is equal to zero.