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- W2792051231 abstract "Let $mathcal R$ be a ring, $mathcal{M}$ be a $mathcal R$-bimodule and $m,n$ be two fixed nonnegative integers with $m+nneq0$. An additive mapping $delta$ from $mathcal R$ into $mathcal{M}$ is called an emph{$(m,n)$-Jordan derivation} if $(m+n)delta(A^{2})=2mAdelta(A)+2ndelta(A)A$ for every $A$ in $mathcal R$. In this paper, we prove that every $(m,n)$-Jordan derivation from a $C^{*}$-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 $(m+n)delta(AB+BA)=2mdelta(A)B+2mdelta(B)A+2nAdelta(B)+2nBdelta(A)$ 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." @default.
- W2792051231 created "2018-03-29" @default.
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- W2792051231 date "2018-03-06" @default.
- W2792051231 modified "2023-09-27" @default.
- W2792051231 title "Characterizations of $(m,n)$-Jordan derivations on some algebras" @default.
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