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Jordan Derivations and Antiderivations of Generalized Matrix Algebras

Let $\mathcal{G}=[A & M N & B]$ be a generalized matrix algebra defined by the Morita context $(A, B,_AM_B,_BN_A, Φ_{MN}, Ψ_{NM})$. In this article we mainly study the question of whether there exist proper Jordan derivations for the generalized matrix algebra $\mathcal{G}$. It is shown that if one of the bilinear pairings $Φ_{MN}$ and $Ψ_{NM}$ is nondegenerate, then every antiderivation of $\mathcal{G}$ is zero. Furthermore, if the bilinear pairings $Φ_{MN}$ and $Ψ_{NM}$ are both zero, then every Jordan derivation of $\mathcal{G}$ is the sum of a derivation and an antiderivation. Several constructive examples and counterexamples are presented.