Generalized Jordan derivations of unital algebras

Abstract

Let $A$ be a unital algebra over a field $F$ with $\mathrm{char}(F)\ne 2$. In this paper we introduce a new concept of a generalized Jordan derivation, covering Jordan centralizers and Jordan derivations, as follows: a linear map $f:A\to A$ is a generalized Jordan derivation if there exist linear maps $g;h:A\to A$ such that $f\left(x\right)\circ y+x\circ g\left(y\right)=h\left(x\circ y\right)$ for all $x,y\in A$ (here $x\circ y=xy+yx$). Our aim is to give the form of map $f$ in terms of the so called quasi Jordan centralizers and quasi Jordan derivations. In addition, a characterization of such maps is presented.