Jordan $\ast$-derivations on Standard Operator Algebras

Abstract

Let $\mathcal{H}$ be a real or complex Hilbert space with $dim(\mathcal{H})>1$ and $A(\mathcal{H})$ be a standard operator algebra on $\mathcal{H}$. If $D:A(\mathcal{H})\to L(\mathcal{H})$ is a linear mapping satisfying $D(A^{n+1})=\sum\limits_{i=0}^{n}A^{i}D(A)(A^\ast )^{n-i}$ for all $A\in A(\mathcal{H})$, then $D$ is a Jordan $\ast$-derivation on $A(\mathcal{H})$. Later, we discuss some algebraic identities on semiprime rings.