The product of derivations on Banach algebras (Q2721587)

An additive mapping \(d\) on a ring \({\mathbb R}\) into itself is said to be a Jordan derivation if \(d(x^2)=d(x)x+xd(x)\) for all \(x\in {\mathbb R}\). The authors prove that : If \(d\) and \(g\) are derivations on a Banach algebra \({\mathbb A}\) over a complex field \({\mathbb C}\) such that \(\alpha d^3+dg\) is a Jordan derivation for some \(\alpha \in {\mathbb C}\), then the product \(dg\) maps \({\mathbb A}\) into the Jacobson readical of \({\mathbb A}\).