Communications of the
Korean Mathematical Society CKMS

Let $A$ be a Banach algebra with $\mbox{rad}(A)$. We show that if there exists a continuous linear Jordan derivation $D$ on $A$, then $$[D(x),x]D(x)^2\in \mbox{rad}(A)$$ if and only if $D(x)[D(x),x]D(x)\in \mbox{rad}(A)$ for all $x\in A$.

Keywords: Jordan derivation, derivation, semiprime ring, Banach algebra, the (Jacobson) radical

MSC numbers: 16N60, 16W25, 17B40