How to prove that this linear operator is nilpotent? Let $A\in M_n(\mathbb{C})$ be an arbitrary matrix , $\mathbb{C}$ is complex fields, and $L$ a mapping that is defined by
$L:M_n(\mathbb{C})\to M_n(\mathbb{C})$, $L(X):=AX+XA$. How can we show that $A$ is nilpotent iff $L$ be nilpotent?
Thanks for any hint.

*

*Look at L(A)

*Look at LL(X), LLL(X), and maybe LLLL(X).

