How to prove that this linear operator is nilpotent?

Question

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.

Answer 1

First, show that for $i=1,2,\ldots$, there exist some positive integers $c_{i0},\ldots,c_{ii}$ such that $L^i(X) = \sum_{j=0}^i c_{ij} A^j X A^{i-j}$.

Now, if $A^k=0$, show that $L^{2k-1}=0$.

Conversely, if $L^i=0$ for some natural number $i$, by considering $L^i(A)$, show that $A^{i+1}=0$.

Answer 2

• Look at L(A)
• Look at LL(X), LLL(X), and maybe LLLL(X).