I have the following question:
Let $V$ be a vector space over a field $F$, and let $A$ be an endomorphism $V\rightarrow V$. Prove there is at most one pair of linear maps $D$ and $N$ such that $D+N=A$, $D$ is diagonalizable, $N$ is nilpotent, and every map that commutes with $D+N$ commutes with $D$ and $N$.
If there are two pairs, $D_1$, $N_1$ and $D_2$, $N_2$ that satisfy these properties, then $D_1-D_2 = N_2-N_1$. Since the sum of commuting diagonalizable and nilpotent matrices are diagonalizable and nilpotent respectively, the theorem is proved if I can show that $D_1, D_2$ and $N_1$, $N_2$ commute. But this is where I'm stuck.
Any help would be greatly appreciated. Thank you!