Prove that $N$ normal and $TN=NT$ implies $T^N=NT^$

Question

Let $T,N:V\to V$ such that $N$ is a normal operator and $TN=NT$. Prove that $T^*N=NT^*$ and $N^*T=TN^*$.

I'd be glad for help because I don't even have an idea how to approach this

Look up "Fuglede's theorem" in any book on advanced linear algebra. — user1551, Jul 22 '15 at 14:20

score 3 · Answer 1 · answered Jul 22 '15 at 13:53

3

Hint: As a consequence of the spectral theorem, there exists a polynomial $p(t)$ such that $N^* = p(N)$.

answered Jul 22 '15 at 13:53

Ben Grossmann

Prove that $N$ normal and $TN=NT$ implies $T^*N=NT^*$