Let $\phi$ be a normal operator on the dimension $n$ Euclidean space, $\psi$ is a linear operator. If $\phi \psi=\psi \phi$, then $\phi^* \psi = \psi \phi^*$, where $\phi^*$ is the adjoint of $\phi$.
I only know I should use then normal condition $\phi \phi^*=\phi^* \phi$, but don't know how. Thanks for any help.
If we are to get a quick proof, it is essential that we exploit the finite dimension of the space. The more general statement is known as Fugelde's theorem.
One proof (using the spectral theorem) is presented here. In short: because of the spectral theorem, we may deduce that there exists a polynomial $p$ such that $\phi^* = p(\phi)$. The rest is easy.
