I am trying to prove the following homework problem: Let $T \in B(H)$ (so $T$ is a bounded operator on a Hilbert space $H$), and let $T = U|T|$ be the polar decomposition of $T$. Prove that if $T$ is a normal operator, then $U|T| = |T|U$ and $U$ is normal.
The definition of $|T|$ is the unique positive square root of $T^*T$. An operator $T$ is normal if $TT^* = T^*T$.
I've been messing around with calculations based on definitions (in my class, we are not assuming that $U$ is unitary, although we call it the polar part of $T$). However, I'm stuck. I found the following link: Polar decomposition normal operator and looked up the theorem of Rudin that is mentioned, but it made use of some things we haven't covered in my class. Can anyone help? I feel like it is staring me in the face and I just can't see it.