$A,B$ are bounded normal operators, there exists invertible operator $T$ such that $TA=BT$. Prove there exists unitary operator $U$ such that $UA=BU$.(Hint: prove $z\mapsto e^{-izB^*}Te^{izA^*}$ is bounded)
I think it is related with polar decomposition and "any unitary operator $U$ can be written as $U=e^{i H}$, where $H$ is bounded positive definite operator". But I do not know how to prove that hint and use it. I appreciate your help!