Normal operator proof - is $TT = T^*T^*$ says T is normal operator?

Question

Is it true what i wrote in title?
if yes, how can I prove it?
I have a question and I reached that point ( I need to show if $T^2=\frac 1 2 (T+T^*)$ so conclude T is normal operator).
I have the answer to that question from my exerciser at study, but I dont want to look at solution yet ( so posted here to ask if it says its normal ).

Answer 1

As a general comment and not to leave an unanswered post, every time you can write $T^*$ in terms of $T$, be it a polynomial or even a continuous/Borel function through functional calculus, the operator will be normal. Simply because $T$ commutes with $f(T)$, and so if $T^*=f(T)$ then $T$ is normal.