normal operators question

[paul2834]

Let it be T,S normal operators (F can be R or C) so that TS=ST.
show that TS is normal operator.

(V is finite dimensional inner product spaces).


My problem here is that the operators aren't necessarily bounded, so I don't know how to prove that TS*=S*T (and I think that's what I need here).
How do I prove this?

 

[Limax]

I think you just have prove this lemma: If \(\displaystyle A\) is an \(\displaystyle n\times n\) complex matrix, then \(\displaystyle A\) is normal if and only if there exists a complex polynomial \(\displaystyle P(X)\) such that \(\displaystyle P(A)\) is the Hermitian adjoint \(\displaystyle A^*\). The result follows from the fact that if \(\displaystyle T\) communtes with \(\displaystyle S\) then it commutes with every power and every scalar multiple of \(\displaystyle S\).