Let $V$ be finite space and $T$ and $S$ normal operator supose $TS=ST$, prove that $S+T$ and $ST$ are normal.
Okay, my ideia was to show that $(S+T)^{*}(S+T)=(S+T)(S+T)^*$ so
$(S+T)^*(S+T)=S*S+S*T+T*S+T*T $
$(S+T)(S+T)^*=S^*S+S^*T+T^*S+T^*T $
my doubt is why can i assume that if $T$ comute with $S$ and $T$ is normal then $T$ comute with $S^*$. Anyone could prove this to me?
You can use one of the approaches given here. Ultimately, you'll need something like either Schur triangularization or the spectral theorem for normal operators.
The fastest approach that builds on top of your work so far is as follows: it suffices to note (using the spectral theorem) that there exists a polynomial $p(x)$ such that $S^* = p(S)$. From there, it follows that $TS^* = S^*T$.
