Everyone, I get stuck in an exercise of Functional Analysis.
Let $T \in B(H)$ (H a complex Hilbert space) and $T^*$ adjoint of $T$. Supose $T$ is a normal operator.
1) Prove that $Ker(T)= Ker(T^*) = R(T)^\perp$ - I've finished this.
2) Using previous proof. If $\alpha$ is an eigenvalue of $T$ then the conjugate $\bar\alpha$ is an eigenvalue of $T^*$.
3) If $\alpha \neq \beta $ are eigenvectors of T, then the associated eigenspaces are orthogonal between them.
2) Let $x \in H$ such that $Tx = \alpha x $.
$$<x,Tx> = <T^*x,x> = \alpha<x,x> = <\bar\alpha x,x>$$
Then we have that
$$<T^*x - \bar \alpha x,x>=0$$
Here I don't know if above implies that $T^*x - \bar \alpha x =0$.
3) I didn't make a great progress here.