Let be T,S two self adjoint linear Operator on a Hilbert Space $\mathcal{H}$ with pure point spectrum.
Then T,S commute if and only if they have a complete set of common eigenvectors.
This is the full statement i want to prove. In the case of bounded Operators this would be fine. My Problem now is, the Unbounded Case.I may tell you my thoughts for the first implication:
Let be $\phi$ an eigenvector from T with eigenvalue $\lambda$, then we know, since T and S commute: $$ TS\phi=ST\phi=S\lambda \phi=\lambda S\phi $$ Now, here is my problem: Can i assume (and why) that $\phi\in D(S)$? Maybe this is quite simple but i dont get it at the Moment...