I have a question. Given the Hilbertspace $H=l^2(\Bbb{N})$ with multiplicationoperator $T_f$, with $f:\Bbb{N}\rightarrow\Bbb{C}$ and $T_f\psi=f\psi$. I want to prove the following statement: Suppose only that $f$ takes only real values. Show that $T_f$ is self adjoint on the domain $D(T_f)=${$\psi\in l^2{\Bbb{N}}:f\psi\in l^2(\Bbb{N})$}.
I think the argumentation is very logical, but follows me is it not enought to say that $T_f^*=T_{\bar{f}}=T_f$ is $f$ is real valued. But wat is the argument for the claim?
Thank you :)