Let $H$ be a complex Hilbert space, $T\in H'$ and $T=T^*$. Here is where I need help: If $\sigma(T)\subset\{0,1\}$ then $T=T^2$.
Using the spectral theorem I know that $\{0,1\} \supset q(\sigma(P)) = \sigma(q(T))$ for $q:x\mapsto x^2$, thus $\sigma(T^2)=\sigma(T)$. How can I continue?