Define for a self-adjoint pure contraction $S$ (remember: $\|S\|\leq1$ and $\pm1\notin\sigma_p(S))$ on a Hilbert space $\mathcal{H}$ the following set: $C_c^*(S):=\{g(S):g\in C_c(\hat{\sigma}(S))\}$ with $\hat{\sigma}(S)=\sigma(S)\cap(-1,1)$. Now i to prove that $\overline{C^*_c(S)\mathcal{H}}=\mathcal{H}$. Can someone give an example of self-adjoint pure contraction? Does someone can give an example where this is true? Maybe the multiplication operator $m_f$ on $l^2(\mathbb{N})$ and $f=\overline{f}$?
Thank you for help :)