Let $\mu $ be a regular Borel measure on $\mathbb C$ with compact support, define $N_\mu$ on $ L^2(\mu)$ by $N_\mu f=zf$ for each $f\in L^2(\mu)$. Prove that $f\in L^2(\mu)$ is star-cyclic vector for $N_\mu$ if and only if $\mu(\{x:\,f(x)=0\})=0$
My attempt
1: I've proved $N_\mu $ is normal
2: If $E$ is the spectral measure for $N_\mu $, then $E(\Delta)=\chi_{\Delta}(N_\mu) $
3: $1$ is a star-cyclic vector for $N_\mu$
4: $N_\mu= \int \lambda \, dE(\lambda) $
Any help is greatly appreciated