spectral measure and normal operators range

Question

Let $N$ be a normal operator with spectral measure $E$. We want to show that if $N=\int z\ dE(z)$ and $ε>0$, then $\operatorname{ran} E(\{z∶ |z|>ε\})⊆\operatorname{ran}N$.

Is this true?

Let $\Delta=\{z∶ |z|>ε\}$ and $f(z)=\frac{1}{z}\chi_\Delta(z)$, $f$ is bounded borel function, we have $\chi_\Delta(z)=zf(z)$, therefore

$E(\Delta)=\int_{\sigma(N)}\chi_\Delta(z)\ dE(z)=\int_{\sigma(N)}zf(z)\ dE(z)=Nf(N) $

Hence $\operatorname{ran}E(\Delta)=\operatorname{ran}N$.

Answer 1

Hint: $E\{z:|z|>\epsilon\} = N f(N)$ where $f(z) = \ldots$