This thread is only Q&A!
Given a Hilbert space $\mathcal{H}$.
Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$
Regard the domain: $$\int|f(\lambda)|^2\mathrm{d}\|E(\lambda)\varphi\|^2<\infty$$
And the calculus: $$\langle f(E)\varphi,\chi\rangle=\int_\mathbb{C} f(\lambda)\mathrm{d}\langle E(\lambda)\varphi,\chi\rangle$$
Then equivalence holds: $$\mathcal{N}f(E)=(0)\iff E\{f=0\}=0$$
Especially one has: $$f(E)^{-1}=f^{-1}(E)$$
How to prove this?