This thread is only Q&A.
Given a Hilbert space $\mathcal{H}$.
Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$
And its spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H}):\quad N=\int\lambda\mathrm{d}E(\lambda)$$
Construct scale functions: $$\Lambda_s:=\sqrt{1+|\mathrm{id}|^2}^s\in\mathcal{B}(\mathbb{C})$$
As well as scale norms: $$\varphi\in\mathcal{D}\Lambda_s(N):\quad\|\varphi\|_s:=\|\Lambda_s(N)\varphi\|$$
And the scale spaces: $$\mathcal{H}_s:=\overline{\mathcal{D}\Lambda_s(N)}^s:=\widehat{\mathcal{D}\Lambda_s(N)}^s$$
Regard a bounded form: $$s\in\mathcal{B}(\mathcal{H}^s,\mathcal{H}^{s'};\mathbb{C})$$
Then it represents as: $$S\in\mathcal{B}(\mathcal{H}^s,\mathcal{H}^{s'}):\quad s(\varphi,\psi)=\langle S\varphi,\psi\rangle_{s'}$$
How can I check this?