Spectral Measures: Constructions

Any constructions welcome!!!

Given a Hilbert space $\mathcal{H}$.

Regard spectral measures: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$

That are additive: $$E\left(\biguplus_kA_k\right)=\sum_kE(A_k)$$

And projection valued: $$E(A)^2=E(A)=E(A)^*$$

As well as complete: $$E(\mathbb{C})=1\quad E(\varnothing)=0$$

How to construct them?

1 Answer

Construction

Given a Hilbert space $\mathcal{H}$.

Regard an ONB $\mathcal{S}$.

Construct spectral measure:* $$E(A)\varphi:=\sum_{\sigma\in\mathcal{S}}1_A(\lambda_\sigma)\langle\sigma,\varphi\rangle\sigma$$

Denote orthonormals: $$\mathcal{S}_A:=\{\sigma\in\mathcal{S}:\lambda_\sigma\in A\}$$

Then it boils down to: $$E(A)\varphi=\sum_{\sigma\in\mathcal{S}_A}\langle\sigma,\varphi\rangle\sigma$$

So they are additive: $$E(A)+E(A')=E(A\uplus A')$$

And projection valued: $$E(A)^2=E(A)=E(A)^*$$

As well as complete: $$E(\mathbb{C})=1\quad E(\varnothing)=0$$

Regard disjoint Borel sets: $$A_k\in\mathcal{B}(\mathbb{C}):\quad A_k\cap A_{k'}=\varnothing$$

Denote for readability: $$A:=\biguplus_{k=1}^\infty A_k\in\mathcal{B}(\mathbb{C})$$

For an element find: $$\varphi\in\mathcal{H}:\quad\#\mathcal{S}_\varphi<\infty\quad(\mathcal{S}_\varphi\subseteq\mathcal{S})$$

Such that one has: $$\sum_{\sigma\in\mathcal{S}_\varphi^\complement}|\langle\sigma,\varphi\rangle|^2=\|\sum_{\sigma\in\mathcal{S}_\varphi^\complement}\langle\sigma,\varphi\rangle\sigma\|^2<\varepsilon$$

Choose the scalars: $$\Lambda_\varphi:=\{\lambda_\sigma:\sigma\in\mathcal{S}_\varphi\}:\quad\#\Lambda_\varphi<\infty$$

Then one can find: $$A_\varphi\cap\Lambda_\varphi:=(A_1\cup\ldots\cup A_K)\cap\Lambda_\varphi\supseteq A\cap\Lambda_\varphi$$

So one finally obtains: $$\|E(A)\varphi-E(A_\varphi)\varphi\|^2\leq2\|\sum_{\sigma\in\mathcal{S}_A\cap\mathcal{S}_\varphi^\complement}\langle\sigma,\varphi\rangle\sigma\|^2+2\|\sum_{\sigma\in\mathcal{S}_{A_\varphi}\cap\mathcal{S}_\varphi^\complement}\langle\sigma,\varphi\rangle\sigma\|^2\\ =2\sum_{\sigma\in\mathcal{S}_A\cap\mathcal{S}_\varphi^\complement}|\langle\sigma,\varphi\rangle|^2+2\sum_{\sigma\in\mathcal{S}_{A_\varphi}\cap\mathcal{S}_\varphi^\complement}|\langle\sigma,\varphi\rangle|^2\leq4\sum_{\sigma\in\mathcal{S}_\varphi^\complement}|\langle\sigma,\varphi\rangle|^2<4\varepsilon$$

Concluding the spectral measure.

*Scalars may repeat: $\lambda_\sigma=\lambda_{\sigma'}$

Representation

Given the Hilbert space $\ell^2(\mathbb{C})$.

Regard the ONB $\{\delta_\lambda\}_{\lambda\in\mathbb{C}}$.

Spectral measure becomes: $$[E(A)\varphi](\xi)=\sum_{\lambda\in\mathbb{C}}1_A(\lambda)\langle\delta_\lambda,\varphi\rangle\delta_\lambda(\xi)\\ =\sum_{\lambda\in\mathbb{C}}1_A(\lambda)\varphi(\lambda)\delta_\lambda(\xi)=1_A(\xi)\varphi(\xi)=[1_A\varphi](\xi)$$

(That is the canonical one.)