Identity for spectral resolution: $dE_{\xi, \xi}= |g|^2 dE_{\eta, \eta}$

Question

Let $(\Omega, \mathcal{F})$ be a measurable space. Let $E: \mathcal{F}\to B(H)$ be a regular resolution of the identity on the Hilbert space $H$, see e.g. Rudin's functional analysis book.

Suppose that $g: \Omega \to \mathbb{C}$ is Borel-measurable and that $\xi \in \mathscr{D}\left(\int_\Omega g dE\right)=: \mathscr{D}_g$. We can then define $$\eta:= \left(\int_\Omega g dE\right)\xi\in H.$$ Is it true that $$dE_{\eta, \eta}= |g|^2 dE_{\xi, \xi}?$$ I.e., if $A\in \mathcal{F}$, do we have $$E_{\eta, \eta}(A)= \int_{A}|g|^2 dE_{\xi, \xi}.$$

I believe I can show this if $g$ is bounded. But I am mostly interested in the case where $g$ is unbounded.

Thanks in advance for your help!

Answer 1

If $g=\sum_k\beta_k\,1_{X_k}$ is simple, $$ \Big\langle\Big(\int_\Omega g\,dE\Big)\xi,\xi\Big\rangle=\int_\Omega g\,dE_{\xi,\xi}=\sum_k\beta_k\,E_{\xi,\xi}(X_k)=\Big\langle\sum_k\beta_k\,E(X_k)\xi,\xi\Big\rangle. $$

Since $g$ is measurable, then there exist simple functions $g_n$ such that $g_n\to g$ and $|g_n|^2\nearrow |g|^2$ (this is easily achievable by finding first simple functions that increase to $(\operatorname{Re}g)^+$, $(\operatorname{Re}g)^-$, $(\operatorname{Im}g)^+$, $(\operatorname{Im}g)^-$, respectively).

For any $\nu$ we have, using monotone convergence in each of the sixteen integrals coming from decomposing $g$ and $dE_{\xi,\nu}$ as a linear combinations of positive functions/measures,

$$ \langle \eta,\nu\rangle=\int_\Omega g\,dE_{\xi,\nu}=\lim_n\int_\Omega g_n\,dE_{\xi,\nu}=\lim_n\Big\langle\Big(\int_\Omega g_n\,dE\Big)\xi,\nu\Big\rangle $$

So, writing $g_n=\sum_k\beta_{n,k}\,1_{X_{n,k}}$, \begin{align} E_{\eta,\eta}(A) &=\langle E(A)\eta,\eta\rangle =\Big\langle E(A)\Big(\int_\Omega g\,dE\Big)\xi,\eta\rangle\\[0.3cm] &=\lim_n\Big\langle E(A)\Big(\int_\Omega g\,dE\Big)\xi,\Big(\int_\Omega g_n\,dE\Big)\xi\Big\rangle\\[0.3cm] &=\lim_n\sum_k\overline{\beta_{n,k}}\Big\langle E(A)\Big(\int_\Omega g\,dE\Big)\xi,E(X_{n,k})\xi\Big\rangle\\[0.3cm] &=\lim_n\sum_k\overline{\beta_{n,k}}\Big\langle \Big(\int_\Omega g\,dE\Big)\xi,E(A)E(X_{n,k})\xi\Big\rangle\\[0.3cm] &=\lim_n\sum_k\overline{\beta_{n,k}}\Big\langle \Big(\int_\Omega g\,dE\Big)\xi,E(A\cap X_{n,k})\xi\Big\rangle\\[0.3cm] &=\lim_n\lim_m\sum_k\sum_j\overline{\beta_{n,k}}\beta_{m,j}\Big\langle E(X_{m,j})\xi,E(A\cap X_{n,k})\xi\Big\rangle\\[0.3cm] &=\lim_n\lim_m\sum_k\sum_j\overline{\beta_{n,k}}\beta_{m,j}\Big\langle E(A\cap X_{n,k})E(X_{m,j})\xi,\xi\Big\rangle\\[0.3cm] &=\lim_n\lim_m\Big\langle \Big(\int_A \overline{g_n}g_m\,dE\Big)\xi,\xi\Big\rangle\\[0.3cm] &=\lim_n\Big\langle \Big(\int_A \overline{g_n}g\,dE\Big)\xi,\xi\Big\rangle\\[0.3cm] &=\Big\langle \Big(\int_A \overline{g}g\,dE\Big)\xi,\xi\Big\rangle\\[0.3cm] &=\int_A|g|^2\,dE_{\xi,\xi}. \end{align}