这是一份umass麻省大学 MATH 523H作业代写的成功案例
Finally, we discuss one useful formula relating the resolvent and spectral projections. It is a matter of computation to see that
$$
f_{z}(x) \rightarrow \begin{cases}0 & x \notin[a, b] \ \frac{1}{2} & x=a \text { or } x=b \ 1 & x \in(a, b)\end{cases}
$$
if $\varepsilon \downarrow 0$ where
$$
f_{z}(x)=\frac{1}{2 \pi i} \int_{a}^{b}\left(\frac{1}{x-\lambda-i \varepsilon}-\frac{1}{x-\lambda+i \varepsilon}\right) d \lambda
$$
Moreover, $\left|f_{t}(x)\right|$ is bounded uniformly in $\varepsilon$, so by the functional calculus, one has:
(Stone’s formula) Let $A$ be a bounded self-adjoint operator. Then
$$
\operatorname{s-lim}{\varepsilon \downarrow 0}(2 \pi i)^{-1} \int{a}^{b}\left[(A-\lambda-i \varepsilon)^{-1}-(A-\lambda+i \varepsilon)^{-1}\right] d \lambda=\frac{1}{2}\left[P_{[a, b]}+P_{(a, b)}\right]
$$
MATH 523H COURSE NOTES ：
Let $A_{1}, \ldots, A_{n}$ be commuting bounded self-adjoint operators on $\mathscr{A}$, a separable Hilbert space.
(a) Let $\Omega_{1}, \ldots, \Omega_{n}$ be Borel sets on $\mathbb{R}$. Prove that $P_{\Omega_{1}}\left(A_{1}\right), P_{\Omega_{2}}\left(A_{2}\right), \ldots, P_{\Omega_{n}}\left(A_{n}\right)$ all commute.
(b) Let $f$ be a function on $\mathbb{R}^{n}$ which is a linear combination of characteristic functions of rectangles (that is, sets of the form $\Omega=\Omega_{1} \times \cdots \times \Omega_{n}$ ). Show that $f$ can be written
$$
f=\sum_{i=1}^{n} c_{i} \chi_{\Omega(t)} \quad \text { with } \Omega^{(t)} \cap \Omega^{(j)}=\varnothing \text { if } i \neq j
$$
(c) For $f$ of the above form, define