Spectral decomposition of a self-adjoint operator. Let $\mathcal{H}$ be a Hilbert space, $A$ be a self-adjoint operator of $B(\mathcal{H})$. If $0\le A\le I$, how to show that there are mutually commutative projections $\{P_n\}_{n=1}^{\infty}$ such that $A=\sum_{n=1}^{\infty} \frac{1}{2^n} P_n$?
I know there is a decomposition for compact normal operator $N\in B(\mathcal{H})$(GTM 209 Theorem 2.8.2):
$$
N=\sum_{n=1}^{\infty} \lambda_n E_n,
$$
where $E_n$ are mutually orthogonal projections onto $H_{\lambda_n}=\{\xi\in \mathcal{H}: N\xi=\lambda_n \xi\}$.
My questions are

*

*Can we prove $A$ is compact by $0\le A\le I$? And how to do that?

*Can I solve this problem by writing
$$
A=\sum_{n=1}^{\infty} \frac{1}{2^n}\cdot \left(2^n\lambda_n E_n\right)\colon=\sum_{n=1}^{\infty} \frac{1}{2^n} P_n
$$
?

Could you please give me more details? I am not sure about my thoughts. Thank you very much!

Let $\mu$ denote the projection valued measure associated to $A$. Note that for any two sets $X,Y\subset\Bbb R$ you have that $\mu(X)$ and $\mu(Y)$ are commuting projections. You can assume by the spectral theorem $A$ is of the form $f\mapsto (x\mapsto x\cdot f(x))$ on $L^2([0,1],d\lambda)$ for some measure $\lambda$, this will help you visualise whats happening.
Note that $A-\frac12\mu([\frac12,1])$ then corresponds to multplication with $x-\frac12 \chi_{[\frac12,1]}(x)$, ie it is $x$ until value $1/2$ and then reset to $0$ after which it grows to $1/2$ again.
Do this visualisation again, what function grows to $\frac14$ and then resets to $0$,  then grows to $\frac14$ and resets and so on? The function would be
$$x-\left(\frac12\chi_{[\frac12,1]}(x) + \frac14\chi_{[\frac14,\frac12)}(x) + \frac14\chi_{[\frac34,1]}(x)\right)$$
now for the next step you want it to reset in intervals of $\frac18$, so you will need to add some more projections. Continue in this way, being more explicit you get that
$$P_1=\mu([\frac12,1]), \quad P_2=\mu([\frac14,\frac12)\,\cup\,[\frac34,1]) \\ 
P_3=\mu([\frac18,\frac14)\,\cup\,[\frac38,\frac12)\,\cup\,[\frac58,\frac34)\,\cup\,[\frac78,1])$$
and so on, it can be viewed as an exercise to get the explicit forms of the sets you are interested in.
