Let $H$ be a complex Hilbert Space and $P$ a projection valued measure on H with compact support (K), let
A:=$\int_{K} \lambda dP_\lambda$, then A is a self adjoint Operator and $P$ is the spectral measure of $A$.
Now this is a part of the spectral theorem for bounded self adjoint operators I had in the lecture. There is one part of the proof I dont understand. At some point, the professor proofs that $\sigma(A)=K$.
He Argues: "$K\subset \sigma(A)$": Let $\lambda \in K$, then $P_\Omega \neq 0$ for all neighborhoods $\Omega$ of $\lambda$.
Then $ P_k$:=$P_{[\lambda - \frac{1}{k},\lambda + \frac{1}{k}]} \neq 0$ for all $k \in \mathbb{N}$,
Then there exist $x_k \in H$ such that $\|x_k \|=1$ and $P_k x_k=x_k$ for all $k$.
Then $\|(\lambda-A)x_k \| =\|(\lambda-A)P_k x_k \|$= $\int_{[\lambda - \frac{1}{k},\lambda + \frac{1}{k}]} (\lambda-t)^2 d\mu_{x_k}$ , where ($\mu_x:=<x,P_\Omega x>$)
the following estimation I dont understand:
$\int_{[\lambda - \frac{1}{k},\lambda + \frac{1}{k}]} (\lambda-t)^2 d\mu_{x_k}$ $\leq \frac{1}{k} \int_{\sigma(A)} d\mu_{x_k}$
Im not sure why we are integrating over the spectrum of A all over sudden, maybe i copied it wrong i guess here should be the support of P (K), but even if that is true I dont understand it.
Can someone help?
Or maybe give another proof?