At the moment, I'm studying the book "Introduction to Spectral Theory" from P.D. Hislop and I.M. Sigal, I arrived at chapter 10 and I'm stuck on two problems there.
Problem 10.1: Let $A$ be a closed operator on $L^2(\mathbb{R}^n$ with $\rho(A)\neq\emptyset$ and let $\chi_B$ be the characteristic function for a set $B\subset\mathbb{R}^n$. Then $A$ is locally compact if for each bounded set $B$, $\chi_B (A-z)^{-1}$ is compact for some $z\in \rho(A)$. Prove that if $\chi_B (A-z)^{-1}$ is compact for some $z\in \rho(A)$, it is compact for all $z\in \rho(A)$.
Problem 10.5: Let $A$ be a closed operator on a Hilbertspace $\mathcal{H}$. We define Weyl sequences and a Weyl spectrum for $A$ by $$W(A):=\{\lambda\in\mathbb{C}\,|\,\exists\{u_n\}\in\,D(A),\,||u_n||=1,\,u_n\overset{w}{\rightarrow}0,\,\text{and }||(A-\lambda)u_n||\rightarrow 0\}.$$ Prove that $\sigma_{\text{ess}}(A)$ and $W(A)$ are closed subsets of $\mathbb{C}$. The hint there that one should construct a new Weyl sequence using a diagonal argument to prove that $W(A)$ contains all its accumulation points did not help me yet.
Regarding (2), the idea is something like: if $\lambda_m \to \lambda$ with $\lambda_m$ in $W(A)$, then for each $\lambda_m$ we have the sequence $u_{m,n}$ as in the definition.
Now $|| (A - \lambda) u_{m,n}|| \leq || (A - \lambda_m) u_{m,n}|| + | \lambda - \lambda_m|$ (using that $||u_{m,n}|| = 1$).
So if we choose some diagonal subsequence of the double sequence $u_{m,n}$ we should get a Weyl sequence witnessing $\lambda$ as an element of $W(A)$.