Exercise 43, § 5

Let E be a complex Hilbert space.

a) Let $u$ be a self-adjoint partial operator on E. For every $\lambda \in \mathbf{R}$, one has $\lambda \in$ Sp$_e(u)$ if and only if there exists a sequence $(x_n)$ in E which converges weakly to 0 in E and satisfies the conditions

lim inf$_{n\rightarrow+\infty}\|x_n\|>0$, lim$_{n\rightarrow+\infty}(u(x_n)-\lambda x_n) = 0$.

b) Let $u$ and $v$ be self-adjoint partial operators on E. If there exists $\lambda \in \mathbf{C}-$ (Sp($u$)$\cup$ Sp($v$)) such that $R(u, \lambda )-R(v, \lambda )$ is compact, then Sp$_e(u) =$ Sp$_e(v)$.

c) Let $u$ be a closed symmetric partial operator on E. Let $u_1$ and $u_2$ be self-adjoint extensions of $u$. If the space Ker($u^*-i$) is finite-dimensional, then Sp$_e(u_1) =$ Sp$_e(u_2)$.

d) Let $u$ be a self-adjoint partial operator on E and $v$ a partial operator compact with respect to $u$ (see the preceding exercise). The partial operator $u+v$ is self-adjoint and Sp$_e(u+v) =$ Sp$_e(u)$.