Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 5 / 2 / 12

Let $(f_n)$ be a singular Weyl sequence for $L$ at $\lambda \in {\Sigma }_{\mathrm{e}}(L)$. Since $K$ is compact and $f_n\rightharpoonup 0$, the preceding result gives $K{f}_n\to 0$. ThereforeThe norms remain one and the weak limit remains zero. The singular Weyl sequence criterion yields $\lambda \in {\Sigma }_{\mathrm{e}}(L+K)$. Apply the same argument to $L+K$ and the compact self-adjoint operator $-K$ for the reverse inclusion. Both operators are bounded and self-adjoint. Thus the Weyl theorem for compact self-adjoint perturbations isThe corrected shifted-range definition is necessary. For the diagonal example in the preceding solutions, a rank-one perturbation changing the entry $2$ to $0$ removes $2$ from the spectrum. The unshifted printed definition had classified $2$ as essential merely because the original range was not closed, so it would make this invariance false.