= Solution
Use the stated Fredholm definition of the <essential spectrum>, which applies to general bounded operators:
$$
\Sigma_{\mathrm e}(L)=\{z\in\mathbb C:L-zI\text{ is not Fredholm}\}.
$$
For each $z$, part (f) applied to $L-zI$ says that $L-zI$ is <Fredholm> exactly when $L+K-zI$ is Fredholm. Taking the complement of these equivalent Fredholm conditions gives
$$
\boxed{\Sigma_{\mathrm e}(L+K)=\Sigma_{\mathrm e}(L).}
$$
This is the general <Weyl theorem for bounded compact perturbations>. Individual isolated <eigenvalues> can change; the Fredholm obstruction, and hence this <essential spectrum>, cannot.
Back to article page
