= Weyl theorem for compact self-adjoint perturbations
{c}
{title2=$\Sigma_{\mathrm e}(L+K)=\Sigma_{\mathrm e}(L)$}
If $L$ is bounded and self-adjoint on a complex <Hilbert space> and $K$ is compact and self-adjoint, then
$$
\Sigma_{\mathrm e}(L+K)=\Sigma_{\mathrm e}(L).
$$
A <singular Weyl sequence> for $L$ stays singular for $L+K$, because <compact operators send weak convergence to norm convergence> and therefore $Kf_n\to0$. Applying the same argument with $-K$ proves the reverse inclusion. Finite-multiplicity isolated <eigenvalues> can move under such perturbations; the <essential spectrum of a bounded self-adjoint operator> is unchanged.
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