Spectral Theorem for Compact Selfadjoint Operators

If $A$ is compact and selfadjoint on a Hilbert space, there is an orthonormal basis of eigenvectors $\{e_k\}$ with $Ae_k=\lambda_k e_k$, $\lambda_k\in\mathbb R$, and $\lambda_k\to0$.

Key properties (paper use):

• Applied to positive Hilbert–Schmidt operators to compute products like $\prod_k (2\lambda_k/(\lambda_k^2+1))^{1/2}$ (Lemma 3.2).
• Ensures existence of “eigensystems” used in continuity arguments (Lemma 2.4).

Example: A diagonal operator on $\ell^2$ with diagonal entries $\lambda_k\to0$.