Spectral Theorem for Compact Selfadjoint Operators - Imported Knowlpedia Content

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 $\lambda_k\to0$ .

Remarks

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).

Examples

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