= Solution
Use complex <inner products> linear in the first argument throughout. The <spectral theorem for compact self-adjoint operators> states that, for a <compact operator> $T=T^*$ on a <Hilbert space>, its nonzero spectral values are real <eigenvalues>, each of finite multiplicity, with no possible accumulation point except zero. Moreover,
$$
\boxed{H=\ker T\oplus\bigoplus_{\lambda\ne0}\ker(T-\lambda I),
\qquad T=\sum_{\lambda\ne0}\lambda P_\lambda,}
$$
where $P_\lambda$ are the mutually orthogonal <eigenspace> <projections>. The operator series converges in <operator norm> when the nonzero <eigenvalues> are listed in decreasing absolute value. The sum is at most countable, even if the <kernel> is nonseparable. Zero need not be an <eigenvalue>.
First prove that a nonzero compact <self-adjoint operator> has an <eigenvalue> of magnitude its <norm>, without assuming a spectral decomposition. Put $r=\|T\|>0$ and choose unit <vectors> $\xi_n$ with $\|T\xi_n\|\to r$. Since $\|T^2\xi\|\leq r\|T\xi\|$, expansion gives
$$
\|(T^2-r^2I)\xi_n\|^2
\leq r^2\bigl(r^2-\|T\xi_n\|^2\bigr)\longrightarrow0.
$$
<Compactness> of $T^2$ supplies a subsequence for which $T^2\xi_n$ converges; the displayed estimate then forces $\xi_n$ itself to converge to a unit <vector> $\xi$ satisfying $T^2\xi=r^2\xi$. The <vectors> $\xi+T\xi/r$ and $\xi-T\xi/r$ are <eigenvectors> for $r$ and $-r$, respectively, whenever nonzero, and they cannot both vanish. This proves the claim.
Self-adjointness gives reality of every <eigenvalue> and <orthogonality> of <eigenspaces> for distinct <eigenvalues>. An <eigenspace> for $\lambda\ne0$ cannot contain an infinite orthonormal sequence: its images under $T$ would have pairwise distance $\sqrt2|\lambda|$, contradicting <compactness>. The same argument proves that there are only finitely many mutually orthogonal <eigenvectors> with <eigenvalues> of magnitude at least any given $\varepsilon>0$. Hence all nonzero <eigenspaces> together have a countable <orthonormal basis>, and their <eigenvalues> tend to zero if there are infinitely many.
Let $K$ be the <orthogonal complement> of the closed span of all these <eigenspaces>. It is invariant under $T$. If $T|_K$ were nonzero, the norm-attaining <eigenvalue> argument applied to that compact <self-adjoint> restriction would produce another nonzero <eigenvector>, a contradiction. Thus $K\subseteq\ker T$, and the converse inclusion follows from <orthogonality>. This establishes the asserted decomposition. On its orthogonal summands the tail of the operator series has <norm> equal to the supremum of the omitted <eigenvalue> magnitudes, proving <norm> convergence.
Finally, a nonzero complex number not among these <eigenvalues> has positive distance from their set and from zero. Inverting $T-zI$ separately on the <eigenspaces> and <kernel> therefore gives a bounded inverse. This proves that no other nonzero spectral values occur and completes the theorem.
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