Let $(X,\Omega,\mu)$ be a measure space, and $k\in L^2(X\times X, \Omega\times \Omega,\mu\times\mu)$. Then it is well-known that 

$$(Kf)(x)=\int k(x,y)f(y)\ d\mu(y)$$

is a compact operator with norm at most $\|k\|\_2$. 

Suppose that $k(x,y)=\overline k(y,x)$. Show that $K$ is self-adjoint and if $\{\mu\_n\}$ are the eigenvalues of $K$, each repeated $\dim \ker (K-\mu\_n)$ times, then $\sum\_1^n |\mu\_n|^2 <\infty.$

This is problem II.6.3 on page 49 of Conway's *A Course in Functional Analysis*, 2nd edition. A proof that the operator is compact is proposition II.4.7 on page 43.

**Hint:** 

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>  Apply the spectral theorem for compact self-adjoint operators. The operator diagonalizes. 
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