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Let $\Omega \subset \mathbb{R}^n$ be bounded. Suppose we have an integral kernel $K: \Omega^2\to \Omega$ with $\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}|K(x,y)|^2dxdy < \infty$. We know that the corresponding Hilbert Schmidt integral operator is compact, meaning there are countably infinite eigenvalues that cluster around zero. What are the decay rates for these eigenvalues?

What I am looking for in particular is, given the eigenvalues $\lambda_i$, does the sum $\sum_i - \lambda_i \ln(\lambda_i)$ converge?

This would be achieved if the eigenvalues had a decay rate of $o(n^{-1-\epsilon})$ for some $\epsilon >0$. This seems likely since we know Hilbert Schmidt operators are trace-class. But it is not guaranteed by that fact (e.g. $\lambda_i = \frac{1}{i\ln(i)}$).

Which of the following conditions are sufficient to prove $\sum_i - \lambda_i \ln(\lambda_i)$ converges (if any)?

  • $K$ is smooth

  • $K$ is piecewise smooth and continuously differentiable

  • $K$ is piecewise smooth and continuous

  • $K$ is bounded

Any references would be appreciated as well.

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