# Decay rates of eigenvalues of Hilbert-Schmidt integral operator

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.