Hilbert-Schmidt determinant

In the previous post, I mentioned the intuition of Fredholm determinant for trace class operator, which is Tr(K)<\infty. When it comes to Hilbert-Schmidt determinant, which is Tr(K^*K)<\infty, that definition(let’s call it det_1) may fail since trace of H-S operator my be infinity. So we need to come up with another definition(let’s call it det_2) for H-S operator that makes sense.

The intuition is that consider the eigen values of a H-S matrix K, say \lambda_1,\lambda_2,dots,notice that for \forall \lambda, 1+\lambda\leq exp(\lambda), so we can define det_2(I+K)=\prod(\frac{1+\lambda_i}{exp(\lambda_i)})=det_1(I+K)\cdot exp(-Tr(K))


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