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 $det_2(I+K)=\prod(\frac{1+\lambda_i}{exp(\lambda_i)})=det_1(I+K)\cdot exp(-Tr(K))$