Assume that $A$ is an infinite matrix and it's a function of the parameter $\epsilon$. I would like to find $\epsilon$ so that the $det(I+A(\epsilon))=0$.
I know if $A$ was a trace class I could use the Fredholm identity, i.e. \begin{equation} det(I+A)=\prod_{n\geq 1} (1+\lambda_{n}) \end{equation}
where $\lambda_{n}$'s are eigenvalues of $A$.
In my case however, $A$ is not a trace class but it is symmetric and I know it's eigenvalues $\lambda_{n}(\epsilon)$. I was wondering if there is an identity similar to this that I can use. Also, how good of an approximation this would be if I truncate the product for a large $n$?