Numerical evaluation of operator determinants

For any integral operator $K$ in the Schatten–von Neumann classes of compact operators and its approximated operator $K_N\sim (N\ge 1)$ obtained by using for example a quadrature or projection method, we show that the convergence of the approximate $p$-modified Fredholm determinants $\text{sideset}{}_{Np}\det (I_N+z{K}_N)$ to the $p$-modified Fredholm determinants $\text{sideset}{}_p\det (I_{\mathcal{H}}+zK)$ is uniform for all $p\ge 1$. As a result, we give the rate of convergences when evaluating at an eigenvalue or at an element of the resolvent set of $K$.