Numerical evaluation of operator determinants

TL;DR

This paper proves uniform convergence of approximate p-modified Fredholm determinants for integral operators in Schatten classes, providing convergence rates when evaluated at eigenvalues or resolvent set elements.

Contribution

It establishes the uniform convergence of approximate p-modified Fredholm determinants for operators in Schatten classes, including convergence rates at eigenvalues and resolvent points.

Findings

Uniform convergence of determinants for all p ≥ 1.

Quantitative convergence rates at eigenvalues.

Applicability to quadrature and projection methods.

Abstract

For any integral operator $K$ in the Schatten--von Neumann classes of compact operators and its approximated operator $K_N\sim(N\ge1)$ obtained by using for example a quadrature or projection method, we show that the convergence of the approximate $p$-modified Fredholm determinants $\sideset{}{_{Np}}\det(I_N+zK_N)$ to the $p$-modified Fredholm determinants $\sideset{}{_p}\det(I_\mathcal{H}+zK)$ is uniform for all $p\ge1$. As a result, we give the rate of convergences when evaluating at an eigenvalue or at an element of the resolvent set of $K$.