On Multiplicative Properties of Determinants

TL;DR

This paper investigates the multiplicative properties of determinants associated with elliptic pseudodifferential operators, providing explicit formulas for their regularized determinants on compact manifolds.

Contribution

It offers a novel computation of the difference between zeta-regularized determinants and Fredholm determinants for elliptic operators with specific perturbations.

Findings

Derived explicit formulas for determinant differences

Extended understanding of multiplicative properties of determinants

Applicable to elliptic pseudodifferential operators on manifolds

Abstract

Let $A$ be an elliptic pseudodifferential operator of positive order on a compact closed manifold, and let $T$ be a pseudodifferential operator of negative order such that $T^m$ is of trace class. We compute $\log\det(A(I+T))-\log\det A-\log\det_m (I+T)$ where first two determinants are zeta function regularized, and the last one is a regularized Fredholm determinant.