Determinants of Classical SG-Pseudodifferential Operators

We introduce a generalized trace functional TR in the spirit of Kontsevich and Vishik's canonical trace for classical SG-pseudodifferential operators on R^n and suitable manifolds, using a finite-part integral regularization technique. This allows us to define a zeta-regularized determinant det A for classical parameter-elliptic SG-operators A of order (μ,m), with μ>0, m≥0. For m=0, the asymptotics of TR exp(-tA) as t\to 0 and of TR (λ-A)^-k$ as |λ|\to∞ are derived. For suitable pairs (A,A_0) we show that det A/det A_0 coincides with the so-called relative determinant det(A,A_0).