  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 (\mu,m), with \mu>0,
m\ge0. For m=0, the asymptotics of TR exp(-tA) as t\to 0 and of TR
(\lambda-A)^{-k}$ as |\lambda|\to\infty 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).

Analysis of PDEs;Functional Analysis