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 \rz^n and suitable manifolds, using a finite-part integral regularization technique. This allows us to define a zeta-regularized determinant \det A for parameter-elliptic operators A\in S^{\mu,m}_\cl, \mu>0, m\ge0. For m=0, the asymptotics of \TR e^{-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).