Regularized determinants for pseudodifferential operators in vector bundles over S¹

We express the ζ-regularized determinant of an elliptic pseudodifferential operator A over S¹ with strongly invertible principal symbol in terms of the Fredholm determinant of an operator of determinant class, canonically associated to A, and local invariants. These invariants are given by explicit formulae involving the principal and subprincipal symbol of the operator. We remark that, generically, elliptic pseudodifferential operators have a strongly invertible principal symbol.