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Regularized determinants for pseudodifferential operators in vector bundles over \(S^ 1\). (English) Zbl 0784.35126

Let \(A:C^ \infty(S^ 1,\mathbb{C}^ N)\to C^ \infty(S^ 1,\mathbb{C}^ N)\) be an elliptic pseudodifferential operator with strongly invertible principal symbol and assume that \(\theta \in S^ 1\) is a principal angle for \(A\) (which means that the ray \(\{re^{i \theta},r \geq 0\}\) does not meet the spectrum of the principal symbol of \(A\) computed at any \((x,\xi)\), \(\xi \neq 0)\). Under some additional assumptions, one can define complex powers \(A_ \theta^{-s}\) and then the \(\zeta\)- regularized determinant of \(A\), which is \(\exp \bigl( -{d \over ds} \text{tr} A_ \theta^{-s} |_{s=0} \bigr)\). This definition is then extended to the case where those additional assumptions are removed.
In this paper the authors express the \(\zeta\)-regularized determinant of \(A\) in terms of a Fredholm determinant of a pseudodifferential operator canonically associated to \(A\), and local invariants given by explicit formulae involving the principal and the subprincipal symbol of \(A\).
This paper continues previous work by the same authors, where the case of differential operators was considered.

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
34L05 General spectral theory of ordinary differential operators
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