## 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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### References:

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