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Geometry of determinants of elliptic operators. (English) Zbl 0920.58061
Gindikin, Simon (ed.) et al., Functional analysis on the eve of the 21st century. Volume I. In honor of the eightieth birthday of I. M. Gelfand. Proceedings of the conference, held at Rutgers University, New Brunswick, NJ, October 24-27, 1993. Boston, MA: Birkhäuser. Prog. Math. 131, 173-197 (1995).
Let \(A: C^\infty(E)\to C^\infty(E)\) be a positive elliptic pseudo-differential operator \((\Psi\text{DO})\) of positive order over a closed manifold \(M\), \(\zeta_A(\xi)= \sum_{\lambda_i\in \sigma(A)} \lambda_i^{-\xi}= \text{tr} (A^{-\xi})\) its zeta-function with meromorphic extension to \(\mathbb{C}\), regular at zero and \(\det_\zeta(A):= \exp(-\zeta_A'(0))\) its regularized determinant.
The authors study the multiplicative anomaly \[ F(A,B):= \text{det}_\zeta (A\cdot B)/ (\text{det}_\zeta(A)\cdot \text{det}_\zeta(B)) \] and related algebraic and geometric objects. In this way they provide an extension of the notion of the zeta-regularized determinant to a larger class of operators, including operators of nonreal orders. The main result of the paper can be formulated as follows. For positive selfadjoint elliptic differential operators on odd-dimensional manifolds holds \(F(A,B)=1\).
For the entire collection see [Zbl 0830.00037].

58J52 Determinants and determinant bundles, analytic torsion
58J40 Pseudodifferential and Fourier integral operators on manifolds
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