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The residue determinant. (English) Zbl 1236.58037
Summary: The purpose of this paper is to present the construction of a canonical determinant functional on elliptic pseudodifferential operators (\(\psi\)-dos) associated to the Guillemin-Wodzicki residue trace. The resulting residue determinant functional is multiplicative, a local invariant, and not defined by a regularization procedure. The residue determinant is consequently a quite different object from the zeta function determinant, which is nonlocal and nonmultiplicative. Indeed, the residue determinant does not arise as the derivative of a trace on the complex power operators and does not depend on a choice of spectral cut. The identification of a certain residue determinant with the index of an elliptic \(\psi\)-do shows the residue determinant to be topologically significant.

MSC:
58J52 Determinants and determinant bundles, analytic torsion
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
35J46 First-order elliptic systems
47G30 Pseudodifferential operators
58J40 Pseudodifferential and Fourier integral operators on manifolds
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