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Proceedings of the London Mathematical Society Advance Access originally published online on March 22, 2007
Proceedings of the London Mathematical Society 2007 94(3):772-812; doi:10.1112/plms/pdm004
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© 2007 London Mathematical Society

Uniqueness of multiplicative determinants on elliptic pseudodifferential operators

Jean-Marie Lescure and Sylvie Paycha

UMR-6620 université Blaise Pascal
Complexe scientifique des Cézeaux
63170 Aubière
France
paycha{at}math.univ-bpclermont.fr
lescure{at}math.univ-bpclermont.fr

Received 22 February 2006.

We describe all the multiplicative determinants on the pathwise connected component of identity in the group of invertible classical pseudodifferential operators on a closed manifold that are continuous along continuous paths and the restriction to zero order operators of which is of class C1. This boils down to a description of all traces on zero order classical pseudodifferential operators, which turn out to be linear combinations of the Wodzicki residue and leading symbol traces introduced in previous work of S. Rosenberg and the second author. Both of these are continuous. Consequently, multiplicative determinants are parametrized by the residue determinant of S. Scott and a new ‘leading symbol determinant’, both of which are expressed in terms of a homogeneous component of the symbol of the logarithm of the operator.


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