Uniqueness of multiplicative determinants on elliptic pseudodifferential operators

Jean-Marie Lescure (1) , Sylvie Paycha (1)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal

We describe all 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 C 1. 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 [W] and leading symbol traces introduced in [PR1], both of which are continuous. Consequently, multi-plicative determinants are parametrized by the residue determinant [W87, Sc] 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.

Domaines: Algèbres d'opérateurs [math.OA]

Jean-Marie Lescure, Sylvie Paycha. Uniqueness of multiplicative determinants on elliptic pseudodifferential operators. Proc. of the London Math. Soc., 2007, 94, pp.772--812. ⟨hal-00483875⟩