Inspired by M. Braverman’s symmetrized determinant, we introduce a symmetrized logarithm logsym for certain elliptic
pseudodifferential operators. The symmetrized logarithm of an operator lies in the odd class whenever the operator does. Using the canonical trace extended to log-polyhomogeneous pseudodifferential operators, we define an associated canonical symmetrized determinant DETsym on odd-class classical elliptic operators in odd dimensions: DETsym = exp ∘ TR ∘ log which provides a canonical description of Braverman’s symmetrized determinant. Using the cyclicity of the canonical trace on odd- class operators, one then easily infers multiplicative properties of this canonical symmetrized determinant.

pseudodifferentiels operateurs, symmetrized trace, symmetrized determinant, holomorphic familly