Determinants of elliptic pseudo-differential operators

Abstract

Determinants of invertible pseudo-differential operators (PDOs) close to positive self-adjoint ones are defined throughthe zeta-function regularization. We define a multiplicative anomaly as the ratio $\det(AB)/(\det(A)\det(B))$ considered as a functionon pairs of elliptic PDOs. We obtained an explicit formula for the multiplicative anomaly in terms of symbols of operators. For a certain natural classof PDOs on odd-dimensional manifolds generalizing the class of ellipticdifferential operators, the multiplicative anomaly is identically $1$. For elliptic PDOs from this class a holomorphic determinant and a determinant for zero orders PDOs are introduced. Using various algebraic, analytic, and topological tools we study local and global properties of the multiplicative anomaly and of the determinant Lie group closely related with it. The Lie algebra for the determinant Lie group has a description in terms of symbols only. Our main discovery is that there is a {\em quadratic non-linearity} hidden in the definition of determinants of PDOs through zeta-functions. The natural explanation of this non-linearity follows from complex-analytic properties of a new trace functional TR on PDOs of non-integer orders. Using TR we easily reproduce known facts about noncommutative residues of PDOs and obtain several new results. In particular, we describe a structure of derivatives of zeta-functions at zero as of functions on logarithms of elliptic PDOs. We propose several definitions extending zeta-regularized determinants to general elliptic PDOs. For elliptic PDOs of nonzero complex orders we introduce a canonical determinant in its natural domain of definition.