The multiplicative anomaly of three or more commuting elliptic operators

Abstract

Zeta-regularized determinants are well-known to fail to be multiplicative. Hence one is lead to study the n-fold multiplicative anomaly M_n(A_1,...,A_n) :=\frac{\det_\zeta\Big(\prod_{i=1}^n A_i\Big)}{\prod_{i=1}^n \det_\zeta(A_i)} attached to n (suitable) operators A_1,...,A_n. We show that if the A_i are commuting pseudo-differential elliptic operators, then their joint multiplicative anomaly can be expressed in terms of the pairwise multiplicative anomalies. Namely M_n(A_1,...,A_n)^{m_1+...+m_n} =\prod_{1\le i<j\le n}M_2(A_i,A_j)^{m_i+m_j}, where m_j is the order of A_j. The proof relies on Wodzicki's 1987 formula for the pairwise multiplicative anomaly M_2(A,B) of two commuting elliptic operators.