2012/11/17 by Victor Castillo-Garate, Castillo-Garate, Victor, Eduardo Friedman +3
Mathematics · Physics and Astronomy · #58J52 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #math-ph #math.FA #math.MP #msc:58J52
paper · pdf · doi:10.48550/arxiv.1211.4117
7 pages
arxiv created 2012/11/17 · arxiv updated 2012/11/20
Zeta-regularized determinants are well-known to fail to be multiplicative. Hence one is lead to study the n-fold multiplicative anomaly Mn(A1,...,An) :=\fracdetζ(∏i=1n Ai)∏i=1n detζ(Ai) attached to n (suitable) operators A1,...,An. We show that if the Ai are commuting pseudo-differential elliptic operators, then their joint multiplicative anomaly can be expressed in terms of the pairwise multiplicative anomalies. Namely Mn(A1,...,An)m1+...+mn =∏1≤ i<j≤ nM2(Ai,Aj)mi+mj, where mj is the order of Aj. The proof relies on Wodzicki's 1987 formula for the pairwise multiplicative anomaly M2(A,B) of two commuting elliptic operators.