an:06008782
Zbl 1246.47012
Ouedraogo, Marie-Fran??oise; Paycha, Sylvie
The multiplicative anomaly for determinants revisited; locality
EN
Commun. Math. Anal. 12, No. 1, 28-63 (2012).
00295768
2012
j
47G30 11M36 35R01 58J40
pseudodifferential operator; noncommutative residue; canonical and weighted traces; zeta and weighted determinants; multiplicative anomaly
The paper deals with the multiplicative anomaly of \(\zeta\)-determinants. The authors review the construction and properties of logarithms of elliptic operators and prove that the expression \(L(A,B) := \log(AB) - \log(A) - \log(B)\) is a finite sum of commutators of zero order classical pseudodifferential operators. Basic properties of weighted traces are also reviewed and it is proved that the canonical and weighted traces as well as the noncommutative residue commute with differentiation on differentiable families of operators with constant order.
Moreover, for two admissible pseudodifferential operators \(A\) and \(B\) (acting on smooth sections of a certain vector bundle and) having non negative orders, the authors prove that the weighted trace of \(L(A,B)\) is a local expression as a finite sum of noncommutative residues, which only depends on the first \(n\) homogeneous components of the symbols of \(A\) and \(B\). In addition, an explicit local expression for the weighted traces of \(L(A,B)\) is derived. This allows the authors to derive an explicit local formula for the multiplicative anomaly of \(\zeta\)-determinants.
Throughout the text, detailed interconnections are drawn with several known results.
Luis Filipe Pinheiro de Castro (Aveiro)