On the logarithm component in trace defect formulas

摘要

In asymptotic expansions of resolvent traces Tr(A( P-lambda)(-1)) for classical pseudodifferential operators on closed manifolds, the coefficient C-0 (A,P) of (-lambda)(-1) is of special interest, since it is the first coefficient containing nonlocal elements from A ; moreover, it enters in index formulas. C-0 (A,P) also equals the zeta function value at zero when P is invertible. C-0(A,P) is a trace modulo local terms, since C-0 (A,P) - C-0(A,P') and C-0 ([A,A'], P) are local. By use of complex powers P-s (or similar holomorphic families of order s ), Okikiolu, Kontsevich and Vishik, Melrose and Nistor showed formulas for these trace defects in terms of residues of operators defined from A,A', log P and log P'.