On the Logarithm Component in Trace Defect Formulas

In asymptotic expansions of resolvent traces Tr(A(P ? λ)^?1) for classical pseudodifferential operators on closed manifolds, the coefficient C ₀(A, P) of ( ? λ)^?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′.

The present paper has two purposes. One is to show how the trace defect formulas can be obtained from the resolvents in a simple way without use of the complex powers of P as in the original proofs. We also give here a simple direct proof of a recent residue formula of Scott for C ₀(I, P). The other purpose is to establish trace defect residue formulas for operators on manifolds with boundary, where complex powers are not easily accessible; we do this using only resolvents. We also generalize Scott's formula to boundary problems.