Abstract
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
(−λ
)
−1
is of special interest, since it is the first coefficient containing nonlocal elements from
A
; on the other hand if
A=I
and
P=
D
∗
D
it gives part of the index of
D
.
C
0
(A,P)
also equals the zeta function value at 0 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
′
,
logP
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 here also give 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.