On the Logarithm Component in Trace Defect Formulas

In asymptotic expansions of resolvent traces Tr(A(P − λ)) for classical pseudodifferential operators on closed manifolds, the coefficient C0(A,P ) of (−λ) is of special interest, since it is the first coefficient containing nonlocal elements from A; on the other hand if A = I and P = DD it gives part of the index of D. C0(A,P ) also equals the zeta function value at 0 when P is invertible. C0(A,P ) is a trace modulo local terms, since C0(A,P )− C0(A,P ) and C0([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 logP . 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 C0(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. Introduction. Consider a classical pseudodifferential operator (ψdo) A of order σ on an n-dimensional smooth compact boundaryless manifold X . When P denotes an auxiliary elliptic ψdo of order m > 0 and, say, positive, one can study the generalized zeta funcion ζ(A, P, s) defined as the meromorphic extension of Tr(AP) to the complex plane, where the complex powers P are defined from the resolvent (P −λ) as in Seeley [S]. It is well-known that ζ(A, P, s) has a Laurent expansion at s = 0, (0.1) ζ(A, P, s) ∼ C−1(A, P )s −1 + C0(A, P ) + ∑ l≥1 Cl(A, P )s , where mC−1(A, P ) equals the noncommutative residue resA (Wodzicki [W], Guillemin [Gu]), and C0(A, P ) equals the canonical trace TRA in particular cases (Kontsevich and Vishik [KV], Lesch [L], recent extension in Grubb [G2]). The coefficient C0(A, P ) is not in general independent of P , but then it is viewed as a “regularized trace” (Melrose and Nistor [MN]) or a “weighted trace” (Cardona, Ducourtioux, Magnot and Paycha [CDMP], [CDP]). In general it satisfies the trace defect formulas C0(A, P )− C0(A, P ) = − 1 m res(A(logP − logP )), (0.2) C0([A,A ], P ) = − 1 m res(A[A , logP ]), (0.3)