The Local and Global Parts of the Basic Zeta Coefficient for Pseudodifferential Boundary Operators

For operators on a compact manifold X with boundary ∂X, the basic zeta coefficient is the regular value at s = 0 of the zeta function Tr(BP 1,T ), where B = P+ + G is a pseudodifferential boundary operator (in the Boutet de Monvel calculus), and P1,T is a realization of an elliptic differential operator P1, having a ray free of eigenvalues. In the case ∂X = ∅, Paycha and Scott showed how the basic zeta coefficient is the sum of a global Hadamard finite-part integral defined from B and a local residue-like term (à la Wodzicki’s noncommutative residue) defined from B logP1. We here establish a generalization to the case ∂X 6= ∅, with similar global and local elements, and moreover several new local residue-like terms coming from the boundary; the logarithm of P1,T plays an important role. For this we develop resolvent methods, since complex powers of realizations do not fit naturally into the Boutet de Monvel calculus. Introduction. The value of the zeta function at s = 0 plays an important role in the analysis of geometric invariants of operators on manifolds. For the zeta function ζ(P1, s) = TrP −s 1 (extended meromorphically to C) of a classical elliptic pseudodifferential operator (ψdo) P1 on a closed manifold X , the value at s = 0 is a fundamental ingredient in index formulas. For the generalized zeta function ζ(A, P1, s) = Tr(AP −s 1 ), there is a pole at s = 0 and the regular value behind it serves as a “regularized trace” or “weighted trace” (cf. e.g. Melrose et al. [MN, MMS], Paycha et al. [CDMP, CDP]); it is likewise important in index formulas. Much is known for the case of closed manifolds: The residue of ζ(A, P1, s) at 0 is proportional to Wodzicki’s noncommutative residue of A ([W], see also Guillemin [Gu]). The regular value at 0 (which we call the basic zeta value) equals the Kontsevich-Vishik [KV] canonical trace in special cases, and in general there are defect formulas for it (formulas relating two different choices of the auxiliary operator P1, and formulas where A is a commutator), in terms of noncommutative residues of related expressions involving logP1. The basic zeta coefficient itself has recently been shown by Paycha and Scott [PS] to satisfy a formula with elements of canonical trace-type integrals (finite-part integrals in the sense of Hadamard) defined from A as well as noncommutative residue-type integrals defined from A logP1. The finite-part integral contributions are global, in the sense that they depend on the full operator; the residue-type contributions are local, in the sense that they depend only on the strictly homogeneous symbols down to a certain order.