The Local and Global Parts of the Basic Zeta Coefficient for Operators on Manifolds with Boundary

For operators on a compact manifold X with boundary ∂X, the basic zeta coefficient C0(B, P1,T ) is the regular value at s = 0 of the zeta function Tr(BP −s 1,T ), where B = P+ +G is a pseudodifferential boundary operator (in the Boutet de Monvel calculus) — for example the solution operator of a classical elliptic problem — and P1,T is a realization of an elliptic differential operator P1, having a ray free of eigenvalues. Relative formulas (e.g. for the difference between the constants with two different choices of P1,T ) have been known for some time and are local. We here determine C0(B, P1,T ) itself (when P1 is of even order), showing how it is put together of local residue-type integrals (generalizing the noncommutative residues of Wodzicki, Guillemin, Fedosov-Golse-LeichtnamSchrohe) and global canonical trace-type integrals (generalizing the canonical trace of Kontsevich and Vishik, formed of Hadamard finite parts). Our formula generalizes that of Paycha and Scott, shown recently for manifolds without boundary. It leads in particular to new definitions of noncommutative residues of expressions involving logP1,T . Since the complex powers of P1,T lie far outside the Boutet de Monvel calculus, the standard consideration of holomorphic families is not really useful here; instead we have developed a resolvent parametric method, where results from our calculus of parameterdependent boundary operators can be used. 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) defined from a classical elliptic pseudodifferential operator (ψdo) P1 on a closed manifold X , having a ray free of eigenvalues, 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] 2000 Mathematics Subject Classification. 35S15, 58J42.