The local and global parts of the basic zeta coefficient for operators on manifolds with boundary

Abstract

For operators on a compact manifold $X$ with boundary $\partial X$, the basic zeta coefficient $C_0(B, P_{1,T})$ is the regular value at $s=0$ of the zeta function $\Tr(B P_{1,T}^{-s})$, 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 $P_{1,T}$ is a realization of an elliptic differential operator $P_1$, having a ray free of eigenvalues. Relative formulas (e.g. for the difference between the constants with two different choices of $P_{1,T}$) have been known for some time and are local. We here determine $C_0(B, P_{1,T})$ itself, showing how it is put together of local residue-type integrals (generalizing the noncommutative residue of Wodzicki, Guillemin, Fedosov-Golse-Leichtnam-Schrohe) 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 $\log P_{1,T}$. Since the complex powers of $P_{1,T}$ 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 parameter-dependent boundary operators can be used.