Zeta and eta functions for Atiyah-Patodi-Singer operators

This paper concerns Dirac-type operatorsP on manifoldsX with boundary which are “product-type” near the boundary. That is, $$P = \sigma \left( {\frac{\partial }{{\partial x_n }} + A} \right)$$ for a unitary morphism σ and a self-adjoint first-order operatorA onbdry(X);x n denotes the normal coordinate. For a realizationP B defined by a boundary operatorB of Atiyah-Patodi-Singer type, the paper gives a complete description of the singularities of the traces of the meromorphic continuations of Γ(s)D(Δ i )−s and Γ(s)DP(Δ i )−s where Δ1 =P B * P B , Δ2 =P B P B * , andD is any differential operator onX which is tangential and independent of 4x n nearbdry(X). This implies expansions for the associated heat kernels and resolvents, containing the usual powers (with both “local” and “global” coefficients) together with logarithmic terms.