COMPLEX POWERS OF RESOLVENTS OF PSEUDODIFFERENTIAL OPERATORS

In a work from 1995 by G. Grubb and R. Seeley, a calculus of weakly parametric pseudodifferential operators on closed manifolds was introduced and used to obtain complete asymptotic expansions of traces of resolvents and heat operators associated with the Atiyah-Patodi-Singer problem. The present paper establishes a generalization allowing not only anisotropic homogeneity in the symbols, but also including symbols of noninteger, even complex, powers of A—λ. The operators in the calculus have complete asymptotic trace expansions in the parameter (when of trace class), with polynomial and logarithmic terms.