An Extension of the Work of V. Guillemin on Complex Powers and Zeta Functions of Elliptic Pseudodifferential Operators

The purpose of this note is to extend the results of V. Guillemin on elliptic self-adjoint pseudodifferential operators of order one, from operators defined on smooth functions on a closed manifold to operators defined on smooth sections in a vector bundle of Hilbert modules of finite type over a finite von Neumann algebra.