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

TL;DR

This paper extends Guillemin's work on complex powers and zeta functions of elliptic pseudodifferential operators to operators acting on sections of vector bundles over von Neumann algebras, broadening the scope of the theory.

Contribution

It generalizes Guillemin's results from smooth functions on closed manifolds to operators on sections of vector bundles over von Neumann algebras.

Findings

Extended the theory of complex powers to new operator classes

Generalized zeta function analysis for operators on vector bundles

Broadened the applicability of elliptic pseudodifferential operator results

Abstract

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.