Functional calculus for Safarov pseudo-differential operators

Abstract

Given a smooth, closed Riemannian manifold $(M,g)$ equipped with a linear connection $\nabla$ (not necessarily metric), we develop the holomorphic functional calculus for operators belonging to the global pseudo-differential classes $\hat{I}{¨}_{\ddot{I},\hat{I}´}^m\left(\hat{I}{\text{c◯}}^{\hat{I}}º,\nabla ,\ddot{I}„\right)$ introduced by Safarov. As a consequence of our main result, we establish a Szegö type-theorem, derive asymptotic expansion of the heat kernel trace, and calculate some associated spectral $\hat{I}\P$-functions.