Functional calculus for Safarov pseudo-differential operators

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 $\Psi_{\rho, \delta}^m\left(\Omega^\kappa, \nabla, \tau\right)$ introduced by Safarov. As a consequence of our main result, we establish a Szeg\"o type-theorem, derive asymptotic expansion of the heat kernel trace, and calculate some associated spectral $\zeta$-functions.