Functional calculus for Safarov pseudo-differential operators

Given a smooth, closed Riemannian manifold ( M , g ) (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 Ψ ρ , δ m ( Ω κ , ∇ , τ ) \Psi _{\rho , \delta }^m\left (\Omega ^\kappa , \nabla , \tau \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 ζ \zeta -functions