Functional calculus for Safarov pseudo-differential operators

<p> Given a smooth, closed Riemannian manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper M comma g right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(M,g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> equipped with a linear connection <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nabla"> <mml:semantics> <mml:mi mathvariant="normal"> ∇ </mml:mi> <mml:annotation encoding="application/x-tex">\nabla</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (not necessarily metric), we develop the holomorphic functional calculus for operators belonging to the global pseudo-differential classes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Psi Subscript rho comma delta Superscript m Baseline left-parenthesis normal upper Omega Superscript kappa Baseline comma nabla comma tau right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi mathvariant="normal"> Ψ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> ρ </mml:mi> <mml:mo>,</mml:mo> <mml:mi> δ </mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mi> κ </mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∇ </mml:mi> <mml:mo>,</mml:mo> <mml:mi> τ </mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\Psi _{\rho , \delta }^m\left (\Omega ^\kappa , \nabla , \tau \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta"> <mml:semantics> <mml:mi> ζ </mml:mi> <mml:annotation encoding="application/x-tex">\zeta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -functions </p>