Let $M$ be a compact smooth manifold without boundary, and let $m \in \mathbb{R}$. Let $A \in \Psi^m_{\mathrm{cl}}(M)$ be a scalar classical pseudodifferential operator acting on $C^\infty(M)$. Assume that $A$ is elliptic, meaning that its principal symbol $\sigma_m(A): T^*M \setminus 0 \to \mathbb{C}$ satisfies