In Higsons's book Analytic K-Homology there is a section (subsection (b) in 2.8 "Geometric Examples of Extensions", starting from page 46) which discusses the following exact sequence called pseudodifferential operator extension (sometimes referred to as symbol sequence):
$0 \longrightarrow \mathcal{K}(L^2(M)) \longrightarrow \Psi^{0}_{phg}(M) \xrightarrow{\sigma_0} C_0(S^*M) \longrightarrow 0$
where $M$ is a smooth compact manifold without boundary, $S^*M$ the unit cosphere bundle, $\mathcal{K}(L^2(M))$ denotes the compact operators on $L^2(M)$, $\Psi^{0}_{phg}(M)$ are the order $0$ classical pseudodifferential operators and $\sigma_0$ is the principal symbol map.
I've seen this statement quite often but always without proof. Could someone give me a reference for this?
Also, it is known that (by Rellich) every negative order pseudodifferential operator extends to a compact operator $L^2(M) \to L^2(M)$. In particular, $\Psi^{-1}_{phg}(M) \subseteq \mathcal{K}(L^2(M))$. How do we prove that $\overline{\Psi^{-1}_{phg}(M)} = \mathcal{K}(L^2(M))$ where on the left side we have the norm closure?