Let $U \subset \mathbb{R}^n$ be open, let $m \in \mathbb{R}$, and for each $r \in \mathbb{R}$ let $\Psi_{\mathrm{ps}}^r(U)$ denote the space of properly supported pseudodifferential operators of order $r$ on $U$, acting on $C_c^\infty(U)$. Define the finite-order properly supported pseudodifferential algebra by

\begin{align*} \Psi_{\mathrm{ps}}^*(U) := \bigcup_{r \in \mathbb{R}} \Psi_{\mathrm{ps}}^r(U) \end{align*}

and define its properly supported smoothing ideal by

\begin{align*} \Psi_{\mathrm{ps}}^{-\infty}(U) := \Psi_{\mathrm{ps}}^*(U) \cap \Psi^{-\infty}(U). \end{align*}

Let

\begin{align*} \mathcal{A}_{\mathrm{ps}}(U) := \Psi_{\mathrm{ps}}^*(U) / \Psi_{\mathrm{ps}}^{-\infty}(U) \end{align*}

be the quotient algebra, with multiplication induced by operator composition and identity class represented by the identity operator $I: C_c^\infty(U) \to C_c^\infty(U)$. If $P \in \Psi_{\mathrm{ps}}^m(U)$ is elliptic, then $[P] \in \mathcal{A}_{\mathrm{ps}}(U)$ is invertible. More precisely, if $Q \in \Psi_{\mathrm{ps}}^{-m}(U)$ is any properly supported two-sided parametrix for $P$, meaning that there exist $R_1, R_2 \in \Psi_{\mathrm{ps}}^{-\infty}(U)$ such that

\begin{align*} QP = I + R_1 \end{align*}

and

\begin{align*} PQ = I + R_2, \end{align*}

then $[Q] \in \mathcal{A}_{\mathrm{ps}}(U)$ is the two-sided inverse of $[P]$:

\begin{align*} [Q][P] = [I] \end{align*}

and

\begin{align*} [P][Q] = [I]. \end{align*}

In particular, the inverse class $[P]^{-1}$ is independent of the chosen properly supported parametrix.