$m\times m$ *matrix pseudodifferential operators* with coefficients in ${\mathcal{A}}$. Its product is determined by the following formula ($n\in{\mathbb{Z}}$, $A\in\operatorname{Mat}_{m\times m}{\mathcal{A}}$): $$\partial^n\circ A=\sum_{k\in{\mathbb{Z}}_+}\binom{n}{k}A^{(k)}\partial^{n-k}\,.
$$ We say that a non-zero $A(\partial)=\sum_{n\leq N}A_n\partial^n\in\operatorname{Mat}_{m\times m}{\mathcal{A}}((\partial^{-1}))$ has *order* $\operatorname{ord}(A(\partial))=N$ if $A_N\neq0$. We denote by $\operatorname{Mat}_{m\times m}{\mathcal{A}}((\partial^{-1}))_N$ the space of all matrix pseudodifferential operators of order less than or equal to $N$. We say that $A(\partial)$ is *monic* if $A_N={\mathbbm{1}}_m$.

The *residue* of $A(\partial)\in\operatorname{Mat}_{m\times m}{\mathcal{A}}((\partial^{-1}))$ is, by definition, $$\label{eq:res}
\operatorname{Res}_{\partial} A(\partial)=A_{-1}\ (=\mbox{ coefficient of }\partial^{-1})\,.$$ The *adjoint* of $A(\partial)\in\operatorname{Mat}_{m\times m}{\mathcal{A}}((\partial^{-1}))$ is, by definition, $$A(\partial)^*=\sum_{n\leq N}(-\partial)^n\circ A_n^t\,,$$ where $A_n^t$ denotes the transpose matrix of $A_n$.

We have the subalgebras $\operatorname{Mat}_{m\times m}{\mathcal{A}}[\partial]$ of *matrix differential operators*, and $\operatorname{Mat}_{m\times m}{\mathcal{A}}[[\partial^{-1}]]\partial^{-1}$ of *matrix integral operators*. Every matrix pseudodifferential operator $A(\partial)$ can be decomposed uniquely as $A(\partial)=A(\partial)_++A(\partial)_-$, where $A(\partial)_+\in\operatorname{Mat}_{m\times m}{\mathcal{A}}[\partial]$ and $A(\partial)_-\in\operatorname{Mat}_{m\times m}{\mathcal{A}}[[\partial^{-1}]]\partial^{-1}$.

We let ${\mathbb{F}}[\partial,\partial^{-1}]\circ\operatorname{Mat}_{m\times m}{\mathcal{A}}\subset\operatorname{Mat}_{m\times m}{\mathcal{A}}((\partial^{-1}))$ be the space of all pseudodifferential operators of the form $\sum_{n\in{\mathbb{Z}}}\partial^n\circ A_n$, where all but finitely many elements $A_n\in\operatorname{Mat}_{m\times m}{\mathcal{A}}$ are zero. We also let $\operatorname{Mat}_{m\times m}{\mathcal{A}}[[\partial,\partial^{-1}]]\supset\operatorname{Mat}_{m\times m}{\mathcal{A}}((\partial^{-1}))$ be the space of all formal series in $\partial,\partial^{-1}$ with coefficients in $\operatorname{Mat}_{m\times m}{\mathcal{A}}$ (it is not an algebra).

Recall the following simple facts about matrix pseudodifferential operators.

\[prop:roots\] Let $A(\partial)$ be a monic matrix 