On overdetermined elliptic systems of pseudodifferential equations on a compact manifold

We consider the overdetermined elliptic system of pseudodifferential equations
$$\begin{equation} Pu=f \end{equation}$$
on a smooth compact manifold $X$. We prove that for system (1) there exists a pseudodifferential compatibility operator $Q$ such that the condition $Qf=0$ is a necessary and sufficient condition for the normal solvability of the system in Sobolev–Slobodetskii spaces.
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