# ellipticity and invertible differential operators

Let $(M,g)$ be a closed, compact Riemannian manifold. Let $P$ be a $2r$th order pseudo-differential operator, where $r \in \Bbb{R}_+$. Suppose that the differential equation $Pu=f$ has a unique $H^r(M)$ solution for all $f$ in the dual space of $H^r(M)$. Does it follow that $P$ is elliptic?

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