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?



ellipticity and invertible differential operators
Yes, this is correct. You can find a more general result in



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> Topics in pseudo-differential operators. 1969 Pseudo-Diff. Operators (C.I.M.E., Stresa, 1968) pp. 167–305 Edizioni Cremonese, Rome
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More precisely see Corollary 1, Chap. IV, page 251 in the above reference.


