Let us consider a self-adjoint elliptic pseudodifferential operator $P \in OPS^2$ on a compact manifold $M$ such that $spec(P) \subset (0, \infty)$. Is the norm $(Pu, u)^{1/2}$ on $H^1(M)$ equivalent to $||u||\_{H^1(M)}$, the usual Sobolev norm? A reference would be greatly appreciated.



Sobolev spaces on compact manifolds
Yes. Take the functional square root $Q:=\sqrt{P}$. It is a 1st order, elliptic, selfadjoint *positive* $\psi$do on $M$ according to the results of Seeley. Then $(Pu,u)^{1/2}=\Vert Qu\Vert\_{L^2}$, $\forall u\in C^\infty (M)$. The operator $Q$ induces a continuous *bijective* linear map


$$Q:H^1(M)\to L^2(M). $$ 


Using the open mapping theorem we deduce that there exist $0<c<C$ such that


$$c \Vert u\Vert\_{H^1}\leq \Vert Qu\Vert\_{L^2}\leq C \Vert u\Vert\_{H^1},\;\;\forall u\in H^1(M). $$


