Let $(M,g)$ be a Riemannian Manifold and $L^2$ the Hilbert space given by the volume form associated to the metric. Let $L\_0^2$ be the subspace which is orthogonal to the constant functions. When is a pseudodifferential operator on $M$ a positive operator on $L^2\_0$?


For second order operators the Laplacian $\Delta$ is the main example.


For order zero, the obvious examples are multiplication by $f$ where $f \in C^\infty(M)$ is a smooth function and $f > 0$. Conversely if $f < 0$ anywhere then it is clear that the multiplication operator is not positive.


If $A$ is positive on $L^2\_0$ then


$(\Delta^{p/2} A \Delta^{p/2} v, v) = (A \Delta^{p/2} v, \Delta^{p/2} v) > 0$


for $v \in L^2\_0$ non-zero. So we can use the Laplacian as a sort of natural way to change the order of a given positive operator. Note that the principle symbol of such an operator is $||\xi||^{p}\sigma(A)(x,\xi)$.


How else can I construct more positive pseudodifferential operators? So far I can only come up with operators whose symbols in a fiber look like $||\xi||^{p}f(x)$. I am looking for "more interesting" symbols, such as those whose restriction to the co-sphere at a point is non-constant.


Ideally of course I would just like a global criterion for a symbol to quantize to a positive operator, but something tells me that this is a hard problem. If it is any easier, I would also be interested in specific examples, like the sphere with the round metric.



Criteria for Positivity of Pseudoddifferential Operators on Manifolds
I think you may be looking for this paper:


Symplectic geometry and positivity of pseudo-differential operators
C. Fefferman† and D. H. Phong


Abstract


In this paper we establish positivity for pseudo-differential operators under a condition that is essentially also necessary. The proof is based on a microlocalization procedure and a geometric lemma.


<http://www.pnas.org/content/79/2/710.short>


Basically, you must require that the principal symbol be positive except in a set of small symplectic capacity (it cannot contain a symplectically embedded unit cube). 


