Atiyah-Singer for pseudodifferential operators via heat kernel? The Atiyah-Singer theorem for Dirac-type operators can be proved using the heat kernel and this proof has an advantage over the proof via K-theory, because the first is local but the latter is not. But the K-theoretic proof has the advantage that it proves Atiyah-Singer not only for Dirac-type operators, but much more generally for elliptic, symmetric pseudodifferential operators (PDOs).

I was wondering whether one could do the heat kernel proof also for pseudodifferential operators?

I know that we can conclude Atiyah-Singer for PDOs from the theorem for Dirac operators by using Poincare duality (i.e., by using that on compact spin$^c$-manifolds every K-homology class can be represented by the class of a twisted Dirac operator).
But I'm asking if the proof via the heat kernel directly generalizes to PDOs? One problem that I can see is that it is often used that $e^{itD}$ has finite propagation speed, and I think that this does not hold anymore if D is merely pseudodifferential (right?). But maybe it is possible to work around that? Or what is the reason why I only encounter heat kernel proofs for Dirac type operators but not for PDOs?
 A: The answer is no. Even for the differentil elliptic operator, the heat kernel method can not give the result.
In the heat method proof, we use McKean-Singer formula,
$$\mathrm{Ind} D= \mathrm{Tr} \left[e^{-tD^*D}-e^{-tDD^*}\right]=\int_M \mathrm{Tr} [p_t(x,x)-q_t(x,x)]dx,$$
where $p_t(x,y), q_t(x,y)$ are the heat kernel.
This is always true, even for PDO.
McKean-Singer conjectured, and proved by a lot of people (Gilkey, Getzler, Bismut...) using different methods, for Dirac operator, when $t\to0$,
$$\mathrm{Tr} [p_t(x,x)-q_t(x,x)]\to a(x). \quad (1)$$
Then we can deduce the Atiyah Singer index theorem
$$\mathrm{Ind} D=\int_X a(x)dx.$$
When $t\to0$, the limit of $\mathrm{Tr} [p_t(x,x)],\mathrm{Tr} [q_t(x,x)]$ do not exist
but the limit of the difference exsit. This is a very deep result relating to the supersymmetry and the structure of Dirac operator. 
Your question is to ask if $\mathrm{Tr} [p_t(x,x)-q_t(x,x)]$ has a limit when $t\to0$ for PDO. This is not true even for general elliptic oprator. 
An example is that for the Kahler manifold $(M,\omega)$, $D=\sqrt{2}(\bar{\partial}+\bar{\partial}^*)$ is a Dirac operator. the limit in (1) exist. But when $M$ is not Kahler, the limit in (1) maybe does not exist. In Bismut's paper, A local index theorem for non Kähler manifolds http://link.springer.com/article/10.1007%2FBF01443359,
he find such limit exist if $\partial \overline{\partial}\omega=0 $, and this is almost a necessary condition.
Let's final remark that the finite propagation speed is not essential, just a trick. 
