For elliptic differential operators $P$ on a compact manifold $M$, we have the formula

$$\mathrm{ind}(D) = \mathrm{tr}(e^{-tP^*P}) - \mathrm{tr}(e^{-tPP^{\star}})$$

I would think that this holds for Pseudo-Differential operators of positive order as well, but no text book states that. Is it not true? If not, what goes wrong?

ellipticps.d.o. on acompactmanifold. – Liviu Nicolaescu Oct 7 '12 at 15:14