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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?

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You only need $P$ to be an elliptic ps.d.o. on a compact manifold. –  Liviu Nicolaescu Oct 7 '12 at 15:14
Ok, two things that I keep forgetting to mention. Thank you! –  Kofi Oct 7 '12 at 23:31
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