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Suppose $X$ is a non-compact manifold. Let $P$ be an order-$0$ pseudodifferential operator on $X$ and $f:L^2(X)\rightarrow L^2(X)$ a compact operator. I'm wondering:

1) Is $P + f$ always a pseudodifferential operator?

2) If so, does it have the same principal symbol as $P$?

Many thanks.

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    $\begingroup$ This would imply that any compact operator is $\Psi$DO. This is not the case, since, for instance, the kernel of a $\Psi$DO is smooth outside the diagonal. So simply take a kernel which is not smooth outside the diagonal, but $L^2$ and you get a compact operator which is not $\Psi$DO. $\endgroup$ – user1688 Jun 15 '17 at 8:50
  • $\begingroup$ I see, that answers the first part. As for the second part, do you know if theres exists a compact pseudodifferential operators of non-negative order? $\endgroup$ – geometricK Jun 15 '17 at 20:52
  • $\begingroup$ See mathoverflow.net/questions/96925/… $\endgroup$ – user1688 Jun 16 '17 at 7:03

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