Let $M$ be a closed manifold and $D$ a Dirac operator on $M$. Then (the closure of) $D^2+1$ has a bounded inverse $$(D^2+1)^{-1}:L^2\rightarrow H^2,$$ where $H^2$ is the second Sobolev space. In fact, $(D^2+1)^{-1}$ is a compact operator on $L^2$ by Rellich's lemma.
Question 1: Is $(D^2+1)^{-1}$ a properly supported pseudodifferential operator if $M$ is non-compact?
Question 2: Is $(D^2+1)^{-1}$ a trace-class operator on $L^2$ (when $M$ is compact)?
Thoughts: I am not sure about the first question, but I believe the answer to the second question is no (although certainly one can raise $(D^2+1)^{-1}$ to a high enough power to get a trace-class operator), although I don't know how to show explicitly that it's not trace-class.
Thanks!