TITLE: When is a Pseudo-differential operator trace class or in Dixmier ideal?
QUESTION [7 upvotes]: Let's denote the set of all Pseudo-differential operators with symbol of “order” $d$  by $\Psi_d(M)$ and Sobolev space on $M$ by $H_s(M)$. It is known that  
If $P\in\Psi_d(M)$ Then $P$ extends to a continuous map $P:H_{s}(M)\to H_{s-d}(M)$ for all $s$. 
Moreover, since the natural inclusion $H_s\to H_t$, for $s>t$ is compact,  $P:H_{s}(M)\to H_{t}(M)$ is compact operator if $t< s-d$.
See for example Lemma 1.3.4, Gilkey's book Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem.
In special case, when $s=0$, $L^2(M)=H_0(M)$, $P:L^2(M)\to L^2(M)$ is continuous if $d\leq 0$. and it is compact if $d<0$.
Now my question is 
when is $P:L^2(M)\to L^2(M)$  trace class? 
and when is it in Dixmier ideal $\mathcal{L}^{1,\infty}(L^2(M))$ or in general $\mathcal{L}^{(p,q)}(L^2(M))$?
Thanks

REPLY [6 votes]: For a comprehensive account of what you are looking for, see the book by Simon Scott ``Traces and determinants of pseudodifferential operators''