Question

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

Answer 1

An operator is trace class whenever it is the product of Hilbert-Schmidt operators. There is a simple characterization of Hilbert-Schmidt operators pseudodifferential operators: a pseudodifferential operator with symbol $a$ is HS if and only if $a$ belongs to $L^2$ of the cotangent bundle (this is equivalent also to the fact that the kernel of the operator is $L^2$).

Answer 2

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