# When is a Pseudo-differential operator trace class or in Dixmier ideal?

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

• It is trace class when it is an operator of order $-k$, $k>\dim M$. For a proof see Section 4.3 of these notes www3.nd.edu/~lnicolae/Pseudo.pdf – Liviu Nicolaescu Jan 26 '13 at 14:43
• Moreover, by the Connes trace formula (alainconnes.org/docs/action88.pdf), if your operator is of order $-k$ for $k = \dim M$, then it is in the Dixmier ideal; indeed, it is measurable (in the sense of the theory of Dixmier traces), and the (unique value of the) Dixmier trace is given by the Wodzicki residue of your operator. – Branimir Ćaćić Jan 28 '13 at 10:30
• Thanks for the useful references. However, I still wondering if there is a bound like $l$ such that pseudo differential operator of the order $d$ is in the Dixmier ideal (not necessary measurable) when $d<l$. of course $l$ should be in $[-k,0)$ where $k=dim M$. – Asghar Ghorbanpour Jan 28 '13 at 17:24

For a comprehensive account of what you are looking for, see the book by Simon Scott Traces and determinants of pseudodifferential operators''
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 $L^2$).