When is a Pseudo-differential operator trace class or in Dixmier ideal? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:15:00Z http://mathoverflow.net/feeds/question/119898 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119898/when-is-a-pseudo-differential-operator-trace-class-or-in-dixmier-ideal When is a Pseudo-differential operator trace class or in Dixmier ideal? Asghar Ghorbanpour 2013-01-25T23:46:59Z 2013-01-28T05:09:59Z <p>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 </p> <p>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 $t&lt; s-d$.</p> <p>See for example Lemma 1.3.4, Gilkey's book <a href="http://books.google.ca/books/about/Invariance_Theory_The_Heat_Equation_and.html?id=RgW9i29_p7sC" rel="nofollow">Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem</a>.</p> <p>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 $d&lt;0$.</p> <p>Now my question is </p> <p><strong>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 $\mathcal{L}^{(p,q)}(L^2(M))$?</strong></p> <p>Thanks</p> http://mathoverflow.net/questions/119898/when-is-a-pseudo-differential-operator-trace-class-or-in-dixmier-ideal/120013#120013 Answer by Bazin for When is a Pseudo-differential operator trace class or in Dixmier ideal? Bazin 2013-01-27T11:33:54Z 2013-01-27T11:33:54Z <p>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 $L^2$).</p> http://mathoverflow.net/questions/119898/when-is-a-pseudo-differential-operator-trace-class-or-in-dixmier-ideal/120079#120079 Answer by Rafe Mazzeo for When is a Pseudo-differential operator trace class or in Dixmier ideal? Rafe Mazzeo 2013-01-28T05:06:53Z 2013-01-28T05:06:53Z <p>For a comprehensive account of what you are looking for, see the book by Simon Scott Traces and determinants of pseudodifferential operators''</p>