Commutator representation of certain smoothing operators

Question

I have a question regarding the classical trace $\text{Tr} \colon \Psi^{-\infty}(S^1)\to \mathbb C$ on pseudodifferential operators of infinite negative order (i.e. smoothing operators), defined over the unit circle $S^1$ and acting on smooth functions. The definition of $\text{Tr}$ is given by $$ \text{Tr}(A) = \int_{S^1} k_A(x,x) $$ where $k_A$ denotes the Schwartz kernel of $A$.

I know that $\text{Tr}$ vanishes on commutators, and I would like to use this fact to study the trace of the an operator $C$ of the form $$ C(\psi) := f(\theta)\frac{\partial}{\partial \theta}\big(g(\theta)e^{\triangle} \,\psi\big) $$ where $f(\theta)$ and $g(\theta)$ are fixed smooth functions on $S^1$ (more details are known about these functions though they are rather technical in nature and I hope that these will not be relevant). Furthermore, $e^\triangle$ denotes the heat operator associated with a Laplace type operator $\triangle$. The question that I would like to understand is whether or not $C$ can generically be written as a (sum of) commutator(s). Hints and suggestions for survey reading would be helpful.

Answer 1

Take $f=g=1$, $\Delta$ the usual Laplace operator on $\mathbb S^1$, then $$ C_0=\partial_\theta e^{\partial_\theta^2}, $$ can be identified to the diagonal infinite matrix $ (ik e^{-k^2})_{k\in \mathbb Z} $ acting on $\ell^2(\mathbb Z)$ and has indeed a null trace. On the other hand, your $C$ is such that $$ C=f\partial_\theta g e^{\partial_\theta^2}= f[\partial_\theta, g ]e^{\partial_\theta^2} +fg \partial_\theta e^{\partial_\theta^2}=(fg'+fg) C_0. $$ Take now $ f=\sum_{k\ge 1}k^{-2} e^{ik\theta},\quad g=\sum_{k\ge 1}k^{-2} e^{-ik\theta}, $ so that we have $$ C(e^{im\theta})=\sum_{k,l\ge 1}k^{-2}l^{-2} e^{ik\theta}\partial_\theta\bigl(e^{-il\theta} e^{-m^2} e^{im\theta}\bigr)= \sum_{k,l\ge 1}k^{-2}l^{-2} e^{ik\theta}i(m-l)e^{-m^2}e^{i(m-l)\theta}, $$ and \begin{multline} \text{trace } C=\sum_{m\in \mathbb Z}e^{-m^2}\sum_{k=l\ge 1}k^{-2}l^{-2}i(m-l) =i\sum_{m\in \mathbb Z}e^{-m^2}\sum_{k\ge 1}k^{-4}(m-k) \\=i\sum_{m\in \mathbb Z}e^{-m^2}m\zeta(4) - i\sum_{m\in \mathbb Z}e^{-m^2}\zeta(3) =- i\zeta(3)\sum_{m\in \mathbb Z}e^{-m^2}\not=0. \end{multline} An iff condition for a null trace of $C$ can be found writing $$ f=\sum_k f_ke^{ik\theta},\quad g=\sum_l g_le^{il\theta}. $$