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.