Let $(M,g)$ be a compact Riemannian manifold and $\Delta_g$ its Laplace operator.
Let $\varphi$ be a test function on $\mathbf{R}_{>0}$. We define the operator on, say, $L^2(M)$
$$T_{\varphi}(u) :=\int_0^{\infty}e^{it\sqrt{\Delta_g}}u\varphi(t)dt.$$
Is $T_{\varphi}$ trace-class? I.e. can one set $$\text{tr}(T_{\varphi}) := \int_0^{\infty}\sum_{\lambda\in\text{Sp}(\Delta_g)}e^{it\sqrt{\lambda}}\varphi(t)dt$$ and prove it converges?
Here $\text{Sp}(\Delta_g)$ is the spectrum of $\Delta_g$, $i$ is the imaginary unit.
Does the integral defined using $e^{t\sqrt{\lambda}}$ as opposed to $e^{it\sqrt{\lambda}}$, converge? In other words, if we defined a variant of $T_{\varphi}$ using $e^{t\sqrt{\Delta_g}}$ as opposed to $e^{it\sqrt{\Delta_g}}$, would $T_{\varphi}$ be traceable?
EDIT: pages 5,6,7 of this paper appear to contain a possible answer in a special case, but the details are terse (to me at least). For example, I'm surprised the series $V_p(t)$ in loc.cit. converges absolutely. I thought it would only as a distribution. My guess is that the reason why one needs distributional convergence is that there will be a convergence problem when $t$ tends to $0$, or perhaps when the absolute value of the eigenvalues of $\Delta_g$ is small (and somehow these are not an issues for $V_p(t)$?)
Any insight would be very helpful!