Absolute convergence of trace-Gibbs bounded correlation series

Assume the Axiom of Choice. Let $\mathcal{H}\mathrm{\ne }\{0\}$ be separable, $K=K^{\ast }$ a densely defined Hamiltonian bounded below with energy unit J, and $\beta$ lie in an open interval $I\subset (0,\mathrm{\infty })$ measured in J${}^{-1}$ on which $e^{-sK}$ is trace class for every $s\in I$. Set $Z_{\beta }=\mathrm{Tr}(e^{-\beta K})$, ${\sigma }_{\beta }=Z_{\beta }^{-1}{e}^{-\beta K}$, and $U(t)=e^{-itK\mathrm{/}\mathrm{\hslash }}$ with $\mathrm{\hslash }>0$ in J s and $t$ in s. By Self-adjoint trace-Gibbs normalization and form-energy optimization, $K$ has a complete orthonormal energy basis $(u_i)$, $K{u}_i=E_i{u}_i$, with Gibbs weights $r_i=e^{-\beta E_i}\mathrm{/}{Z}_{\beta }>0$ and ${\sum }_i{r}_i=1$. For bounded $A,B$, put $A_{ij}=⟨A{u}_j,u_i⟩$, $B_{ij}=⟨B{u}_j,u_i⟩$, and ${\omega }_{ji}=(E_j-E_i)\mathrm{/}\mathrm{\hslash }$. Then $C_{AB}(t):=\mathrm{Tr}({\sigma }_{\beta }U(t{)}^{\ast }AU(t)B)={\sum }_{i,j}{r}_i{A}_{ij}{B}_{ji}{e}^{-i{\omega }_{ji}t}$ converges absolutely and uniformly on $\mathbb{R}$, with ${\sum }_{i,j}{r}_i\mathrm{\mid }{A}_{ij}\mathrm{\mid }\mathrm{\mid }{B}_{ji}\mathrm{\mid }\le \mathrm{\parallel }A\mathrm{\parallel }\mathrm{\parallel }B\mathrm{\parallel }$. Consequently $S_{AB}:=2\pi {\sum }_{i,j}{r}_i{A}_{ij}{B}_{ji}{\delta }_{{\omega }_{ji}}$ is a finite complex Borel measure with total variation at most $2\pi \mathrm{\parallel }A\mathrm{\parallel }\mathrm{\parallel }B\mathrm{\parallel }$, and $C_{AB}(t)=(2\pi {)}^{-1}\int e^{-i\omega t}d{S}_{AB}(\omega )$. A Dirac measure ${\delta }_{\nu }$ assigns mass one to the point $\nu$. When $A,B$ carry units $[A],[B]$, the atomic masses of the finite measure $S_{AB}$ have units $[A][B]$. For every smooth compactly supported complex-valued function $\phi :\mathbb{R}\to \mathbb{C}$, the distribution represented by $S_{AB}$ acts as $⟨S_{AB},\phi ⟩={\int }_{\mathbb{R}}\phi (\omega )d{S}_{AB}(\omega )$. In the Fourier convention, the generalized density $S_{AB}(\omega )$ relative to $d\omega$ has units $[A][B]$ multiplied by seconds; it is a sum of Dirac distributions, not an ordinary function density.