of Theorem \[main\] {#s-proof}
=========================

We present the argument for $\varepsilon =1$, the case $\varepsilon =-1$ can be treated similarly. We first prove (\[product-s\]) in (a), and next (\[product-ns\]), (\[product-ns2\]) in (b).

\(a) Let us begin with the symmetric product case and prove the formula (\[product-s\]) with $\varepsilon =1$. We will check the convergence in (\[product-s\]) on an arbitrary compact $t$-interval in the closed right half-line $[0,\infty)$. The proof for $t$-intervals in the closed left half-line $(-\infty, 0]$ is analogous, and in addition, it can be included in the case $\varepsilon = -1$ with the convergence in (\[product-s\]) on compact $t$-intervals of the closed right half-line $[0,\infty)$.

Put $Q(t) := I-P(t)$ and $Q:= Q(0) = I- P(0) = I-P$, where $I$ is the identity operator on ${\cal H}$. Since $H$ is nonnegative by assumption, there exists a spectral measure $E(d\lambda)$ on the nonnegative real line such that $H = \int _{0-}^{\infty} \lambda\,
E(d\lambda)$. For $\zeta \in {\mathbb{C}}$ with $\hbox{\rm
Re}\,\zeta \geq 0$ and $\tau > 0$, we put $$\label{red-evol}
 F(\zeta, \tau) = P(\tau)\,e^{-\zeta\tau H}P(\tau)\,,$$ which is a contraction, and $$\label{red-evol-der}
 S(\zeta,\tau) = \tau^{-1}[I - F(\zeta,\tau)] =
 \tau^{-1}[I- P(\tau)\,e^{-\zeta\tau H}P(\tau)],$$ which exists as a bounded operator on ${\cal H}$ with $\hbox{\rm
