\to 0$ strongly in $L^2_\mathrm{loc}([0,\infty);\, {\cal
H})$, in other words, for all $f \in {\cal H}$ and every finite $T>0$ we have $$\label{inv-gen}
 \int_0^T \|(I + S(it ,\tau))^{-1}f - (I + itH_P)^{-1}Pf\|^2\, dt
  \rightarrow 0\,, \quad \tau \to 0\,.$$

We postpone the proof of Lemma \[L1\] to the next section. For the moment we will accept its claim and use it to show that it implies the symmetric case (\[product-s\]) of the product formula in Theorem \[main\].

To this end, let $\{m_n\}$ be a strictly increasing sequence of positive integers, i.e. a subsequence of the sequence of all positive integers. We have only to show that there exists a subsequence $\{n'\}$ in any such sequence $\{m_n\}$ along which (\[product-s\]) holds. Then by a standard argument we can conclude that (\[product-s\]) actually holds along the sequence of all positive integers $n$. For if this were not the case, there would exist a subsequence $\{n'\}$ of strictly increasing positive integers along which (\[product-s\]) does not converge. However, we see that there is a subsequence $\{n''\}$ of $\{n'\}$ along which the convergence takes place to the same limit, which is a contradiction.

Fix $\{m_n\}$ and $f \in {\cal H}$. Lemma \[L1\] holds, in 