\infty$, because $I =
P(1/n) + Q(1/n)$ and $P(\tau)
\stackrel{\scriptsize{s}}{\longrightarrow}P$ as $\tau \rightarrow
0$. It follows that $ \int_0^T \|Q(1/n)(e^{-itH/n}P(1/n))^n f\|^2
\, dt \longrightarrow 0$ as $n \rightarrow \infty$. Thus for any $v(\cdot) \in L^2_\mathrm{loc}([0,\infty); {\cal H})$ and every $T>0$ we have, again by (\[product-s\]), $$\begin{aligned}
 &&\int_0^T \langle v(t), (e^{-itH/n}P(1/n))^n f \rangle  \, dt\\
 &=&\int_0^T \langle  P(1/n)v(t),
              (P(1/n)e^{-itH/n}P(1/n))^n f \rangle  \, dt \\
 &&\qquad\quad
     + \int_0^T \langle Q(1/n)v(t),
               Q(1/n)(e^{-itH/n}P(1/n))^n f \rangle  \, dt\\
 &&\longrightarrow
      \int_0^T \langle v(t), e^{-itH_P}Pf \rangle  \, dt\end{aligned}$$ as $n\rightarrow \infty$. It means that $\{(e^{-itH/n}P(1/n))^n
f\}$ converges to $e^{-itH_P}Pf$ weakly in $L^2_\mathrm{loc}([0,\infty); {\cal H})$ together with all the seminorms, and therefore the convergence is strong in $L^2_\mathrm{loc}([0,\infty); {\cal H})$. This yields the formula (\[product-ns2\]).

It remains to prove Lemma \[L1\] on which the above arguments were based.

Proof of Lemma \[L1\] {#s-proof-of-L1}
=====================

To demonstrate (\[inv-gen\]), we shall use the Vitali theorem – see, e.g., [@HP] – for holomorphic functions and employ arguments analogous to those used in Kato’s paper [@Ka2] for the self-adjoint Trotter product formula with the form sum of a pair of nonnegative self-adjoint operators. We do it in three steps.

*I.* In the first step we will show the following lemma.

\[L41\] For 