\rangle
 = \langle v, P\hat{u}(t) \rangle
  + \langle t^{1/2}H^{1/2}Pv, h(t) \rangle,$$ because by spectral theorem $$\begin{aligned}
 \|H(t\tau)^{1/2}(P(\tau)\!-\! P)v\|
  &\!=\!&  \|H(t\tau)^{1/2}(I+H)^{-1/2}(I+H)^{1/2}(P(\tau)\!-\!P)v\|\\
  &\!\leq\!& \|(I+H)^{1/2}(P(\tau)\!-\! P)v\|\,,\end{aligned}$$ which tends to zero since $P(\tau) \stackrel{\scriptsize{s}}
{\rightarrow} P$ as $\tau \to 0$, $\:D[H^{1/2}P(\tau)] \supset
D[H^{1/2}P]$ and $\|H^{1/2}P(\tau)v\| \to \|H^{1/2}Pv\|$ for $v
\in D[H^{1/2}P]$ by assumption[^1]. Hence $H^{1/2}P\hat{u}(t)\in D[H^{1/2}P]$ and $$\begin{aligned}
 \label{fu-id}
Pf &=& P\hat{u}(t) + t^{1/2}(H^{1/2}P)^* h(t)
= \hat{u}(t) + t(H^{1/2}P)^* (H^{1/2}P) \hat{u}(t) \nonumber\\
&=& \hat{u}(t) + tH_P \hat{u}(t)\,,\end{aligned}$$ because $D[H^{1/2}P]$ is supposed to be dense. Applying once again the standard argument mentioned after Lemma \[L1\] to all the three families we conclude that the weak convergence in (\[weakconv\]) takes place independently of a sequence $\{\tau_n(t)\}$ chosen.

On the other hand, we infer from (\[ufinprod\]) that $$\begin{aligned}
 \langle \hat{u}(t),f \rangle
&\geq& \liminf \|\hat{u}(t,\tau)\|^2
  + \liminf \tau^{-1}\|Q(\tau)\hat{u}(t,\tau)\|^2 \\
 &&\qquad\quad
     + \liminf \|t^{1/2}H(t\tau)^{1/2}P(\tau)\hat{u}(t,\tau)\|^2 \\
&\geq& \|\hat{u}(t)\|^2 + \|g_0(t)\|^2 + \|h(t)\|^2 \\
&=&  \|\hat{u}(t)\|^2 + \|t^{1/2}H^{1/2}P\hat{u}(t)\|^2 \\
&=& \|\hat{u}(t)\|^2 + t\|H_P^{1/2}\hat{u}(t)\|^2\end{aligned}$$ with $\liminf$ taken along $\tau \rightarrow 0$. Since by (\[fu-id\]) the left-hand side of the above inequality is equal to $$\langle \hat{u}(t),f \rangle = \langle \hat{u}(t),Pf \rangle
= \|\hat{u}(t)\|^2 + t\langle \hat{u}(t), H_P\hat{u}(t) \rangle
= \|\hat{u}(t)\|^2 + t\|H_P^{1/2}\hat{u}(t)\|^2,$$ we see that the norms of these vectors converge to 