the norms of their limit vectors. It allows us to conclude that the ${{\mathcal H}}$-valued families in question, $\{\hat{u}(t,\tau)\},\,
\{\tau^{-1/2}Q(\tau)\hat{u}(t,\tau)\}\,$ and $\{t^{1/2}H(t\tau)^{1/2} P(\tau) \hat{u}(t,\tau)\}$ converge to $\hat{u}(t)$, $0$ and $t^{1/2}H^{1/2}P\hat{u}(t)$ strongly in ${\cal H} $, respectively, as $\tau\rightarrow 0$. In particular, we have shown that $Pf = (I+tH_P)\hat{u}(t)$ and $\hat{u}(t,\tau)
\stackrel{\scriptsize{s}}{\longrightarrow} \hat{u}(t) = (I+
tH_P)^{-1}Pf$, or (\[realconv\]). This proves Lemma \[L41\].

*II.* Next, for a fixed $\tau >0$, the function $\zeta\mapsto F(\zeta,\tau)$ is holomorphic in the open right half-plane $\hbox{\rm Re}\, \zeta>0$ and uniformly bounded in norm by one. This makes it possible to mimick the argument of Feldman [@Fe], which is reproduced in Chernoff’s book [@Ch2 p. 90], see also [@Fr], to conclude by means of the Vitali theorem (see, e.g., [@HP Thm 3.14.1]) that for $\hbox{\rm Re}\, \zeta>0$ $$(I+ S(\zeta, \tau))^{-1}
 \stackrel{\scriptsize{s}}{\longrightarrow} (I+ \zeta H_P)^{-1}P
 \quad \mathrm{as} \quad \tau \rightarrow 0$$ holds uniformly on compact subsets of $\hbox{\rm Re}\, \zeta >0$.

At the boundary $\hbox{\rm Re}\, \zeta =0$, or $\zeta = it$ with $t$ real, $(I+ S(\zeta, \tau))^{-1}$ still converges as $\tau \to
0$ but in a weaker sense only. Using the argument of [@Fe] based on the Poisson kernel, we can check that for each pair of 