a fixed $\zeta = t>0$, $$\label{realconv}
 (I+ S(t,\tau))^{-1} \stackrel{\scriptsize{s}}{\longrightarrow}
 (I + tH_P)^{-1}P \quad \mathrm{as} \quad \tau \to 0\,.$$

[*Proof:*]{} The argument will be analogous to that in [@Ka2], and indeed, validity of the result in the particular case when our projection-valued function is constant is remarked in [@Ka2 Eq. (5.2), p. 194].

For $\zeta = t\tau >0$ we have from (\[Htau\]) $H(t\tau) = (t\tau)^{-1}[I- e^{-t\tau H}]$, which is a bounded, nonnegative and self-adjoint operator on ${\cal H}$. It allows us to rewrite $$\begin{aligned}
S(t,\tau) &\!=\!& \tau^{-1}[I-  P(\tau)(I-t\tau
H(t\tau))P(\tau)] \\
         &\!=\!& \tau^{-1}Q(\tau) + t P(\tau)H(t\tau)P(\tau)\,,\end{aligned}$$ which is in this case also a bounded and nonnegative self-adjoint operator. To prove (\[realconv\]) take any $f \in {\cal H}$ and put $\hat{u}(t,\tau) := (I+ S(t,\tau))^{-1}f$, so that $$\label{fdecomp}
 f =(I + S(t,\tau))\hat{u}(t,\tau) = [I + \tau^{-1}Q(\tau)
 + tP(\tau)H(t\tau)P(\tau)]\hat{u}(t,\tau)\,.$$ Then we have $$\label{ufinprod}
 \langle \hat{u}(t,\tau),f \rangle = \|\hat{u}(t,\tau)\|^2
                + \tau^{-1}\|Q(\tau)\hat{u}(t,\tau)\|^2
    + t\|H(t\tau)^{1/2}P(\tau)\hat{u}(t,\tau)\|^2.$$ Thus the families $\{\hat{u}(t,\tau)\},\,
\{\tau^{-1/2}Q(\tau)\hat{u}(t,\tau)\}$ and $\{t^{1/2}H(t\tau)^{1/2} P(\tau) \hat{u}(t,\tau)\}$ are all bounded by $\|f\|$ for all $t > 0$, uniformly as $\tau\to 0$, and therefore they are weakly compact in ${\cal H}$. It follows that for each fixed $t > 0$ there exists a sequence $\{\tau_n (t)\}$ with $\tau_n(t) \to 0$ as $n\to 