I am reading B. Simon's "Kato's inequality and the comparison of semigroups", and I am having troubles understanding a part of the proof of Theorem 1 therein, that goes as follows:
Let $A,B$ be semibounded self-adjoint operators on some $L^2(M, \mu)$, such that $\mathrm e ^{-t A} f \ge 0$ for all $f \ge 0$. Denote by $D(A)$ the domain of $A$, and by $Q(A)$ the domain of the quadratic form associated to $A$. If $| \mathrm e ^{-t B} u | \le \mathrm e ^{-t A} |u|$ for all $u$, then $u \in D(B)$ implies $|u| \in Q(A)$.
The proof should be elementary and straightforward, but I am having difficulties with a certain passing to the limit. Specifically, if $u \in D(B)$, then it is clear from the general theory of $1$-parameter semigroups that
$$\lim _{t \to 0} \ \left( u, \frac {1 - \mathrm e ^{-t B}} t u \right) = (u, Bu) \ .$$
(Parantheses denote scalar product.)
What is not clear to me, though, is why
$$\lim _{t \to 0} \ \left( |u|, \frac {1 - \mathrm e ^{-t A}} t |u| \right) = (|u|, A |u|) \ .$$
More specifically, why does the limit even exist? And why does $|u| \in D(A)$, in order to be able to write $A |u|$? Or am I misreading Simon's ideas?
