This is a problem and solution from Rene Schilling's Brownian Motion.
Let $(P_t)_{t\ge 0}$ and $(T_t)_{t\ge 0}$ be two Feller semigroups with generators $A$ and $B$, respectively.
Show that $\frac{d}{ds} P_{t-s}T_s = -P_{t-s}AT_s + P_{t-s} BT_s$.
From the solution below, I'm having trouble seeing how we get $III \to -P_{t-s}AT_su$ as $h\to 0$. Indeed, for $AT_su$ to make sense, we would need $T_su \in D(A)$, however, how do we guarantee this? We have
$$III=-\frac{P_h -id}{h}P_{t-s-h}T_su.$$
The only theorem I see from this book relevant to this part is that if $u\in D(A)$ then $P_tu\in D(A)$ and $(d/dt) P_tu=AP_tu = P_tAu$. But to apply this, we would need to ensure that $P_{t-s-h}T_su \in D(A),$ or $T_su \in D(A)$ and taking $P_{t-s-h}$ on the left side. How can we get this limit?
