We define strong generator of the semigroup $P_t$ as
$ A f := \lim_{t \to 0} \frac{P_tf-f}{t}$ if the limit exists and is in $C_0$
and weak generator of semigroup $P_t$ as
$ A_w f(x) := \lim_{t \to 0} \frac{P_tf(x)-f(x)}{t}$ if the limit exists and is in $C_0$ (Pointwise limit).
If also we assume, that operator $\frac{P_tf(x)-f(x)}{t}$ is uniformly bounded, then is it true that $A_w = A$?