\] Let Assumptions \[A0.0\] and \[assumption on weight for generation\] hold. Then $G^{(w)}=\overline{A^{(w)}+B^{(w)}}$ is the generator of a substochastic $C_0$-semigroup, $(S^{(w)}(t))_{t \ge 0}$, on $\ell_w^0$. 