Let $E$ be a countable set in the discrete topology.
Let $(T_t)_{t \geq 0}$ be a Feller semigroup on $E$, i.e. a strongly continuous semigroup of operators on $\mathcal{C}_0(E)$ (in the topology of uniform convergence) such that
$f \geq 0 \Rightarrow T_t f \geq 0$ for all $t$
$ \| T_t \| \leq 1$ for all $t$
Does it follow that the domain of the generator of $(T_t)_{t \geq 0}$ is the whole space $\mathcal{C}_0(E)$?
Could you please give me the answer and at least some hint for proof or counterexample?