Suppose $A^* : D(A^*)\subset C_0(\mathbb{R}^d)\rightarrow C_0(\mathbb{R}^d)$ is the generator of a strongly continuos semigroup. Does there exists an operator $A:D(A)\subset X \rightarrow X$ for some Banach space $X$ such that $A$ is the generator of a strongly continuos semigroup and $A^*$ is its adjoint?
The major obstacle here is that there $C_0(\mathbb{R}^d)$ is not the dual space of some Banach space $X$. Maybe considering a compactification of $\mathbb{R}^d$ one can get rid off this problem and find a proper operator $A$. But I'm not sure if this is possible.