Let
- $E$ be $\mathbb R$-Banach space
- $A$ be a subspace of $E\times E$ and $$\mathcal D(A):=\left\{x\in E\mid\exists y\in E:(x,y)\in A\right\}$$
- $(T(t))_{t\ge0}$ be a contractive $C^0$-semigroup on $\overline{\mathcal D(A)}$
$A$ is considered as being a multivalued operator. What's meant by "the closure $\overline A$ of $A$ is a generator of $(T(t))_{t\ge0}$"?
Is this related to the concept of a "full generator"? My assumption is that the meaning of the sentence is $$\overline A=\left\{(f,g)\in\overline{\mathcal D(A)}\times\overline{\mathcal D(A)}:T(t)f-f=\int_0^tT(s)g\:{\rm d}s\text{ for all }t\ge0\right\}.\tag1$$
Is that correct? Is that what it's meant?
If so, what can we say about existence and uniqueness of $(T(t))_{t\ge0}$ and $A$? And is $\overline A$ "single-valued" (which means that $(0,y)\in\overline A$ implies $y=0$)?
Remark: I've seen the terminology used in Theorem 8.2 of Chapter 4 in Markov Processes: Characterization and Convergence by Ethier and Kurtz:
