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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:

Theorem 8.2

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  • $\begingroup$ How many questions run through your mind in a given day? $\endgroup$ – mathworker21 Nov 14 '18 at 12:00
  • $\begingroup$ Can you give some more context and/or an example where this kind of terminology is used? $\endgroup$ – MaoWao Nov 14 '18 at 12:31
  • $\begingroup$ @MaoWao It's used in Theorem 8.2 of Chapter 4 in Markov Processes: Characterization and Convergence by Ethier and Kurtz. $\endgroup$ – 0xbadf00d Nov 14 '18 at 12:46
  • $\begingroup$ There is a notion of pre-semigroups for which multi-valued generators make sense (defined in the expected way). However, I don't see how this fits into the context of this theorem. In fact, what you wrote down is exactly the usual single-valued generator of $(T_t)$, if I'm not mistaken. $\endgroup$ – MaoWao Nov 14 '18 at 13:42
  • $\begingroup$ @MaoWao No, it's not. If $E=\mathcal L^\infty(\mathbb R)$ and $$(T(t)f)(x):=f(x+t)\;\;\;\text{for }x\in\mathbb R\text{ and }f\in E.$$ Then, $$(0,g)\in\left\{(f,g)\in E\times E:T(t)f-f=\int_0^tT(s)g\:{\rm d}s\text{ for all }t\ge0\right\}$$ for all $g\in E$ with $g=0$ almost everywhere. $\endgroup$ – 0xbadf00d Nov 14 '18 at 13:53

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