Let $E$ be a $\mathbb R$-Banach space and $(\mathcal D(A),A)$ be a densely-defined dissipative linear operator on $E$.
In the book of Engel and Nagel, I've found the following verison of the Lumer-Phillips theorem:
Version 1: The closure $(\mathcal D(\overline A),\overline A)$ of $(\mathcal D(A),A)$ is the generator of a contraction semigroup if and only if $\lambda\operatorname{id}_{\mathcal D(A)}-A$ has dense range for some (and hence all) $\lambda>0$.
Now, in the book of Pazy, I've found a different version:
Version 2: The range of $\lambda\operatorname{id}_{\mathcal D(A)}-A$ is $E$ for some $\lambda>0$ if and only if $(\mathcal D(A),A)$ is the generator of a strongly continuous contraction semigroup.
Question 1: How are these two versions related? For example, if $(\mathcal D(A),A)$ is the generator of a contraction semigroup, version 1 only yields that $\lambda\operatorname{id}_{\mathcal D(A)}-A$ has dense range for all $\lambda>0$. How do we see that these ranges are actually the whole space $E$? Clearly, the crucial thing must be that not only the closure $(\mathcal D(\overline A),\overline A)$ but $(\mathcal D(A),A)$ itself is the generator of a contraction semigroup.
Question 2: In version 1, there is no strong continuity claim on the semigroup. Is this a mistake in the book or are we not able to conclude the strong continuity of the contraction semigroup generated by $(\mathcal D(\overline A),\overline A)$ if $\lambda\operatorname{id}_{\mathcal D(A)}-A$ has dense range for some $\lambda>0$?
Best Answer