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 $\lambda>0$?

  1. If $A$ is a closed dissipative operator, then the range of $\lambda-A$ is closed for all $\lambda>0$. In fact, if $(\lambda x_n-A x_n)$ is Cauchy, then $$ \|x_n-x_m\|\leq \frac 1 \lambda\|(\lambda-A)(x_n-x_m)\|. $$ Hence $(x_n)$ is Cauchy and thus $(Ax_n)$ is Cauchy as well. Since $A$ is closed, $\lim_{n\to\infty}x_n\in D(A)$ and $$ \lim_{n\to\infty}(\lambda-A)x_n=(\lambda-A)\lim_{n\to\infty} x_n\in \mathrm{ran}(\lambda-A). $$
  2. If a contraction semigroup has a densely defined generator, then it is strongly continuous. This result is also somewhere in the book of Engel an Nagel, but the argument is quite simple: Trajectories starting in the domain of the generator are clearly continuous, and the uniform bound $\|T_t\|\leq 1$ allows one to extend this continuity to all trajectories starting in the closure of the domain of the generator.

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