# How are the two versions of the Lumer-Phillips theorem given in the books of Engel/Nagel and Pazy related?

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.