Let $X$ Banach space. As mentioned in the answer to question
Are m-dissipative operators closed?
a dissipative operator $A \colon D(A) \subset X \to X$ is closed if and only if there is $\lambda_0>0$ such that $R(\lambda_0 I-A)$ is closed (hence $R(\lambda I-A)$ is closed for every $\lambda>0$). This implies that if $A$ is m-dissipative (i.e. dissipative and $R(\lambda I-A)=X$ for some $\lambda$) then it is closed.
This is in contradiction with [R. Philips, DISSIPATIVE OPERATORS AND HYPERBOLIC SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS] where at p 201 in the footnote 6 the author provides an example of a maximally dissipative operator on a Hilbert space which is not closed.
Where is the problem? In Hilbert spaces m-dissipativity is equivalent to maximally dissipativity.