Is a dissipative closed operator maximal dissipative?

Question

This is a follow up to the following questions about m-dissipativity:

Are m-dissipative operators closed?

Are m-dissipative operators closed (II)?

Is a dissipative closed operator $A:D(A)\subset X\rightarrow X$ in the Hilbert space $X$ maximal dissipative?

Answer 1

The answer is No. For maximal dissipativity we need $A$ dissipative and the range of $\lambda I -A$ should coincide with $X$ for some $\lambda>0$. This is the case if $0$ is in the resolvent set of $A$, i.e. if $A$ is a closed and boundedly invertible operator.

Requiring closedness results in the range of $\lambda I -A$ being a subset of $X$.