A linear operator which is the generator of a strongly continuous contraction semigroup is dissipative

Question

I am reading the book of Kurtz and Ethier, and in the proof of the Hille Yosida theorem, I do not see why if a linear operator is the generator of a strongly continuous contraction semigroup it is dissipative. For this statement (in my eyes) no proof is given.

Answer 1

Denote by $A$ the generator. Note that, by Proposition 2.1 (Ethier-Kurtz), we have

$$\|\lambda (\lambda-A)^{-1} g\| \leq 1$$

for any $g$ and $\lambda>0$. Thus

$$\|g\| = \|(\lambda-A)^{-1}(\lambda-A) g\| \leq \|(\lambda-A)g\|$$

for any $g \in D(A)$ and $\lambda>0$.