Consider [Yosida, Functional analysis, reprint of the edition 1980] at p.251, in particular the proof of the corollary (of Lumer Philips theorem) which states that if $A \colon D(A) \subset X \to R(A)\subset X $ is a densely defined closed linear operator on a Banach space $X$ and such that $A$, $A^*$ are both dissipative, then $A$ generates a contraction semigroup.
In this short proof Yosida proves that $(I-A)^{-1} \colon R(I-A) \subset X \to X$ is closed and continuous. Then he wants to prove that $R(I-A)=X$ and he does it by contradiction.
But the fact that $R(I-A)=X$ does not immediately follow from the closed graph theorem? Indeed by the closed graph th. as $(I-A)^{-1} \colon R(I-A) \subset X \to X$ is closed and continuous, we should have immediately $X=D((I-A)^{-1})=R(I-A)$. Am I wrong?