Let $H$ be a $\mathbb R$-Hilbert space and $A$ be a closed linear operator on $H$. Can we conclude that $A$ is an isomorphism between $\mathcal D(A)$ and $H$?
My intuition is that this is clearly wrong, but this seems to be what's been claimed in this book after the proof of Lemma IV.5.3. While they particularly consider the Stokes operator, they claim that the isomorphismness follows by the fact that $A$ is closed.
Any bounded opearator whose domain is the whole of $H$ is closed. So the zero operator is a counter-example.