I am reading the book "Methods of Modern Mathematical Physics, Vol 1", and going through the chapter VIII (Unbounded operators) I just wondered if the relation $(\ker A)^{ \perp}=\overline{A^{*}(D(A^{*}))}$ reamins true, for some possibly unbounded operator $A:D(A)\subset H \to H$, densely defined. I know the proof when $A$ is closed, but I struggle finding a proof or a counter example in the remaining case.
Edit: My proof was wrong when $A$ is closable, but for $A$ closed goes as follows:
$\ker A = \ker A^{**} = R(A^{*})^{\perp} \implies \ker A^{\perp}= \overline{R(A^{*})}$.