If $A:D\subset H \rightarrow H$ on a Hilbert space $H$ is a unbounded densely defined operator. Then its adjoint exists and
$$\left\langle \begin{pmatrix}x \\ Ax \end{pmatrix},\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}\begin{pmatrix}y \\ A^*y \end{pmatrix}\right\rangle=0$$
Where $V:= \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}$ is a unitary operator on $H\times H$.
Therefore $G_{A^*}\subseteq (V G_A)^\perp$.
How to prove $G_{A^*}\supseteq (V G_A)^\perp$?
Here $G_A$ and $G_{A^*}$ denote the graph of $A$ and $A^*$.