Suppose $A:\dom A(\sbe E)\rar E$ is a closed linear operator on a Hilbert space $E$. Then we have an orthogonal decomposition:
$$
E\oplus E
=\{(x,-Ax)+(A^*y,y):x\in\dom A,\,y\in\dom A^*\}
=\G(-A)\oplus\G(A^*)~.
$$
The inner product on $E\oplus E$ is given by $\la(x_1,x_2),(y_1,y_2)\ra=\la x_1,y_1\ra+\la x_2,y_\ra$.
1. For1 $x\in\dom A$ and $y\in\dom A^*$ we have:
$$
\la(x,-Ax),(A^*y,y)\ra=\la x,A^*y\ra-\la Ax,y\ra=0
$$
and thus the spaces are orthogonal.
2. Now suppose $(z,y)$ is orthogonal to $\G(-A)$, the for all $x\in\dom A$:
$$
0=\la(x,-Ax),(z,y)\ra=\la x,z\ra-\la y,Ax\ra
$$
i.e. $x\mapsto\la y,Ax\ra$ is continuous and therefore: $A^*y=z$.
3. Since $A$ is closed, the space $\G(-A)$ is closed and thus
$$
E\oplus E=\G(-A)\oplus\G(-A)^\perp=\G(-A)\oplus\G(A^*)~.
$$