  Given two linear operators $S$ and $T$ acting between Hilbert spaces
$\mathscr{H}$ and $\mathscr{K}$, respectively $\mathscr{K}$ and $\mathscr{H}$
which satisfy the relation \begin{equation*} \langle Sh, k\rangle=\langle h,
Tk\rangle, \quad h\in\dom S, \ k\in\dom T, \end{equation*} i.e., according to
the classical terminology of M.H. Stone, which are adjoint to each other, we
provide necessary and sufficient conditions in order to ensure the equality
between the closure of $S$ and the adjoint of $T.$ A central role in our
approach is played by the range of the operator matrix $M_{S,
T}=\begin{pmatrix} 1_{\dom S} & -T S & 1_{\dom T} \end{pmatrix}.$ We obtain, as
consequences, several results characterizing skewadjointness, selfadjointness
and essential selfadjointness. We improve, in particular, the celebrated
selfadjointness criterion of J. von Neumann.
