On operators which are adjoint to each other

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.

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