  In general, it is a non trivial task to determine the adjoint $S^*$ of an
unbounded operator $S$ acting between two Hilbert spaces. We provide necessary
and sufficient conditions for a given operator $T$ to be identical with $S^*$.
In our considerations, a central role is played by the operator matrix
$M_{S,T}=\left(\begin{array}{cc} I & -T\\ S & I\end{array}\right)$. Our
approach has several consequences such as characterizations of closed, normal,
skew- and selfadjoint, unitary and orthogonal projection operators in real or
complex Hilbert spaces. We also give a self-contained proof of the fact that
$T^*T$ always has a positive selfadjoint extension.
