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,M-T = (I -T S I). 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.