  We provide necessary and sufficient conditions for a pair $S,T$ of Hilbert
space operators in order that they satisfy $S^*=T$ and $T^*=S$. As a main
result we establish an improvement of von Neumann's classical theorem on the
positive self-adjointness of $S^*S$ for two variables. We also give some new
characterizations of self-adjointness and skew-adjointness of operators, not
requiring their symmetry or skew-symmetry, respectively.
