Range-kernel characterizations of operators which are adjoint of each other

We provide necessary and sufficient conditions for a pair $S,T$ of Hilbert space operators in order that they satisfy $S^{\ast }=T$ and $T^{\ast }=S$. As a main result we establish an improvement of von Neumann’s classical theorem on the positive self-adjointness of $S^{\ast }S$ 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.