  For a {bounded} non-negative self-adjoint operator acting in a complex,
infinite-dimensional, separable Hilbert space H and possessing a dense range R
we propose a new approach to characterisation of phenomenon concerning the
existence of subspaces M\subset H such that M\capR=M^\perp\capR=\{0\}. We show
how the existence of such subspaces leads to various {pathological} properties
of {unbounded} self-adjoint operators related to von Neumann theorems
\cite{Neumann}--\cite{Neumann2}. We revise the von Neumann-Van
Daele-Schm\"udgen assertions \cite{Neumann}, \cite{Daele}, \cite{schmud} to
refine them. We also develop {a new systematic approach, which allows to
construct for any {unbounded} densely defined symmetric/self-adjoint operator T
infinitely many pairs of its closed densely defined restrictions T_k\subset T
such that \dom(T^* T_{k})=\{0\} (\Rightarrow \dom T_{k}^2=\{0\}$) k=1,2 and
\dom T_1\cap\dom T_2=\{0\}, \dom T_1\dot+\dom T_2=\dom T.
