Around the Van Daele--Schmüdgen theorem

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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⊂ H such that M=M⊥={0}. We show how the existence of such subspaces leads to various pathological properties of unbounded self-adjoint operators related to von Neumann theorems Neumann--Neumann2. We revise the von Neumann-Van Daele-Schmüdgen assertions Neumann, Daele, 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 Tₖ⊂ T such that (T* Tₖ)={0} (⇒ Tₖ²={0}) k=1,2 and T₁∩ T₂={0}, T₁+ T₂= T.