ar X iv : 1 31 2 . 65 02 v 1 [ m at h . FA ] 2 3 D ec 2 01 3 AROUND THE VAN DAELE – SCHMÜDGEN THEOREM

For a bounded non-negative self-adjoint operator acting in a complex, infinitedimensional, 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 ∩R = M ∩R = {0}. We show how the existence of such subspaces leads to various pathological properties of unbounded self-adjoint operators related to von Neumann theorems [31]–[33]. We revise the von Neumann-Van Daele-Schmüdgen assertions [31], [39], [36] 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 〈T1, T2〉 of its closed densely defined restrictions Tk ⊂ T such that dom (T ∗Tk) = {0} (⇒ domT 2 k = {0}) k = 1, 2 and domT1 ∩ domT2 = {0}, domT1+̇domT2 = domT .