Around the Van Daele–Schmüdgen Theorem

Abstract

For a bounded non-negative self-adjoint operator acting in a complex, infinite-dimensional, separable Hilbert space \({\mathcal{H}}\) and possessing a dense range \({\mathcal{R}}\) we propose a new approach to characterisation of phenomenon concerning the existence of subspaces \({\mathfrak{M}\subset \mathcal{H}}\) such that \({\mathfrak{M}\cap\mathcal{R}=\mathfrak{M}^\perp\cap\mathcal{R}=\{0\}}\). We show how the existence of such subspaces leads to various pathological properties of unbounded self-adjoint operators related to von Neumann theorems (J Reine Angew Math 161:208–236, 1929; Math Ann 102:49–131, 1929; Math Ann 102:370–427, 1929). We revise the von Neumann–Van Daele-Schmüdgen assertions (J Reine Angew Math 161:208–236, 1929; J Oper Theory 11:379–393, 1984; Can J Math 36:1245–1250, 1982) 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 \({\langle T_1 , T_2 \rangle}\) of its closed densely defined restrictions \({T_k\subset T}\) such that \({{\rm dom}(T^* T_{k})=\{0\} (\Rightarrow{\rm dom} T_{k}^2=\{0\})\, k=1,2}\) and \({{\rm dom} T_1\cap{\rm dom} T_2=\{0\}, {\rm dom} T_1\dot+{\rm dom} T_2={\rm dom}T}\).