Around the Van Daele–Schmüdgen Theorem

Yury Arlinskii̇̆ (1) , Valentin A. Zagrebnov (2)

1 Department of Mathematical Analysis
2 I2M - Institut de Mathématiques de Marseille

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.

Mots clés: lifting; shorted operator; Operator range; von Neumann theorem; Van Daele--Schmüdgen theorem; parallel addition; Friedrichs extension; Krein extension

Domaines: Analyse fonctionnelle [math.FA]

Yury Arlinskii̇̆, Valentin A. Zagrebnov. Around the Van Daele–Schmüdgen Theorem. Integral Equations and Operator Theory, 2015, 81 (1), pp.53-95. ⟨10.1007/s00020-014-2143-z⟩. ⟨hal-00922017⟩