Around the Van Daele--Schmüdgen theorem

2013/12/23 by Arlinskii, Yury, Zagrebnov, Valentin
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1312.6502

Abstract

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\capR=M^⊥\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 \citeNeumann--\citeNeumann2. We revise the von Neumann-Van Daele-Schmüdgen assertions \citeNeumann, \citeDaele, \citeschmud 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⊂ T such that \dom(T^* T_k)=\0\ (⇒ \dom T_k²=\0\) k=1,2 and \dom T₁∩\dom T₂=\0\, \dom T₁+\dom T₂=\dom T.

Related

• Some Remarks on the Operator T^*T
• On the unitary equivalence of absolutely continuous parts of self-adjoint extensions
• On a class of Model Hilbert Spaces
• Notes on the Krupa-Zawisza Ultrapower of Self-Adjoint Operators
• Non-negative perturbations of non-negative self-adjoint operators