Around the Van Daele-Schmudgen Theorem

摘要

For a bounded non-negative self-adjoint operator acting in a complex, infinite-dimensional, separable Hilbert space and possessing a dense range we propose a new approach to characterisation of phenomenon concerning the existence of subspaces such that . 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-Schmudgen 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 of its closed densely defined symmetric/self-adjoint operator T infinitely many pairs < T-1, T-2 > of its closed densely defined restrictions T-k subset of T such that dom (T* T-k) = {0}(double right arrow dom T-k(2) = {0}) k = 1, 2 and dom T-1 boolean AND dom T-2 = {0}, dom T-1 + dom T-2 = domT.