Nonseparability and von Neumann's theorem for domains of unbounded operators (Q2835240)

The article deals with the modification of the classical von Neumann theorem on unbounded self-adjoint operators in a separable Hilbert space. This theorem asserts that every unbounded self-adjoint operator \(A\) in a separable Hilbert space \(H\) is unitarily equivalent to an operator \(B\) such that \({\mathcal D}(A) \cap {\mathcal D}(B) = \{0\}\). The examples presented by the authors show that the statement of this theorem does not extend to the nonseparable case. However, the statement of the von Neumann theorem can be formulated as the property of the range \({\mathcal R}(A)\) of the operator \(A\), moreover, the following statement is true: if \({\mathcal R}\) is a nonclosed operator range (i.e., the range of a bounded operator on \(H\)), then there exists a unitary operator \(U\) such that \(U{\mathcal R} \cap {\mathcal R} = \{0\}\). The main result of the authors is the following: if \(H\) is an infinite-dimensional Hilbert space, then the following properties of the range \({\mathcal R}\) are equivalent: (i) there exists a unitary operator \(U\) such that \(U{\mathcal R} \cap {\mathcal R} = \{0\}\) and (ii) for every closed subspace \(K \subset {\mathcal R}\), the inequality \(\dim K \leq \dim K^+\) holds. Also, it is proved that the set of operators \(T \in L(H)\) which admit a unitary operator \(U\) such that \(U {\mathcal R}(T) \cap {\mathcal R}(T) = \{0\}\) is closed with respect to the uniform operator norm, and some similar results. In the article, one can find a number of interesting examples; however, there are some vague places, in particular, the class of operators under consideration is vague.