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

TL;DR

This paper investigates the limitations of von Neumann's theorem for unbounded operators in nonseparable Hilbert spaces and characterizes when operator ranges can be transformed to have trivial intersection with their unitary images.

Contribution

It extends the understanding of von Neumann's theorem by characterizing operator ranges that can be made disjoint via unitary transformations in general Hilbert spaces.

Findings

Von Neumann's theorem does not extend directly to nonseparable spaces.

A characterization of operator ranges admitting a unitary making them disjoint.

Analysis of stability properties of such operator ranges.

Abstract

A classical theorem of von Neumann asserts that every unbounded self-adjoint operator $A$ in a separable Hilbert space $H$ is unitarily equivalent to an operator $B$ in $H$ such that $D(A)\cap D(B)=\{0\}$. Equivalently this can be formulated as a property for nonclosed operator ranges. We will show that von Neumann's theorem does not directly extend to the nonseparable case. In this paper we prove a characterisation of the property that an operator range $\mathcal{R}$ in a general Hilbert space $H$ admits a unitary operator $U$ such that $U\mathcal{R}\cap\mathcal{R}=\{0\}$. This allows us to study stability properties of operator ranges with the aforementioned property.