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

Abstract

A classical theorem of von Neumann asserts that every unbounded self-adjoint operator $A$ in a \textit{separable} Hilbert space is unitarily equivalent to an operator $B$ 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 $\cR$ in a general Hilbert space 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.