  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.
