# Journal of Operator Theory

Volume 75, Issue 2, Spring 2016 pp. 367-386.

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

**Authors**:
A.F.M. ter Elst (1) and Manfred Sauter (2)

**Author institution:**(1) Department of Mathematics, The University of Auckland, Private bag 92019,
Auckland 1142, New Zealand

(2) Institute of Applied Analysis, Ulm University,
89069 Ulm, Germany

**Summary: **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.

**DOI: **http://dx.doi.org/10.7900/jot.2015apr29.2073

**Keywords: **operator range, nonseparable Hilbert space, disjoint operator ranges,
von Neumann's theorem

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