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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.

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

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