Journal of Operator Theory
Volume 75, Issue 2, Spring 2016 pp. 367-386.
Nonseparability and von Neumann's theorem for domains of unbounded operatorsAuthors: 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)∩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 UR∩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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