NONSEPARABILITY AND VON NEUMANN'S THEOREM FOR DOMAINS OF UNBOUNDED OPERATORS

A classical theorem of von Neumann asserts that every unbounded self-adjoint operator A in a separable Hilbert space is unitarily equivalent to an operator B such that D (A) boolean AND 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 R in a general Hilbert space admits a unitary operator U such that U R boolean AND R = {0}. This allows us to study stability properties of operator ranges with the aforementioned property.