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

Abstract

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.

References in corpus (1)

• A generalisation of the form method for accretive forms and operators

Cited by in corpus (2)

• Everything is possible for the domain intersection dom T \cap dom T*
• On the mappings connected with parallel addition of nonnegative operators