On Borel equivalence relations related to self-adjoint operators

TL;DR

This paper analyzes the Borel complexity of the domain equivalence relation on self-adjoint operators in a Hilbert space, showing it is a universal $K_{\sigma}$ relation and exploring spectral properties of generic operators.

Contribution

It determines the exact Borel complexity of the domain equivalence relation and establishes its universality among $K_{\sigma}$ relations, also examining spectral characteristics of generic operators.

Findings

The domain equivalence relation is $F_{\sigma}$ but not $K_{\sigma}$.

It is continuously bireducible with the orbit equivalence relation of $ ext{ extlbrackdbl} ext{ exttt{l}^{ ext{ exttt{ extb}}}} ext{ extgreater}$ on $ ext{ exttt{R}}^{ ext{ exttt{ extb}}}$.

Generic self-adjoint operators have purely singular continuous spectrum equal to $ ext{ exttt{R}}$.

Abstract

In a recent work, the authors studied various Borel equivalence relations defined on the Polish space ${\rm{SA}}(H)$ of all (not necessarily bounded) self-adjoint operators on a separable infinite-dimensional Hilbert space $H$. In this paper we study the domain equivalence relation $E_{\rm{dom}}^{{\rm{SA}}(H)}$ given by $AE_{\rm{dom}}^{{\rm{SA}}(H)}B\Leftrightarrow {\rm{dom}}{A}={\rm{dom}}{B}$ and determine its exact Borel complexity: $E_{\rm{dom}}^{{\rm{SA}}(H)}$ is an $F_{\sigma}$ (but not $K_{\sigma}$) equivalence relation which is continuously bireducible with the orbit equivalence relation $E_{\ell^{\infty}}^{\mathbb{R}^{\mathbb{N}}}$ of the standard Borel group $\ell^{\infty}=\ell^{\infty}(\mathbb{N},\mathbb{R})$ on $\mathbb{R}^{\mathbb{N}}$. This, by Rosendal's Theorem, shows that $E_{\rm{dom}}^{{\rm{SA}}(H)}$ is universal for $K_{\sigma}$ equivalence relations. Moreover, we show that generic self-adjoint operators have purely singular continuous spectrum equal to $\mathbb{R}$.