On Borel equivalence relations related to self-adjoint operators

Abstract

In a recent work, the authors studied various Borel equivalence relations defined on the Polish space $\mathrm{SA}(H)$ 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 $E_{\mathrm{dom}}^{\mathrm{SA}(\mathrm{H})}$ given by $A{E}_{\mathrm{dom}}^{\mathrm{SA}(\mathrm{H})}B\iff \mathrm{dom}A=\mathrm{dom}B$ and determine its exact Borel complexity: $E_{\mathrm{dom}}^{\mathrm{SA}(\mathrm{H})}$ 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 $E_{\mathrm{dom}}^{\mathrm{SA}(\mathrm{H})}$ is universal for $K_{\sigma }$ equivalence relations. Moreover, we show that generic self-adjoint operators have purely singular continuous spectrum equal to $\mathbb{R}$.

10 pages, added more detail of the proof of Proposition 3.8 after the referee's suggestion