On Borel equivalence relations related to self-adjoint operators

In a recent work, the authors studied various Borel equivalence relations defined on the Polish space $\mathrm{S}\mathrm{A}(H)$ 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 $E_{\mathrm{d}\mathrm{o}\mathrm{m}}^{\mathrm{S}\mathrm{A}(H)}$ given by $A{E}_{\mathrm{d}\mathrm{o}\mathrm{m}}^{\mathrm{S}\mathrm{A}(H)}B\iff \mathrm{d}\mathrm{o}\mathrm{m}A=\mathrm{d}\mathrm{o}\mathrm{m}B$ and determine its exact Borel complexity: $E_{\mathrm{d}\mathrm{o}\mathrm{m}}^{\mathrm{S}\mathrm{A}(H)}$ is an $F_{\sigma }$ (but not $K_{\sigma }$) equivalence relation which is continuously bireducible with the orbit equivalence relation $E_{{\mathrm{\ell }}^{\mathrm{\infty }}}^{{\mathbb{R}}^{\mathbb{N}}}$ of the standard Borel group ${\mathrm{\ell }}^{\mathrm{\infty }}={\mathrm{\ell }}^{\mathrm{\infty }}(\mathbb{N},\mathbb{R})$ on ${\mathbb{R}}^{\mathbb{N}}$. This, by Rosendal’s Theorem, shows that $E_{\mathrm{d}\mathrm{o}\mathrm{m}}^{\mathrm{S}\mathrm{A}(H)}$ is universal for $K_{\sigma }$ equivalence relations. Moreover, we show that generic self-adjoint operators have purely singular continuous spectrum equal to $\mathbb{R}$.