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 $\mathbb{ R}$.
Categories: math.LO math.FA math.SP