Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 ${\rm{SA}}(H)$ 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 $\mathbb{R}$.

35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con
16
The categories of this paper are: stat
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
20
The categories of this paper are: acc-phys
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga
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The categories of this paper are: patt-sol
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The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc
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The categories of this paper are: funct-an
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The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph