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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 SA(H){\rm{SA}}(H) of all (not necessarily bounded) self-adjoint operators on a separable infinite-dimensional Hilbert space HH. In this paper we study the domain equivalence relation EdomSA(H)E_{\rm{dom}}^{{\rm{SA}}(H)} given by AEdomSA(H)B⇔domA=domBAE_{\rm{dom}}^{{\rm{SA}}(H)}B\Leftrightarrow {\rm{dom}}{A}={\rm{dom}}{B} and determine its exact Borel complexity: EdomSA(H)E_{\rm{dom}}^{{\rm{SA}}(H)} is an FΟƒF_{\sigma} (but not KΟƒK_{\sigma}) equivalence relation which is continuously bireducible with the orbit equivalence relation Eβ„“βˆžRNE_{\ell^{\infty}}^{\mathbb{R}^{\mathbb{N}}} of the standard Borel group β„“βˆž=β„“βˆž(N,R)\ell^{\infty}=\ell^{\infty}(\mathbb{N},\mathbb{R}) on RN\mathbb{R}^{\mathbb{N}}. This, by Rosendal's Theorem, shows that EdomSA(H)E_{\rm{dom}}^{{\rm{SA}}(H)} is universal for KΟƒK_{\sigma} equivalence relations. Moreover, we show that generic self-adjoint operators have purely singular continuous spectrum equal to R\mathbb{R}.Comment: 10 pages, added more detail of the proof of Proposition 3.8 after the referee's suggestio

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