On Borel equivalence relations related to self-adjoint operators

2014/05/05 by Ando, Hiroshi, Matsuzawa, Yasumichi
#FOS: Mathematics #Functional Analysis (math.FA) #Logic (math.LO) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1405.0860

Abstract

In a recent work, the authors studied various Borel equivalence relations defined on the Polish space \rmSA(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_\rmdom^\rmSA(H) given by AE_\rmdom^\rmSA(H)B⇔ \rmdomA=\rmdomB and determine its exact Borel complexity: E_\rmdom^\rmSA(H) is an F_σ (but not K_σ) equivalence relation which is continuously bireducible with the orbit equivalence relation E_ℓ^∞^ℝ^ℕ of the standard Borel group ℓ^∞=ℓ^∞(ℕ,ℝ) on ℝ^ℕ. This, by Rosendal's Theorem, shows that E_\rmdom^\rmSA(H) is universal for K_σ equivalence relations. Moreover, we show that generic self-adjoint operators have purely singular continuous spectrum equal to ℝ.

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