On Borel equivalence relations related to self-adjoint operators

TY - JOUR

T1 - On Borel equivalence relations related to self-adjoint operators

AU - Ando, Hiroshi

AU - Matsuzawa, Yasumichi

PY - 2015

Y1 - 2015

N2 - In a recent work, we initiated the study of Borel equivalence relations defined on the Polish space SA(H) of self-adjoint operators on a Hilbert space H, focusing on the difference between bounded and unbounded operators. In this paper, we show how the difficulty of specifying the domains of self-adjoint operators is reflected in Borel complexity of associated equivalence relations. More precisely, we show that the equality of domains, regarded as an equivalence relation on SA(H), is continously bireducible with the orbit equivalence relation of the standard Borel group ℓ∞(N) on NN. Moreover, we show that generic self-adjoint operators have purely singular continuous spectrum equal to ℝ.

AB - In a recent work, we initiated the study of Borel equivalence relations defined on the Polish space SA(H) of self-adjoint operators on a Hilbert space H, focusing on the difference between bounded and unbounded operators. In this paper, we show how the difficulty of specifying the domains of self-adjoint operators is reflected in Borel complexity of associated equivalence relations. More precisely, we show that the equality of domains, regarded as an equivalence relation on SA(H), is continously bireducible with the orbit equivalence relation of the standard Borel group ℓ∞(N) on NN. Moreover, we show that generic self-adjoint operators have purely singular continuous spectrum equal to ℝ.

U2 - 10.7900/jot.2014may24.2030

DO - 10.7900/jot.2014may24.2030

M3 - Journal article

SN - 0379-4024

VL - 74

SP - 183

EP - 194

JO - Journal of Operator Theory

JF - Journal of Operator Theory

IS - 1

ER -