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 -