ON BOREL EQUIVALENCE RELATIONS RELATED TO SELF-ADJOINT OPERATORS

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 (N) on RN. Moreover, we show that generic self-adjoint operators have purely singular continuous spectrum equal to R.