The Weyl-von Neumann theorem and Borel complexity of unitary equivalence modulo compacts of self-adjoint operators

The Weyl-von Neumann theorem asserts that two bounded self-adjoint operators A, B on a Hilbert space H are unitarily equivalent modulo compacts, i. e. uAu* + K = B for some unitary u is an element of U(H) and compact self-adjoint operator K, if and only if A and B have the same essential spectrum: sigma(ess)(A) = sigma(ess)(B). We study, using methods from descriptive set theory, the problem of whether the above Weyl-von Neumann result can be extended to unbounded operators. We show that if H is separable infinite dimensional, the relation of unitary equivalence modulo compacts for bounded self-adjoint operators is smooth, while the same equivalence relation for general self-adjoint operators contains a dense G(delta)-orbit but does not admit classification by countable structures. On the other hand, the apparently related equivalence relation A similar to B double left right arrow there exists u is an element of U(H) [u(A-i)-(1)u*-(B-i)(-1) is compact] is shown to be smooth.