The Weyl–von Neumann theorem and Borel complexity of unitary equivalence modulo compacts of self-adjoint operators. Issue 6 (29th October 2015)

• Record Type: Journal Article
• Authors: Ando, Hiroshi; Matsuzawa, Yasumichi
• Is Part Of: Proceedings. Volume 145:Issue 6(2015)
• Page Start: 1115
• Page End: 1144
• Publication Date: 2015-10-29
• Subjects: Weyl–von Neumann theorem, -- self-adjoint operators, -- turbulence; Primary 47B25, -- Secondary 03E15; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0308210515000293
• Languages: English
• ISSNs: 0308-2105

Abstract

Abstract : 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 𝜖 u ( H ) and compact self-adjoint operator K, if and only if A and B have the same essential spectrum: σess ( A ) = σess ( 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 Gδ -orbit but does not admit classification by countable structures. On the other hand, the apparently related equivalence relation A ~ B ⇔ ∃ u 𝜖 U ( H ) [ u(A-i) –1 u * - ( B-i ) –1 is compact] is shown to be smooth.