Weyl-von Neumann Theorem and Borel Complexity of Unitary Equivalence Modulo Compacts of Self-Adjoint Operators

DOI HANDLE Open Access

Abstract

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 2 U(H) and compact self-adjoint operator K, if and only if A and B have the same essential spectra: ess(A) = ess(B). In this paper we consider to what extent the above Weyl-von Neumann's result can(not) be extended to unbounded operators using descriptive set theory. We show that if H is separable in nite-dimensional, this equivalence relation for bounded self-adjoin operators is smooth, while the same equivalence relation for general self-adjoint operators contains a dense G -orbit but does not admit classi cation by countable structures. On the other hand, apparently related equivalence relation A B , 9u 2 U(H) [u(A 􀀀 i) 􀀀1u 􀀀 (B 􀀀 i) 􀀀1 is compact], is shown to be smooth.

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Details 詳細情報について

  • CRID
    1390290699772187648
  • NII Article ID
    120006459742
  • DOI
    10.14943/84197
  • HANDLE
    2115/69857
  • Text Lang
    en
  • Data Source
    • JaLC
    • IRDB
    • CiNii Articles
  • Abstract License Flag
    Allowed

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