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

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Title: Weyl-von Neumann Theorem and Borel Complexity of Unitary Equivalence Modulo Compacts of Self-Adjoint Operators
Authors: Ando, Hiroshi Browse this author
Matsuzawa, Yasumichi Browse this author
Keywords: Weyl-von Neumann Theorem
Self-adjoint operators
Turbulence.
Issue Date: 30-Apr-2014
Journal Title: Hokkaido University Preprint Series in Mathematics
Volume: 1053
Start Page: 1
End Page: 20
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.
Type: bulletin (article)
URI: http://hdl.handle.net/2115/69857
Appears in Collections:理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics

Submitter: 数学紀要登録作業用

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