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

Published online by Cambridge University Press:  29 October 2015

Hiroshi Ando*
Affiliation:
Erwin Schrödinger International Institute for Mathematical Physics, 2 Stock, Boltzmanngasse 9, 1090 Wien, Austria
Yasumichi Matsuzawa
Affiliation:
Department of Mathematics, Shinshu University, 6-Ro, Nishi-nagano, Nagano 380-8544, Japan (myasu@shinshu-u.ac.jp)
*
*Present address: Department of Mathematics and Informatics, Chiba University, 1–33 Yayoicha, Inage, Chiba, 263-8522, Japan (hiroando@s.chiba-u.ac.jp)
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Abstract

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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 -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)–1u* - (B-i)–1 is compact] is shown to be smooth.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2015