2023-03-27T05:01:37Zhttps://eprints.lib.hokudai.ac.jp/dspace-oai/requestoai:eprints.lib.hokudai.ac.jp:2115/698572022-11-17T02:08:08Zhdl_2115_45007hdl_2115_116Weyl-von Neumann Theorem and Borel Complexity of Unitary Equivalence Modulo Compacts of Self-Adjoint OperatorsAndo, HiroshiMatsuzawa, Yasumichiopen accessWeyl-von Neumann TheoremSelf-adjoint operatorsTurbulence.410Weyl-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.Department of Mathematics, Hokkaido University2014-04-30engdepartmental bulletin paperVoRhttps://doi.org/10.14943/84197http://hdl.handle.net/2115/6985710.14943/84197Hokkaido University Preprint Series in Mathematics1053120https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/69857/1/pre1053.pdfapplication/pdf200.11 KB2014-04-30