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- W1818754909 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 $uin mathcal{U}(H)$ and compact self-adjoint operator $K$, if and only if $A$ and $B$ have the same essential spectra: $sigma_{rm{ess}}(A)=sigma_{rm{ess}}(B)$. 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 infinite-dimensional, this equivalence relation for bounded self-adjoin operators is smooth, while the same equivalence relation for general self-adjoint operators contains a dense $G_{delta}$-orbit but does not admit classification by countable structures. On the other hand, apparently related equivalence relation $Asim BLeftrightarrow exists uin mathcal{U}(H) [u(A-i)^{-1}u^*-(B-i)^{-1}$ is compact], is shown to be smooth. Various Borel or co-analytic equivalence relations related to self-adjoint operators are also presented." @default.
- W1818754909 created "2016-06-24" @default.
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- W1818754909 date "2014-02-27" @default.
- W1818754909 modified "2023-09-26" @default.
- W1818754909 title "Weyl-von Neumann Theorem and Borel Complexity of Unitary Equivalence Modulo Compacts of Self-Adjoint Operators" @default.
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- W1818754909 doi "https://doi.org/10.48550/arxiv.1402.6947" @default.
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