2014/02/27 by Hiroshi Ando, Ando, Hiroshi, Yasumichi Matsuzawa +1
Mathematics · #03E15 #34L05 #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Logic (math.LO) #Operator Algebras (math.OA) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1402.6947
openalex publication_date 2014/02/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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∈ U(H) and compact self-adjoint operator K, if and only if A and B have the same essential spectra: σ_\rmess(A)=σ_\rmess(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 Gδ-orbit but does not admit classification by countable structures. On the other hand, apparently related equivalence relation A∼ B⇔ ∃ u∈ U(H) [u(A-i)-1u^*-(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.