On Borel equivalence relations related to self-adjoint operators
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In a recent work, we initiated the study of Borel equivalence relations defined on the Polish space ${\rm{SA}}(H)$ of self-adjoint operators on a Hilbert space $H$, focusing on the difference between bounded and unbounded operators. In this paper, we show how the difficulty of specifying the domains of self-adjoint operators is reflected in Borel complexity of associated equivalence relations. More precisely, we show that the equality of domains, regarded as an equivalence relation on ${\rm{SA}}(H)$, is continously bireducible with the orbit equivalence relation of the standard Borel group $\ell^{\infty}(\mathbb{N})$ on $\mathbb{R}^{\mathbb{N}}$. Moreover, we show that generic self-adjoint operators have purely singular continuous spectrum equal to $\mathbb{R}$.
Original language | English |
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Journal | Journal of Operator Theory |
Volume | 74 |
Issue number | 1 |
Pages (from-to) | 183-194 |
ISSN | 0379-4024 |
DOIs | |
Publication status | Published - 2015 |
ID: 138510048