On Borel equivalence relations related to self-adjoint operators
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On Borel equivalence relations related to self-adjoint operators. / Ando, Hiroshi; Matsuzawa, Yasumichi.
I: Journal of Operator Theory, Bind 74, Nr. 1, 2015, s. 183-194.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - On Borel equivalence relations related to self-adjoint operators
AU - Ando, Hiroshi
AU - Matsuzawa, Yasumichi
PY - 2015
Y1 - 2015
N2 - 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 $\mathbb{R}$.
AB - 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 $\mathbb{R}$.
U2 - 10.7900/jot.2014may24.2030
DO - 10.7900/jot.2014may24.2030
M3 - Journal article
VL - 74
SP - 183
EP - 194
JO - Journal of Operator Theory
JF - Journal of Operator Theory
SN - 0379-4024
IS - 1
ER -
ID: 138510048