# Journal of Operator Theory

Volume 74, Issue 1, Summer 2015 pp. 183-194.

On Borel equivalence relations related to self-adjoint
operators

**Authors**:
Hiroshi Ando (1) and Yasumichi Matsuzawa (2)

**Author institution:**(1) Department of Mathematical Sciences, University
of Copenhagen, Universitetsparken 5, Copenhagen \O, DK-2100, Denmark

(2) Department of Mathematics, Faculty of Education, Shinshu University,
6-Ro, Nishi-nagano, Nagano, 380--8544, Japan

**Summary: **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 $\mathbb{R}$.

**DOI: **http://dx.doi.org/10.7900/jot.2014may24.2030

**Keywords: **unbounded self-adjoint operators, Borel equivalence
relations

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