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