Weyl-von Neumann Theorem and Borel Complexity of Unitary Equivalence Modulo Compacts of Self-Adjoint Operators

TL;DR

This paper investigates the classification complexity of unitary equivalence of self-adjoint operators modulo compacts, revealing smoothness for bounded cases and complexity for unbounded cases using descriptive set theory.

Contribution

It extends the Weyl-von Neumann theorem to unbounded operators and analyzes the Borel complexity of related equivalence relations.

Findings

Bounded self-adjoint operators' equivalence is smooth.

Unbounded operators' equivalence relation is complex, with dense G_delta orbit but no classification by countable structures.

A related equivalence relation involving resolvents is shown to be smooth.

Abstract

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 $u\in \mathcal{U}(H)$ and compact self-adjoint operator $K$, if and only if $A$ and $B$ have the same essential spectra: $\sigma_{\rm{ess}}(A)=\sigma_{\rm{ess}}(B)$. 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 $G_{\delta}$-orbit but does not admit classification by countable structures. On the other hand, apparently related equivalence relation $A\sim B\Leftrightarrow \exists u\in \mathcal{U}(H)\ [u(A-i)^{-1}u^*-(B-i)^{-1}$ is compact], is shown to be smooth. Various Borel or co-analytic equivalence relations related to self-adjoint operators are also presented.