Borel equivalence relations in the space of bounded operators

TL;DR

This paper investigates different equivalence relations among bounded operators on a Hilbert space, demonstrating their complexity and non-classifiability using turbulence theory and orbit equivalence relations.

Contribution

It introduces new results on the non-classifiability of certain operator equivalences using Hjorth's turbulence theory and extends these results to projections.

Findings

Modulo Schatten p-class and compact equivalences are not classifiable by countable structures.

Modulo finite rank equivalence is not reducible to any Polish group orbit equivalence.

Results also apply to the space of projection operators.

Abstract

We consider various notions of equivalence in the space of bounded operators on a Hilbert space, in particular modulo finite rank, modulo Schatten $p$-class, and modulo compact. Using Hjorth's theory of turbulence, the latter two are shown to be not classifiable by countable structures, while the first is not reducible to the orbit equivalence relation of any Polish group action. The results for modulo finite rank and modulo compact operators are also shown for the restrictions of these equivalence relations to the space of projection operators.