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Borel equivalence relations in the space of bounded operators

Iian B. Smythe

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.

Borel equivalence relations in the space of bounded 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 -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.

Paper Structure

This paper contains 10 sections, 23 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounded operators on Hilbert spaces
  4. Borel equivalence relations
  5. Topology and Borel structure on $\mathcal{B}(H)$
  6. Equivalence relations in $\mathcal{B}(H)$
  7. Equivalence relations in $\mathcal{P}(H)$
  8. Equivalence modulo finite rank
  9. Essential equivalence
  10. Further questions

Key Result

Theorem 1

For $T$ and $S$ bounded self-adjoint operators on a Hilbert space as above, the following are equivalent:

Theorems & Definitions (40)

  • Theorem : Weyl--von Neumann vN35
  • Theorem : Ando--Matsuzawa MR3427601
  • Theorem : Kechris--Sofronidis MR1855842
  • Theorem : Ando--Matsuzawa MR3427601
  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Proposition 2.1: 3.3.5 in MR971256
  • Lemma 2.2
  • Lemma 2.3
  • ...and 30 more