On the relation between an operator and its self-commutator

TL;DR

This paper establishes a connection between an operator and its self-commutator, generalizing key theorems in operator theory and extending results to infinite-dimensional operators and various norms.

Contribution

It introduces a new theorem linking operators to their self-commutators under approximation conditions, extending classical results to broader contexts including $C^*$-algebras and infinite rank operators.

Findings

Generalizes the Brown--Douglas--Fillmore theorem

Extends Huaxin Lin's theorem to infinite rank operators

Refines existing theorems and extends to Schatten norms

Abstract

Our main result is a theorem saying that a bounded operator $A$ on a Hilbert space belongs to a certain set associated with its self-commutator $[A^*,A]$, provided that $A-zI$ can be approximated by invertible operators for all complex numbers $z$. The theorem remains valid in a general $C^*$-algebra of real rank zero under the assumption that $A-zI$ belong to the closure of the connected component of unity in the set of invertible elements. This result implies the Brown--Douglas--Fillmore theorem and Huaxin Lin's theorem on almost commuting matrices. Moreover, it allows us to refine the former and to extend the latter to operators of infinite rank and other norms (including the Schatten norms on the space of matrices). The proof is based on an abstract theorem, which states that a normal element of a $C^*$-algebra of real rank zero satisfying the above condition has a resolution of the identity associated with any open cover of its spectrum.