Around the closures of the set of commutators and the set of differences of idempotent elements of $\mathcal{B}(\mathcal{H})$

TL;DR

This paper characterizes the norm-closures of sets of commutators and differences of idempotent and orthogonal projection operators on various Hilbert spaces, including finite and infinite dimensions.

Contribution

It provides a detailed description of the closures of these sets and their intersections with compact operators across different Hilbert space settings.

Findings

Closure of commutators of idempotents characterized

Closure of differences of idempotents described

Closures of commutators and differences of projections analyzed

Abstract

We describe the norm-closures of the set $\mathfrak{C}_{\mathfrak{E}}$ of commutators of idempotent operators and the set $\mathfrak{E} - \mathfrak{E}$ of differences of idempotent operators acting on a finite-dimensional complex Hilbert space, as well as characterising the intersection of the closures of these sets with the set $\mathcal{K}(\mathcal{H})$ of compact operators acting on an infinite-dimensional, separable Hilbert space. Finally, we characterise the closures of the set $\mathfrak{C}_{\mathfrak{P}}$ of commutators of orthogonal projections and the set $\mathfrak{P} - \mathfrak{P}$ of differences of orthogonal projections acting on an arbitrary complex Hilbert space.