Around the closures of the set of commutators and the set of differences of idempotent elements of B(H)

Abstract

We describe the norm-closures of the set CE of commutators of idempotent operators and the set C-C of differences of idempotent operators on a finite-dimensional complex Hilbert space, as well as characterise the intersection of the closures of these sets with the set K(H) of compact operators acting on an infinite-dimensional complex separable Hilbert space H. Finally, we characterise the closures of the set CP of commutators of orthogonal projections and the set B-B of differences of orthogonal projections acting on a complex separable Hilbert space.