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

Abstract

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