Around the closure of the set of commutators of idempotents in B(H): Biquasitriangularity and factorisation

In this paper, we continue our study of the norm-closure of the set CE of bounded linear operators acting on a complex, infinite-dimensional, separable Hilbert space H which may be expressed as the commutator of two idempotent operators. In particular, we identify which biquasitriangular operators belong to the norm-closure Image 1 of CE, and we exhibit an index obstruction to membership in Image 1. Finally, we consider factorisations of bounded linear operators on H as sums and products of elements in CE and related sets.