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.