solved in Arlen Brown, Carl Pearcy, Multiplicative commutators of operators , Canad. J. Math.  18 (1966), 737–749, MR200720 . Theorem 5 on page 746 states that a normal invertible operator is a commutator if and only if it is not of the form $\lambda \cdot \mathrm{id} + C$ with $C$ a compact operator