The main result is that the commutators on are the operators not of the form with and K compact. We generalize Apostol's technique [C. Apostol, Rev. Roumaine Math. Appl. 17 (1972) 1513–1534] to obtain this result and use this generalization to obtain partial results about the commutators on spaces which can be represented as for some or . In particular, it is shown that every compact operator on is a commutator. A characterization of the commutators on is given. We also show that strictly singular operators on are commutators.