Commutators on ℓ1

The main result is that the commutators on ℓ1 are the operators not of the form λI+K with λ≠0 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 X which can be represented as for some 1⩽p<∞ or p=0. In particular, it is shown that every compact operator on L1 is a commutator. A characterization of the commutators on ℓp1⊕ℓp2⊕⋯⊕ℓpn is given. We also show that strictly singular operators on ℓ∞ are commutators.