Commutators on 1

The main result is that the commutators on 1 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 X ( ∞i =0 X)p 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 p1 ⊕ p2 ⊕···⊕ pn is given. We also show that strictly singular operators on ∞ are commutators. © 2009 Elsevier Inc. All rights reserved.