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Dosev, Detelin T.
Journal of functional analysis, 06/2009, Letnik: 256, Številka: 11Journal Article
The main result is that the commutators on ℓ 1 are the operators not of the form λ I + 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 L 1 is a commutator. A characterization of the commutators on ℓ p 1 ⊕ ℓ p 2 ⊕ ⋯ ⊕ ℓ p n is given. We also show that strictly singular operators on ℓ ∞ are commutators.
