Commutators on $\ell_1$

The main result is that the commutators on $\ell_1$ are the operators not of the form $��I + K$ with $��\neq 0$ and $K$ compact. We generalize Apostol's technique (1972, Rev. Roum. Math. Appl. 17, 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 $\displaystyle \X\simeq (\bigoplus_{i=0}^{\infty} \X)_{p}$ for some $1\leq p
17 pages. Submitted to the Journal of Functional Analysis