  The main result is that the commutators on $\ell_1$ are the operators not of
the form $\lambda I + K$ with $\lambda\neq 0$ 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 $1\leq p<\infty$ or $p=0$. In
particular, it is shown that every compact operator on $L_1$ is a commutator. A
characterization of the commutators on
$\ell_{p_1}\oplus\ell_{p_2}\oplus...\oplus\ell_{p_n}$ is given. We also show
that strictly singular operators on $\ell_{\infty}$ are commutators.
