Commutators on ${\ell }_1$

Abstract

The main result is that the commutators on ${\ell }_1$ are the operators not of the form $\lambda I+K$ with $\lambda \ne 0$ 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 $1\le p<\infty$ 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.

17 pages. Submitted to the Journal of Functional Analysis

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