Commutators on $\ell_1$

TL;DR

This paper characterizes the operators that are commutators on  and related spaces, extending Apostol's technique, and shows that certain classes of operators, including compact and strictly singular operators, are commutators.

Contribution

It generalizes Apostol's method to identify commutators on  and other spaces, providing new characterizations and partial results for various operator classes.

Findings

All compact operators on  are commutators.

Characterization of commutators on  and direct sums of  spaces.

Strictly singular operators on  are commutators.

Abstract

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.