Journal of Operator Theory

Volume 48, Issue 3, Supplementary 2002 pp. 503-514.

Commutators of operators on Banach spaces

Authors: Niels Jakob Laustsen
Author institution: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England

Summary: We study the commutators of operators on a Banach space $\mathcal{B}(\mathcal{X})$ to gain insight into the non-commutative structure of the Banach algebra $\mathcal{B}(\mathcal{X})$ of all (bounded, linear) operators on $\mathcal{B}(\mathcal{X})$. First we obtain a purely algebraic generalization of Halmos's theorem that each operator on an infinite-dimensional Hilbert space is the sum of two commutators. Our result applies in particular to the algebra $\mathcal{B}(\mathcal{X})$ for $\mathcal{B}(\mathcal{X}) = C([0,1])$, $\mathcal{B}(\mathcal{X}) = \ell_p$, and $\mathcal{B}(\mathcal{X}) = L_p([0,1])$, where $1\leq p\leq\infty$. Then we show that each weakly compact operator on the $p^{\rm th}$ James space $\mathcal J_p$, where $1 < p < \infty$, is the sum of three commutators; a key step in the proof of this result is a characterization of the weakly compact operators on $\mathcal J_p$ as the set of operators which factor through a certain reflexive, complemented subspace of $\mathcal J_p$.

Keywords: Commutators, operators on Banach spaces, Banach algebras, traces

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