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~

$\spx$ to gain insight into the non-commutative structure of the Banach algebra

$\allop(\spx)$ of all (bounded, linear) operators on~

$\spx$ . 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

$\allop(\spx)$ for

$\spx = c_0$ ,

$\spx = C([0,1])$ ,

$\spx = \ell_p$ , and

$\spx = L_p([0,1])$ , where

$1\leq p\leq\infty$ . Then we show that each weakly compact operator on the

$p^{\rm th}$ James space

$\spj_p$ , 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

$\spj_p$ as the set of operators which factor through a certain reflexive, complemented subspace of

$\spj_p$ . %On the other hand, the identity operator %on

$\spj_p$ has distance at least 1 to any sum of commutators. %It follows that each trace on

$\allop(\spj_p)$ is a scalar multiple of %the character on

$\allop(\spj_p)$ induced by the quotient homomorphism %of

$\allop(\spj_p)$ onto

$\allop(\spj_p)/\wcompactop(\spj_p)$ .

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