Commutators of operators on Banach spaces.

Laustsen, Niels J. (2002) Commutators of operators on Banach spaces. Journal of Operator Theory, 48 (3). pp. 503-514.

Abstract

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