Representation of commutators on Schatten $p$-classes

Abstract

Let ${\mathcal{C}}_p$ be the Schatten $p$-class of ${\ell }_2$ for $1<p<\infty$, and let ${\mathcal{T}}_p$ be the closed subspace of ${\mathcal{C}}_p$ consisting of all lower triangular matrices. In this paper, we show that a bounded linear operator $T$ on ${\mathcal{C}}_p$ is a commutator if and only if $T$ is not of the form $\lambda I+K$, where $0\ne \lambda \in \mathbb{C}$ and $K$ is a ${\mathcal{C}}_p$-strictly singular operator. It is done by showing that a bounded linear operator $T$ on ${\mathcal{T}}_p$ is not a commutator if and only if $T$ has the same form.