Commutators on $L_p$, $1\le p<\infty$

Abstract

The operators on $\LP=L_p[0,1]$, $1\le p<\infty$, which are not commutators are those of the form $\lambda I+S$ where $\lambda \ne 0$ and $S$ belongs to the largest ideal in $\opLP$. The proof involves new structural results for operators on $\LP$ which are of independent interest.