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

TL;DR

This paper characterizes non-commutator operators on $L_p[0,1]$, showing they are precisely those of the form $ ext{scalar} imes I + S$, with $S$ in the largest ideal, supported by new structural results.

Contribution

It provides a complete characterization of non-commutator operators on $L_p$ spaces, introducing new structural insights into the operator algebra on these spaces.

Findings

Operators not being commutators are exactly scalar multiples of the identity plus an ideal operator.

New structural results for operators on $L_p$ spaces are established.

The results clarify the structure of the largest ideal in the operator algebra on $L_p$.

Abstract

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