Commutators on $L_p$, $1\le p<\mathrm{\infty }$

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