  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.
