Commutators on $L_p$, $1\le p<\infty$
  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. 
Detelin Dosev, William B. Johnson, Gideon Schechtman
math.FA