On some algebra of continuous linear operators

We present a criterion for an operator on $L_p$ to belong to the set $I_p$ of all sums of integral operators on $L_p$ and multiplication operators by functions in $L_\infty$. We describe the closure of $I_p$ in the operator norm. We prove that the set $L_{p,1}$ of all sums of multiplication operators and operators on $L_p$ mapping the unit ball of $L_p$ into compact subsets of $L_1$ is a Banach algebra.