On the factorization of integral operators on spaces of summable functions

We consider the factorization $I-K=(I-U^+)(I-U^-)$, where $I$ is the identity operator, $K$ is an integral operator acting on some Banach space of functions summable with respect to a measure $\mu$ on $(a,b)\subset(-\infty,+\infty)$ continuous relative to the Lebesgue measure,
\begin{equation*} (Kf)(x)=\int^b_ak(x,t)f(t)\mu(dt),\qquad x\in(a,b), \end{equation*}
and $U^\pm$ are the desired Volterra operators. A necessary and sufficient condition is found for the existence of this factorization for a rather wide class of operators $K$ with positive kernels and for Hilbert–Schmidt operators.