Two classes of operators related to the perturbation classes problem

Let $SS$ and $SC$ be the strictly singular and the strictly cosingular operators acting between Banach spaces, and let $P\Phi_+$ and $P\Phi_+$ be the perturbation classes for the upper and the lower semi-Fredholm operators. We study two classes of operators $\Phi S$ and $\Phi C$ that satisfy $SS\subset \Phi S \subset P\Phi_+$ and $SC\subset \Phi C\subset P\Phi_-$. We give some conditions under which these inclusions become equalities, from which we derive some positive solutions to the perturbation classes problem for semi-Fredholm operators.