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 $P{\mathrm{\Phi }}_{+}$ and $P{\mathrm{\Phi }}_{+}$ be the perturbation classes for the upper and the lower semi-Fredholm operators. We study two classes of operators $\mathrm{\Phi }S$ and $\mathrm{\Phi }C$ that satisfy $SS\subset \mathrm{\Phi }S\subset P{\mathrm{\Phi }}_{+}$ and $SC\subset \mathrm{\Phi }C\subset P{\mathrm{\Phi }}_{-}$. 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.