Perturbation classes for semi-Fredholm operators on Lp(μ)-spaces

We study the perturbation class PΦ+(Lp,Y) for the upper semi-Fredholm operators when Φ+(Lp,Y) is non-empty. We show that it coincides with the strictly singular operators when 1⩽p<2 and Y satisfies certain condition, and when 2⩽p⩽∞ and Y is arbitrary. As a consequence, we derive similar results for PΦ−(X,Lq) when 1⩽q<∞.