Perturbation classes for semi-Fredholm operators on -spaces

We study the perturbation class P Φ + ( L p , Y ) for the upper semi-Fredholm operators when Φ + ( L p , 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 P Φ − ( X , L q ) when 1 ⩽ q < ∞ .