Abstract: We study the perturbation class P ?+(L p , Y ) for the upper semi-Fredholm operators when ?+(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 ??(X, Lq ) when 1  q < ?.

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