Strictly singular operators on Lp spaces and interpolation

Abstract

We study the class Vp of strictly singular non-compact operators on Lp spaces. This allows us to obtain interpolation results for strictly singular operators on Lp spaces. Given 1 ≤ p < q ≤ ∞, it is shown that if an operator T bounded on Lp and Lq is strictly singular on Lr for some p ≤ r ≤ q, then it is compact on Ls for every p < s < q.