Troitsky, Classes of strictly singular operators and their products

Abstract. V. D. Milman proved in [18] that the product of two strictly singular operators on Lp[0, 1] (1 6 p < ∞) or on C[0, 1] is compact. In this note we utilize Schreier families Sξ in order to define the class of Sξ-strictly singular operators, and then we refine the technique of Milman to show that certain products of operators from this class are compact, under the assumption that the underlying Banach space has finitely many equivalence classes of Schreier-spreading sequences. Finally we define the class of Sξ-hereditarily indecomposable Banach spaces and we examine the operators on them. 1