m at h . FA ] 1 S ep 2 00 6 CLASSES OF STRICTLY SINGULAR OPERATORS AND THEIR PRODUCTS

V. D. Milman proved in [14] that the product of two strictly singular operators on L p [0, 1] (1 p < ∞) or on C[0, 1] is compact. In this note we utilize Schreier families S ξ in order to define the class of S ξ-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 S ξ-hereditarily indecomposable Banach spaces and we examine the operators on them. 1. Classes of strictly singular operators Recall that a bounded operator T from a Banach space X to a Banach space Y is called strictly singular if its restriction to any infinite-dimensional subspace is not an isomorphism. That is, for every infinite dimensional subspace Z of X and for every ε > 0 there exists z ∈ Z such that T z < εz. We say that T is finitely strictly singular if for every ε > 0 there exists n ∈ N such that for every subspace Z of X with dim Z n there exists z ∈ Z such that T z < εz. In particular, for 1 p < q ∞ the inclusion operator i p,q from ℓ p to ℓ q is finitely strictly singular. X = Y we will write K(X), SS(X), and F SS(X). It is known that these sets are norm closed operator ideals in L(X), the space of all bounded linear operators on X, see [14, 23] for more details on these classes of operators. It is well known that K(X) ⊆ F SS(X) ⊆ SS(X). We provide the proof for completeness. The second inclusion is obvious. To prove the first inclusion, suppose that T is not finitely strictly