  V. D. Milman proved in \cite{Milman:70} that the product of two strictly
singular operators on $L_p[0,1]$ ($1\le p<\infty$) or on $C[0,1]$ is compact.
In this note we utilize Schreier families $\S_\xi$ in order to define the class
of $\S_\xi$-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 $\S_\xi$-hereditarily indecomposable Banach spaces and we examine the
operators on them.
