This is an announcement for the paper "Classes of strictly singular operators and their products" by George Androulakis, Gleb Sirotkin, and Vladimir G. Troitsky. Abstract: 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 $\mathcal{S}_\xi$ in order to define the class of $\mathcal{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 ${\mathcal S}_\xi$-hereditarily indecomposable Banach spaces and we examine the operators on them. Archive classification: Functional Analysis Mathematics Subject Classification: 47B07, 47A15 The source file(s), compactproducts.tex: 76155 bytes, is(are) stored in gzipped form as 0609039.gz with size 22kb. The corresponding postcript file has gzipped size 102kb. Submitted from: giorgis@math.sc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0609039 or http://arXiv.org/abs/math.FA/0609039 or by email in unzipped form by transmitting an empty message with subject line uget 0609039 or in gzipped form by using subject line get 0609039 to: math@arXiv.org.