Classes of strictly singular operators and their products

Androulakis, George; Dodos, Pandelis; Sirotkin, Gleb; Troitsky, Vladimir G.

Classes of strictly singular operators and their products

Authors

Androulakis, George

Dodos, Pandelis

Sirotkin, Gleb

Troitsky, Vladimir G.

Description

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 $§_ξ$ in order to define the class of $§_ξ$-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 $§_ξ$-hereditarily indecomposable Banach spaces and we examine the operators on them.