[Banach] Abstract of a paper by George Androulakis, Gleb Sirotkin, and Vladimir G. Troitsky

[Dale Alspach]

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

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Submitted from: giorgis at math.sc.edu

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