Compactness of compositions of strictly singular operators on direct sums of Baernstein, Schreier and $\ell_p$-spaces

Abstract

Let $X$ be the direct sum of finitely many Banach spaces chosen from the following three families: (i) the Baernstein spaces $B_p$ for $1<p<\infty$; (ii) the $p$-convexified Schreier spaces $S_p$ for $1\le p<\infty$; (iii) the sequence spaces $\ell_p$ for $1\le p<\infty$ (and $c_0$). We show that the quotient algebra of strictly singular by compact operators on $X$ is nilpotent; that is, there is a natural number $k$, dependent only on the collections of direct summands from each of the three families, such that: - every composition of $k+1$ strictly singular operators on $X$ is compact; - there are $k$ strictly singular operators on $X$ whose composition is not compact.