Compactness of compositions of strictly singular operators on direct sums of Baernstein, Schreier and $\ell_p$-spaces
Niels Jakob Laustsen, Henrik Wirzenius
TL;DR
The authors study the nilpotency of the quotient algebra of strictly singular by compact operators on finite direct sums of Baernstein spaces B_p, p-convexified Schreier spaces S_p, and ℓ_p (with c_0 allowance). They prove that there exists an index k (depending only on the chosen summands) such that every composition of k+1 strictly singular operators is compact, while there exist k strictly singular operators whose composition is not compact, rendering the quotient algebra nilpotent of index k+1. The proof combines a matrix representation of operators on finite direct sums with a careful case analysis of summand types and the known compactness properties of compositions in these spaces, and it uses formal inclusion maps between complemented subspaces to certify the sharpness of the nilpotency index. The paper also corrects a previously claimed Seifert-type result by showing the existence of non-compact strictly singular maps between B_p and B_q (p≠q) and discusses the role of dominated bases in this context, highlighting the precise structure of nilpotent behavior in these direct-sum constructions.
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.