Let X X be the direct sum of finitely many Banach spaces chosen from the following three families: (i) the Baernstein spaces B p Bₚ for 1 > p > ∞ 1>p> ; (ii) the p p -convexified Schreier spaces S p Sₚ for 1 ⩽ p > ∞ 1 p> ; (iii) the sequence spaces ℓ p ₚ for 1 ⩽ p > ∞ 1 p> (and c 0 c₀). We show that the quotient algebra of strictly singular by compact operators on X X is nilpotent; that is, there is a natural number k k, dependent only on the collections of direct summands from each of the three families, such that: every composition of k + 1 k+1 strictly singular operators on X X is compact; there are k k strictly singular operators on X X
Laustsen et al. (Fri,) studied this question.