Compactness of compositions of strictly singular operators on direct sums of Baernstein, Schreier and ℓ𝑝-spaces

Let X X be the direct sum of finitely many Banach spaces chosen from the following three families: (i) the Baernstein spaces B p B_p for 1 > p > ∞ 1>p>∞ ; (ii) the p p -convexified Schreier spaces S p S_p for 1 ⩽ p > ∞ 1 p>∞ ; (iii) the sequence spaces ℓ p _p for 1 ⩽ p > ∞ 1 p>∞ (and c 0 c_0 ). 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