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

Let X be the direct sum of finitely many Banach spaces chosen from the following three families: (i) the Baernstein spaces Bp for 1<p<∞; (ii) the p-convexified Schreier spaces Sp for 1≤p<∞; (iii) the sequence spaces lp for 1≤p<∞ (and c0). 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.