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

Laustsen, Niels and Wirzenius, Henrik (2025) Compactness of compositions of strictly singular operators on direct sums of Baernstein, Schreier and lp-spaces. Proceedings of the American Mathematical Society. ISSN 0002-9939 (In Press)

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Abstract

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.