Closed ideals of operators on the Baernstein and Schreier spaces

We study the lattice of closed ideals of bounded operators on two families of Banach spaces: the Baernstein spaces B p for 1 < p < ∞ and the Schreier spaces S p for 1 ⩽ p < ∞ . Our main conclusion is that there are 2 c many closed ideals that lie between the ideals of compact and strictly singular operators on each of these spaces, and also 2 c many closed ideals that contain projections of infinite rank. Counterparts of results of Gasparis and Leung using a numerical index to distinguish the isomorphism types of subspaces spanned by subsequences of the unit vector basis for the higher-order Schreier spaces play a key role in the proofs, as does the Johnson–Schechtman technique for constructing 2 c many closed ideals of operators on a Banach space.