Closed ideals of operators on the Baernstein and Schreier spaces

Abstract

We study the lattice of closed ideals of bounded operators on two families of Banach spaces: the Baernstein spaces $B_p$ for $1<p<\infty$ and the p-convexified Schreier spaces $S_p$ for $1⩽p<\infty$. 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 classical Schreier space $S_1$ and its higher-order variants 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.