Closed ideals of operators on the Baernstein and Schreier spaces

TL;DR

This paper investigates the complex lattice structure of closed ideals of bounded operators on Baernstein and Schreier Banach spaces, revealing a vast number of distinct ideals between well-known classes.

Contribution

It demonstrates the existence of continuum many closed ideals between compact and strictly singular operators on these spaces, using advanced techniques and prior results.

Findings

2^{ ext{c}} many closed ideals between compact and strictly singular operators

2^{ ext{c}} many closed ideals containing projections of infinite rank

Utilizes numerical index and Johnson-Schechtman techniques

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 Schreier spaces $S_p$ for $1\le p<\infty$. Our main conclusion is that there are $2^{\mathfrak{c}}$ many closed ideals that lie between the ideals of compact and strictly singular operators on each of these spaces, and also $2^{\mathfrak{c}}$ 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 $2^{\mathfrak{c}}$ many closed ideals of operators on a Banach space.