Closed ideals of operators on the Baernstein and Schreier spaces

Laustsen, Niels and Smith, James (2025) Closed ideals of operators on the Baernstein and Schreier spaces. Journal of Mathematical Analysis and Applications. ISSN 0022-247X (In Press)

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Abstract

We study the lattice of closed ideals of bounded operators on two families of Banach spaces: the Baernstein spaces Bp for 1<p<∞ and the Schreier spaces Sp for 1≤p<∞. Our main conclusion is that there are 2 to the continuum many closed ideals that lie between the ideals of compact and  strictly singular operators on each of these spaces, and also 2 to the continuum 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 to the continuum many closed ideals of operators on a Banach space.