Closed Ideals in the Algebra of Bounded Operators on Baernstein and Schreier spaces

Smith, James and Laustsen, Niels (2026) Closed Ideals in the Algebra of Bounded Operators on Baernstein and Schreier spaces. PhD thesis, Lancaster University.

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Abstract

This thesis is broadly centered around the study of closed ideals within the Banach algebra B(E) of bounded linear operators on E, where E is a Banach space from the following two families: the Baernstein spaces Bp (1 < p < ∞) and p-convexified Schreier spaces Sp (1 ≤ p < ∞). In Chapters 2 and 3 we study the complemented subspaces in Baernstein and pconvexified Schreier spaces, where our main goal is to demonstrate that a continuum c of pairwise non-isomorphic complemented subspaces exist in the Baernstein and pconvexified Schreier spaces. This had already been established for the Schreier space S1 by Gasparis and Leung, with the aid of a numerical index that characterizes when two subsequences of the unit vector basis are equivalent in S1. We surprisingly find that the same index works for the Baernstein spaces and p-convexified Schreier spaces, producing analogous results. In Chapter 4, we study the closed ideals of B(E) generated by projections. The findings represent the main results of this thesis; B(E) has at least c maximal ideals and 2c closed ideals. In Chapter 5 we focus on the ideal of strictly singular operators on a Baernstein or p-convexified Schreier space. Our main finding is that the composition of two strictly singular operators is compact. We then study several applications of this result. In Chapter 6, we study operator algebra structure which can arise from 1- unconditional bases when equipped with the coordinate-wise multiplication. It was shown by Varopoulos that ℓp is an operator algebra in the coordinate-wise multiplication. We show that many more Banach spaces are operator algebras in this respect. Whether this property extends to all 1-unconditional bases, remains unknown.