Factorization of operators in classical Banach spaces

The central goal of this project is to obtain factorization results in Banach spaces, including mixed norm Lebesgue and martingale spaces as well as classical counterexamples of Banach space theory such as the dual of the James tree space, the dual of the Hagler tree space and the Ghoussoub- Maurey space. For the most part, these spaces are non-separable, which adds to the difficulty of the factorization problems due to a lack of a proper basis. Instead of decomposing the spaces into finite dimensional building blocks (Bourgains localization method) which creates its own set of challenges, our strategy is to work directly in the infinite dimensional non-separable spaces by exploiting their geometry. A parallel and closely related focus of this project is the investigation of finite dimensional quantitative factorization results, which are not only an important research object themselves, but are also useful for obtaining infinite dimensional results. We propose to solve the quantitative factorization problem by using a probabilistic block basis on which a given operator is a perturbation of a diagonal operator. This project`s last research objective is to extend the factorization result of Drewnowski and Roberts in the quotient of sequence spaces to the quotient of natural martingale generalizations of these sequence spaces. Our approach is to adapt the techniques of Drewnowski and Roberts to fit the martingale setting and to combine them with combinatorics of dyadic intervals.

The project made substantial contributions in creating a systematic framework for the factorization of operators on large classes of Banach spaces. The framework facilitates the determination of crucial properties of Banach spaces, including their primarity and whether the bounded operators have a unique maximal ideal. A special emphasis was put on Banach spaces which possess a Schauder bases and their dual spaces, as well as on the classical bi-parameter Lebesgue spaces $L^p(L^q)$, $1 \leq p,q < \infty$. The following achievements of this project are particularly noteworthy: *) Proving that the bi-parameter Lebesgue spaces $L^1(L^p)$, $1 < p < \infty$ are primary and thereby solving a 40~year old problem. *) Taking a major step towards the primarity of $L^p(L^1)$, $1 < p < \infty$ (also a 40~year old problem) by analyzing Haar multipliers on bi-parameter spaces. *) Creating a unified framework in terms of two-player games to achieve factorization results for Banach spaces with a Schauder basis, and *) further development of this framework to provide concrete, verifiable conditions which imply for example the uniqueness of the maximal ideal of operators for an even broader class of Banach spaces.