Geometry of Banach spaces and their spaces of operators

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The research project is comprised of a broad class of problems in the geometry of Banach spaces and operator theory. Banach spaces, in particular function spaces and sequence spaces, are an important tool used either directly or indirectly in scientific fields such as engineering and physics. Hilbert spaces, which are a special type of Banach spaces, and operators acting on such spaces, provide a theoretical framework that can be used to study problems in quantum mechanics and computer science. Therefore, there exists potential practical value in studying the fairly abstract notion of a Banach space and its space of operators. The project is centered around the study of old problems, as well as ones that have emerged from recent developments in the theory. Although it is formulated in terms of Banach space theory, many of the problems studied are related to other areas of mathematics, such as descriptive set theory and operator theory. The solution of those problems will require the combination of techniques from such areas as combinatorics, set theory and topology.The project revolves around the investigation of a variety of problems. Emphasis is given on the local and asymptotic behavior of basic sequences, which can in fact have far reaching implications in the properties of operators on certain spaces. Some notions that are studied are those of spreading models, finite block representability, and script-L-infinity spaces. As already mentioned, one of the main intentions is to deduce properties of operators. There are two main approaches that can be used in this setting. The first one is the construction of spaces with hereditary heterogeneous asymptotic structure. This method was previously used by the principal investigator and S. Argyros to construct the first example of a separable reflexive Banach space with the invariant subspace property. The question as to whether the separable Hilbert space has this property as well, is one of the central problems of operator theory, in fact of mathematics. Further study of this method can possibly yield further examples of spaces which resemble Hilbert spaces, for example uniformly convex spaces, with the invariant subspace property. The second approach is related to the study of script-L-infinity spaces. This approach is based on the method developed by S. Argyros and R. Haydon to construct the first known Banach space with the scalar-plus-compact property. This means that every bounded linear operator on this space is a compact perturbation of a scalar multiple of the identity. This method has its roots in a construction of J. Bourgain and F. Delbaen. Further study of this method may lead to the characterization of spaces with the aforementioned property.