Banach Spaces and Operators on them

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

Abstract Schlumprecht The author will work on several open problems in Geometrical Functional Analysis. The first problem asks whether or not every infinite dimensional Banach space admits a non-trivial operator, i.e. an operator that is not a compact perturbation of a multiple of the identity. The principal investigator proposes to explore several interesting variations on this theme, each with a distinctly structure-theoretical flavor. A counter example to aforementioned question would present the first example of an infinite dimensional Banach space for which it is known that every operator on it has an invariant subspace. Secondly, he presents an approach to the most general formulation of the invariant subspace problem, namely: does every adjoint operator have a nontrivial invariant subspace? The third part of the proposal deals with quasi-greedy bases, one of the mathematical ideas that help provide a theoretical groundwork for image compression and reconstruction and asks for the existence of quasi-greedy bases in each Banach space. Banach spaces, their geometric and topological structure, as well as operators on them, provide a natural framework for studying dynamical systems, differential equations, physics and multiresolution analysis. For example, it is pointed out in the proposal how Banach space theoretical ideas relate to the theory of image compression and reconstruction.