Problems in the General Theory of Banach Spaces

Kent State University, Kent OH

Investigators

Abstract

PI: Lomonosov Proposal Number: 0139900 ABSTRACT The principal investigator will conduct research on two projects which relate to both Operator Theory and Geometry of Banach spaces. The recent solution of the complex version of the Klee's Conjecture using ideas from Operator Theory suggests further research on support points and convex sets. Some new geometric approaches will be used for investigating the Invariant Subspace Problem in general Banach Spaces. The principal investigator recently solved a long-standing problem, constructing a convex body which has no tangent hyper-planes. It is important to understand under which conditions such a strange phenomenon can exist. Another project starts with the fundamental fact that every matrix has an eigenvector. A natural extension of this to infinite-dimensional spaces would be that a linear operator has an invariant subspace. We will investigate whether this statement is true in a number of important cases.