Banach Spaces and their Applications

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

Abstract Kalton In this proposal, the aim is to study a number of problems related to Banach space theory and its applications. Part of the project deals with the theory of extensions of one Banach space by another. For example it is planned to study the problem of classifying those Banach spaces such that every minimal extension is trivial. This problem can be reformulated as a problem in approximation theory. It is also planned to study applications to problems in the nonlinear theory of Banach spaces, related to problems of existence of uniform homeomorphisms between two Banach spaces or between their unit balls. In a somewhat different direction the proposer plans to continue his ongoing research on Rademacher-bounded families of operators with applications to sectorial operators and semigroups. The theory of extensions can be considered in the following terms. Suppose we are given a centrally symmetric solid in three or more dimensions and we are allowed only to compute its cross-section by a slice in some directions and the shadow cast by the body in the perpendicular directions. This gives us some information about the solid, but does not permit complete reconstruction. The idea of studying extensions is to obtain more complete information about the body under certain additional conditions. Many questions of importance in mathematical analysis and applications, although not formally expressed in this way, can be visualized in terms of extensions. Sectorial operators are of importance in the basic theory of partial differential equations of evolution type; such equations are very important in physical applications. Typically a sectorial operator is a differential operator acting on some suitable space of functions. By understanding the properties of sectorial operators and the semigroups they generate one gains a better understanding of the behavior of solutions of certain partial differential equations.