Problems in Modern Analysis

Kent State University, Kent OH

Investigators

Abstract

The proposer will work on Operator Theory problems, problems related to invariant subspaces and to stability. The proposer has developed a new technique to construct invariant subspaces by using different kinds of extremal vectors. This has, so far, led to new and unified constructions of invariant subspaces for large classes of operators, including operators commuting with compact or normal operators. It has also led to strengthenings of so called Two Sequences Theorems, which are connected to invariant subspaces. There are several directions to refine and improve this techniques and the proposer intends to do so. The study of stability under perturbations of cyclicity, supercyclicity or hypercyclicity is still in its beginning, both for operators and vectors. This is strongly connected to the study of which properties are carried by large resp. small classes of operators. The proposer intends to carry these studies further. The study of operators and their invariant subspaces can be seen as a generalization of linear algebra, where matrices and their eigenvectors are studied. Although these are problems in "pure" mathematics, they are closely connected to the many applications of linear algebra to Physics, Biology, Genetics, Economics and other sciences as well as to the applications in industry. Operators can be seen as infinite matrices and are a suitable tool for the study of large and complex systems depending on an unbounded number of parameters.