Mathematical Sciences: Problems in Modern Analysis

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9401582 Enflo The proposers plan to work on two projects involving bounded operators on Hilbert space. The first project is to study orbits of cyclic vectors of operators. In particular, different regularity properties of these orbits. The second project involves a study of some aspects of invariant subspaces using recently developed techniques of the proposers. The focus here is on quasinilpotent operators. Operators on Hilbert space represent an infinite dimensional generalization of matrices. It is a well known fact that given a matrix there is always a vector such that the span of iterates of the matrix action on this vector is not the whole vector space. This project is focused on the analogous question in the infinite dimensional case. Appropriately formulated, this question is called the invariant subspace problem. It has been the focus of much research on operator theory. The question is an attempt to decompose operators into building `locks and is central to the analysis of the structure of operators. ***