9623211 HADWIN The principal investigator plans to continue his work in operator theory in the areas of reflexivity, approximation, and invariant subspaces. We have developed a very general way of viewing reflexivity, and we have used it to discover new results and easier ways of proving old ones. We are now developing techniques that make it possible to prove that a subspace is reflexive or hyperreflexive by looking at small "pieces" of it. We hope to prove that a subspace with a strictly separating vector and a strongly disjoint strictly separating subspace must be hyperreflexive. We are studying approximation by characterizing closures of certain classes of operators and by determining when there are unique approximants. Our approximation results aid in our study of stable invariant subspaces of operators. We are working on a notion of ranked operator spaces, which, combined with our work on strong limits of similarities, should help determine the closures of joint similarity orbits in finite-dimensional spaces. The theory of operators on Hilbert space is a part of mathematics that generalizes to infinite dimensions the basic ideas of linear algebra. The growth of this subject was stimulated by, and has applications to the study of integral and differential equations, quantum mechanics and control theory. The general goal of operator theory is to understand the structure of various classes and collections of operators. The principal investigator plans to approach this goal by continuing his work on approximation (i.e., which - and how closely - operators can be approximated by ones that are well understood), reflexivity and hyperreflexivity (operators determined by their invariant subspaces), and developing simple algebraic techniques to solve difficult analysis problems in operator theory.
