Invariant Subspaces in Spaces of Analytic Functions

University Of Tennessee Knoxville, Knoxville TN

Investigators

Abstract

Richter and Sundberg will continue their research on spaces of analytic functions with their naturally associated operators. The best and most completely understood example in this field is the unilateral shift, which simply takes every element in an orthogonal basis of a Hilbert space indexed by the nonnegative integers and "shifts" it to the element with one higher index. This operator is the simplest example of a nonnormal operator with genuinely infinite dimensional properties and its study has been of major importance in Operator Theory. The unilateral shift is modelled by the operation of multiplication by the coordinate function z on the Hardy space of complex analytic functions on the unit disc in the complex plane. It is this modelling that has been at the heart of the study of the unilateral shift and that has led to our remarkably thorough understanding of it, and also to important results in the theory of general contraction operators. There has been much research since the 1980's that has shown that the ideas used in connection with the study of the unilateral shift have interesting and important extensions to the study of operators modelled in function spaces other than the Hardy space. Among the important examples of such spaces are the Dirichlet Space, the Bergman Space, and the weighted Bergman spaces. Richter and Sundberg will continue to study these and other spaces with a view especially to a better understanding of their lattices of subspaces invariant under the operation of multiplication by z, as well as related questions concerning zero sets, nontangential limiting behavior, and polynomial approximations. The proposed work involves several areas of Pure and Applied Mathematics. Operator Theory as a branch of Functional Analysis, arose in the 1880's in the study of Partial Differential Equations arising in Physics and Engineering, and became increasingly important in the twentieth century with the advent of Quantum Mechanics. Complex Analysis is a subject with a long and distinguished history and a wide applicability - it has in fact important applications in almost every area of Mathematics as well as many areas of Physics. In particular, Complex Analysis has been of importance in Operator Theory from its inception and the investigations and connections between these areas continues to be a very fruitful area of research. One important source of connections is the modelling of operators by natural operations on spaces of analytic functions. The study of such operators on a space called the Hardy space has been of great importance in both Pure and Applied Mathematics. It is at the heart of a certain useful approach in Control Theory, and area of importance in electrical Engineering and the design of guidance systems. Work by a number of researchers since the 1980's (including the present authors), has shown the ideas involved in these studies have important applicability to an extensive class of function spaces and operators.