Operator Theory and Analytic Functions

Abstract

The theory of analytic functions plays a central role in operator theory. It has been a source of methods, examples and problems, and has led to numerous important results. Weighted shifts (which we shall see in the sequel) have been studied with analytic function theory approach. In this thesis, inspired by the work A. L. Shields, we give excellent exposition of an interplay between weighted shift operators and analytic functions. Essential ingredients of the considerations therein were viewing a weighted shift operator as ”multiplication by z” on a Hilbert space consisting of formal power/Laurent series and showing that the structure of this space is in fact analytic. This enabled using multiplication operators and bounded point evaluations, tools known to be very powerful in variety of problems in operator theory