Polynomial approximation and Carleson measures on a general domain and equivalence classes of subnormal operators

his thesis consists of eight chapters. Chapter 1 contains the preliminaries: the background, notation and results needed for this work.;In Chapter 2 we study the problem of when ;In Chapter 3 we generalize the definition of a Carleson measure to an arbitrary simply connected domain. Let G be a bounded simply connected domain with harmonic measure ;In Chapter 4 we describe some special simply connected domains and describe how they are related to each other via various types of polynomial approximation.;In Chapter 5 we study the various equivalence classes of subnormal operators under the relations of unitary equivalence, similarity and quasisimilarity under the assumption that G is a normal domain.;In Chapter 6 we characterize the Carleson measures on a finitely connected domain. We are able to push our techniques in the latter setting to characterize those subnormal operators similar to the shift on the closure of R(K) in ;In Chapter 7 we illustrate our results by looking at their implications when G is a crescent. Several interesting function theory problems are studied.;In Chapter 8 we study arclength and harmonic measures. Let G be a Dirichlet domain with a countable number of boundary components. Let...