| In this thesis we explore some of the properties of composition operators acting on weighted Hardy-Hilbert spaces. Much of our work is related to invariant subspaces of composition operators.;Chapter two deals with some properties of the invariant subspaces for the operators under consideration; it is divided into two main parts, according to where the Denjoy-Wolff point lies.;In chapter three we deal with orbit reflexivity, and show that every composition operator and also every adjoint of every composition operator is orbit reflexive. We also find some sufficient conditions for a general operator to be orbit reflexive, and then apply them to composition operators.;Chapter four discusses the Berezin symbol on general H 2(beta) spaces. In particular, we discuss the cases when the generating functions are one-to-one. We show that this is the case for a large number of spaces that we have studied so far, and then characterize normal, self-adjoint and unitary composition operators acting on these kinds of spaces, with connection to their Berezin symbol.;We begin with a detailed introductory chapter, where we give all the definitions and develop the framework and the tools which we will be using. We find out that we need to impose three natural conditions on the space.;Finally, in our fifth and last chapter, we summarize the questions that arose while working on the dissertation. We then solve some of these questions in the appendix.;The results of chapter three and the appendix can be found in the joint paper with Atzmon, Mahvidi and Rosenthal [1], and those of chapter four in the paper [16].;This thesis was typed using LATEX. |
