Composition operators on weighted Hardy spaces

For a Hilbert space H of functions analytic in the unit disc D and {dollar}phi{dollar} an analytic function that maps D into itself the composition operator C{dollar}sb{lcub}phi{rcub}{dollar} is defined by C{dollar}sb{lcub}phi{rcub}{dollar} f = f {dollar}circ phi{dollar}, for f {dollar}in{dollar} {dollar}H{dollar}. Composition operators on the classical Hardy space H{dollar}sp2{dollar} have been studied extensively. Results on some types of weighted Hardy spaces (like Bergman and Dirichlet spaces) were obtained recently.; The purpose of this thesis is to study the behaviour of composition operators on general weighted Hardy spaces H{dollar}sp2(beta){dollar}. In such a general context even some basic questions, like boundedness, are sometimes hard to answer. There are some nice classes of functions (for example, rotations, or functions that are analytic on the closed unit disc and have supremum strictly smaller than one) that induce bounded composition operators on all weighted Hardy spaces. On some spaces (for example, the ones that satisfy the relation H{dollar}sp{lcub}infty{rcub}(beta){dollar} = H{dollar}sp2(beta){dollar} and that contain the induction function), having supremum strictly smaller than one is sufficient for the function to induce not only a bounded operator but even one that is in the trace class. It is not known if this is true in the case of general weighted Hardy spaces.; Disc automorphisms of the unit disc play an important role in the question of compactness of composition operators, especially in determining the position of the Denjoy-Wollf point of the inducing map and the spectrum of the operator. Results concerning this are contained in the third chapter.; Disc automorphisms are also very interesting in the case of the classical Hardy space H{dollar}sp2{dollar}. We prove that they induce cyclic composition operators.; Finally, we look at the spaces S{dollar}sb{lcub}rm a{rcub}{dollar} for a {dollar}>{dollar} 1, which are a natural continuation of the family of Bergman and Dirichlet spaces. If 1 {dollar}{dollar} 3/2 we prove that any function in the space that maps the unit disc into itself induces a composition operator that is well defined. Concerning compactness, if a {dollar}>{dollar} 1/2 then C{dollar}sb{lcub}phi{rcub}{dollar} is compact if and only if C{dollar}sb{lcub}phi{rcub}{dollar} is in the trace class, which happens if and only if {dollar}phi{dollar} belongs to S{dollar}sb{lcub}rm a{rcub}{dollar} and the supremum of {dollar}phi{dollar} is strictly smaller than one.