Linear fractional composition operators on weighted Hardy spaces

This thesis consists of two parts. Chapters 2 and 3 comprise the first part. Its foundation is a 1988 paper by Carl Cowen (4). Chapter 4 is the second part. Its motivation is a 1987 paper by Nordgren, Rosenthal and Wintrobe (10). In his paper, Cowen found a formula expressing the adjoint of any linear fractional composition operator on the Hardy space as a product of Toeplitz operators and another linear fractional composition operator. We build upon this in two different directions. In chapter 2, a similar formula is obtained for the weighted Bergman spaces. This is used to calculate the norm of any composition operator with affine symbol. In chapter 3, Cowen\u27s adjoint formula is used to give a unitary equivalence relating composition operators on different weighted Hardy spaces. This result is then applied to some composition operators on the S$\sb{a}$ spaces. We find the spectrum of any linear fractional composition operator whose symbol has exactly one fixed point of multiplicity one on the unit circle. In chapter 4, a model is obtained for invertible hyperbolic and parabolic composition operators on H$\sp2$. This model shows that the adjoints of these composition operators are similar to block Toeplitz matrices constructed with weighted bilateral shifts and rank one operators