Relating composition operators on different weighted Hardy spaces

In a 1988 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. In this paper, we use Cowen's adjoint formula to give a unitary equivalence relating composition operators on different weighted Hardy spaces. This result is then applied to some composition operators on the Sa 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.