The spectra of composition operators from linear fractional maps acting upon the Dirichlet space

Let φ:D→D be a non-constant linear fractional transformation (necessarily of the form φ(z)=az+bcz+d). Let D denote the Dirichlet space of analytic functions. We determine the spectrum of the composition operator Cϕ:D→D defined by Cϕ(f)=f∘φ. Eigenfunctions for the operator Cϕ:H2→H2 frequently do not belong to the space D. However, spectral results for the operator Cϕ:D→D, much like those that have already been demonstrated for the operator Cϕ:H2→H2, are presented in this paper.