The essential norms and spectra of composition operators

This paper gives a complete characterization of the spec-tra of composition operators acting on H ∞ in the case that the symbol ϕ has an interior fixed point. This is done after it proves that the essential norm of a composition operator acting on H ∞ is either 1 or 0. 1. Introduction. Throughout this paper, D denotes the unit disk {z: |z | < 1}, ϕ denotes an analytic self-map of D, Cϕ is the composition operator defined by Cϕ(f) = f ◦ϕ, ‖Cϕ‖e and ρe represent the essential norm and essential spectral radius of Cϕ respectively, and H(D) is the space of analytic functions on D