Spectra of Some Composition Operators

If H is a Hilbert space of holomorphic functions on the unit ball BN in CN and φ is a non-constant holomorphic map of the unit ball into itself, the composition operator Cφ is the operator on H defined by Cφf = f ∘ φ. In this paper, we give spectral information for bounded composition operators on some weighted Hardy spaces under the condition that φ is univalent and has a fixed point in the ball. When H is the usual Hardy space or a standard weighted Bergman space on the unit disk, this information shows that the spectrum of the composition operator is the disk centered at 0 whose radius is the essential spectral radius of the operator together with some isolated eigenvalues.