SPECTRA OF SOME COMPOSITION OPERATORS

If H is a Hilbert space of holomorphic functions on the unit ball B-N in C-N and phi is a non-constant holomorphic map of the unit ball into itself, the composition operator C-phi, is the operator on H defined by C(phi)f=f circle phi. In this paper, we give spectral information f or bounded composition operators on some weighted Hardy spaces under t he condition that phi is univalent and has a fixed point in the ball. When H is the usual Hardy space or a standard weighted Bergman space o n the unit disk, this information shows that the spectrum of the compo sition operator is the disk centered at 0 whose radius is the essentia l spectral radius of the operator together with some isolated eigenval ues. (C) 1994 Academic Press, Inc.