The Essential Norm and Spectrum of a Weighted Composition Operator on H^∞ (BN)

For φ a holomorphic self-map of the unit ball BN of C^N, and u in H^∞ (BN) (the Banach space of bounded holomorphic functions on BN), we investigate the essential norm and spectrum of the weighted composition operator uCφ acting on the space H^∞ (BN). For φ univalent, not unitary on any slice, and fixing a point of BN, we obtain a complete characterization of the spectrum of uCφ.