Spectrum of Compact Weighted Composition Operators on the Weighted Hardy Space in the Unit Ball

Abstract

Let BN be the unit ball in the N-dimensional complex space, for ψ, a holomorphic function in BN, and ϕ, a holomorphic map from BN into itself, the weighted composition operator on the weighted Hardy space H2(β,BN) is given by (Cψ,ϕ)f=ψ(z)f(ϕ(z)), where f∈H2(β,BN). This paper discusses the spectrum of Cψ,ϕ when it is compact on a certain class of weighted Hardy spaces and when the composition map ϕ has only one fixed point inside the unit ball.