Composition operators on holomorphic function spaces in several complex variables - Clahane, Dana Dwight (ProQuest Dissertations & Theses, 2000)
In this dissertation, we study boundedness, compactness, and spectra of composition operators acting on Banach spaces of holomorphic functions on domains in [special characters omitted]. In Chapter 1, we list some preliminary facts and notation. In Chapter 2, we give purely function-theoretic conditions on a self-map of the unit ball Bn in [special characters omitted] such that C is compact on the weighted Bergman spaces [special characters omitted] for p > 0 and α > −1. In doing so, we prove two auxiliary results. First, we prove a Carleson measure-type theorem that relates α-Carleson measures and weighted holomorphic Sobolev spaces. Second, we prove a comparison result for bounded (respectively, compact) holomorphic composition operators. This result states that if a composition operator is bounded (respectively, compact) on the Hardy space Hp( Bn), then the operator is also bounded (respectively, compact) on the weighted Bergman spaces [special characters omitted] for p > 0 and α > −1. We show that a similar result compares boundedness (respectively, compactness) of holomorphic composition operators on [special characters omitted] and [special characters omitted] for −1 < α < β. In Chapter 3, we extend results of J. Caughran/H. Schwartz [CaS] and B. MacCluer [Ma2] by showing that on the Hardy space H 2(D) or the weighted Bergman space [special characters omitted] of a bounded symmetric domain containing the origin in [special characters omitted], the spectrum of a compact (or power-compact) composition operator induced by a holomorphic self-map of D consists of the set containing 0, 1, and all possible products of eigenvalues of ′(z0), where z 0 ∈ D is the fixed point of in D. The uniqueness and existence of this fixed point is proven under very general conditions. In particular, we are able to determine spectra in the case of Hp (Δn) and [special characters omitted], where Δn is the unit polydisk. In Chapter 4, we prove a multivariable extension of a result due to K. Madigan in [Mad], wherein it is shown that for analytic maps of the unit disk Δ ⊂ [special characters omitted], boundedness of C on the analytic Lipschitz spaces Liphα(Δ), for 0 < α < 1, is equivalent to the condition supz∈D 1-z 21-fz 1-a f′z <∞. </display-math>