Composition operators on holomorphic function spaces in several complex variables

In this dissertation, we study boundedness, compactness, and spectra of composition operators acting on Banach spaces of holomorphic functions on domains in Cn .;In Chapter 1, we list some preliminary facts and notation. In Chapter 2, we give purely function-theoretic conditions on a self-map &phis; of the unit ball Bn in Cn such that C&phis; is compact on the weighted Bergman spaces ApaBn for p > 0 and alpha > -1. In doing so, we prove two auxiliary results. First, we prove a Carleson measure-type theorem that relates alpha-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 ApaBn for p > 0 and alpha > -1. We show that a similar result compares boundedness (respectively, compactness) of holomorphic composition operators on ApaBn and Apb Bn for -1 < alpha < beta.;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 ApaD of a bounded symmetric domain containing the origin in Cn , the spectrum of a compact (or power-compact) composition operator induced by a holomorphic self-map &phis; of D consists of the set containing 0, 1, and all possible products of eigenvalues of &phis; '(z0), where z 0 ∈ D is the fixed point of &phis; 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 (Deltan) and ApaDn , where Deltan 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 &phis; of the unit disk Delta ⊂ C , boundedness of C&phis; on the analytic Lipschitz spaces Liphalpha(Delta), for 0 < alpha < 1, is equivalent to the condition supz∈D 1-z 21-fz 1-a f'z <infinity.