Operators on the Bloch and Bergman spaces of several complex variables

The main subjects studied in this dissertation are the isometries of little Bloch spaces on the unit polydisk in {dollar}{lcub}bf C{rcub}sp{lcub}n{rcub}{dollar} and the weak compactness of composition operators on the Bergman space of strongly pseudoconvex domains.; Isometries of different holomorphic function spaces have been studied by many authors. For the Hardy spaces and Bergman spaces of bounded symmetric domains, the characterizations of isometries on them have been established. For the Bloch space, Cima and Wogen found the structure of the isometries in the case of the unit disk in 1980. Then in 1989, Krantz and Ma studied the isometries of the little Bloch space on the unit ball in {dollar}{lcub}bf C{rcub}sp{lcub}n{rcub}{dollar} and obtained the result that every such isometry is given by composition with a Mobius transformation. In this dissertation, we showed that under a mild assumption, the isometries of the little Bloch space on the unit polydisk have the same structure. This is done in the second chapter.; The second subject of the dissertation is motivated by Sarason's recent result on the equivalence of weak and norm compactness of composition operators on the Hardy space {dollar}{lcub}cal H{rcub}sp1(partial D){dollar}. More recently, Li and Russo extended Sarason's result to the case of strongly pseudoconvex domains in {dollar}{lcub}bf C{rcub}sp{lcub}n{rcub}{dollar}. In this dissertation, I considered the composition operators on the Bergman space {dollar}Asp1(Omega){dollar}, where {dollar}Omega{dollar} is a strongly pseudoconvex domain or a bounded symmetric domain of tube type. Chapter 3 of this dissertation is devoted to the proof, in this context, of the result similar to that of Sarason. The proof requires duality theorems for the Bergman space. We give a proof for these dualities in the case of strongly pseudoconvex domains in chapter 3.