Compactness Of Toeplitz Operators On Weighted Bergman Spaces Of The Unit Polydisk

Let A_φ~p(D~n) denote the Banach space consisting of all analytic functions in the unit polydisk D~n that are also p-integrable. By using polydisk function theory and Schur theorem, this paper deals mainly with the compactness of Toeplitz operators on A_φ~p(D~n) . In particular, some basic problems of operator theory, such as the bounded projection and duality in function spaces , and the compactness of a bounded operator S or Toeplitz operators on A_φ~p(D~n) if they satisfied some inte-grable conditions are discussed.Chapter 1 is concerned with the basic structure and properties of A_φ~p(D~n). The basic properties of Toeplitz operator , Berezin transform , and Mobius transform on D~n are also discussed.Chapter 2 is devoted to studying the bounded projections and duality in function spaces . We prove that P is a bounded projection from L_φ~p(D~n) onto A_φ~p(D~n). Using this projection, we show that the dual of A_φ~p(D~n) is topologically isomorphic to A_φ~p(D~n) Chapter 3 is concentrated at characting the compactness of a bounded operator S or Toeplitz operators on the weighted Bergman space A_φ~p(D~n) if they satisfied some integrable conditions. We prove that S is compact if and only if its Berezin transform vanishes on the topological boundary of the unit polydisk, and prove that a finite sum of finite product of Toeplitz operators is compact if and only if its Berezin transform vanishes on the topological boundary of the unit polydisk.