多复变函数与算子及算子代数的若干问题

The project is on the study of operators and algebras on analytic function spaces for several complex variables. Using operator method, we research some important problems.which are relative to function theory in several complex variables. Using composition operator technique, a problem posed by W.Rudin is solved in some special cases, and the extension properties of analytic maps are discussed. In addition, we define Toeplitz operators on Dirichlet spaces and study systematically their properties and structure, some new.phenomenonna are found. These operators and algebras are important examples in general operators and algebras. On the.other hand, the automorphism group of the Toeplitz C*-algebra on the polydisk is characterized completely. In aspect of structure of Toeplitz operators, the commutants of Toeplitzoperators with symbols which are finite Blaschke products are characterized, and some reducible properties of these operators are discussed.

函数空间上的算子及生成的代数在当代核心数学的许多分支中起重要作用，它为一般算子代数与K-理论以及其它数学分支提供有用的信息。本项目主要研究高维函数空间上托普利兹C*代数的可解扩张与I-型因子及自同构群等问题，探讨高维哈代空间上复合算子及其代数以及多复变函数中的有意义的问题，可望对沟通算子理论与其它数学分支的关系起一定作用。