Truncated Toeplitz Operators And Multiplication Operators On Bergman Space

Operator theory is a very important part of functional analysis.Many mathematicans all over the world focus on the field of operator theory on analytic function spaces.Toeplitz operator forms an important class of operators on analytic function spaces.Although the scientific research of Toeplitz operators has lasted for more than half a century,many results have been got,there are still many problems we can do.On one hand,this is because Toeplitz operators are of importance in connection with a variety of problems in von Neumann algebra,noncommutative geometry,random matrix,information and control theory,quantum mechanics,and several other fields.On the other hand,our study on the Toeplitz operator and Toeplitz algebra plays an important role in the development of Mathematics,Physics,Engineering and Technology.In this thesis,we will focus on truncated Toeplitz operators on Model spaces and multiplication operators on Bergman spaces.Firstly,we study the compactness of truncated Toeplitz operators on the Model spaces via product of Hankel operators on Hardy space.It is well known that the bounded Toeplitz operators on Hardy space have the unique symbol,while the symbols for truncated Toeplitz operators are not unique.Baranov,Chalendar,Fricain constructed a bounded truncated Toeplitz operator with an unbounded symbol.So we just consider the truncated Toeplitz operator with bounded symbol.Using the product of Hankel operators on Hardy space and function algebra,we get a necessary and sufficient condition on compactness of truncated Toeplitz operators on the Model spaces.As a consequence,we can see the previous results of compact truncated Toeplitz operators are just a special case for our theorem.Second,we study the von Neumann algebras generated by the multiplication operators on Bergman spaces.We mainly concerns the von Neumann algebras induced by a tuple of multiplication operators on Bergman spaces which arise essentially from holomorphic proper maps over higher dimensional domains.We study the structures and abelian properties of the related von Neumann algebras,and in interesting cases they turn out to be tightly related to a Riemann manifold.There is a close interplay between operator theory,geometry and complex analysis.Finally,the main results of the thesis are summarized.Moreover,we pose some problems we will study in the future.