(M,?)-Berezin Transforms And Toeplitz Operators

An important branch of the operator theory is the operator theory in function spaces.The core issue of the operator theory in function spaces is how to characterize the properties of operators and operator algebras by the properties of their symbol functions.In this paper,we study the dense of the Toeplitz operator algebra and the compactness of a special class operators.At the same time,we do some research on the complex symmetric Toeplitz operators on the Hardy and Bergman spaces.The paper is organized as follows:In the first chapter,we introduce the research background of operator theory in function spaces.The algebraic properties of Toeplitz operators are summarized,such as commutativity,compactness and denseness.We also introduce the research background and proposition for complex symmetric operators.In Chapter 2,we give the definition of the(m,?)-Berezin transform and study its basic properties.Moreover,we find the(m,?)-Berezin transform is one-to-one.In Chapter 3,we prove that every bounded linear operator on weighted Bergman space over the polydisk can be approximated by Toeplitz operators under some conditions.The main tool here is the(m,?)-Berezin transform.Moreover,we find a class bounded operators which can be approximated in norm by Toeplitz operators without any conditions.We use the(m,?)-Berezin transform to characterize the compactness of this class.Finally,we define the radialization of the operator and function and obtain the relationship between the radialization and the(m,?)-Berezin transform.Chapter 4 introduces the complex symmetric property of Toeplitz operators on the Hardy and Bergman spaces.We find the Toeplitz operator with the Berezin transform of complex sym-metric operator on the Hardy space is complex symmetric with the same conjugation.However,this property disappears on the Bergman space.