Compact Toepitz Operators On Weighted Bergman Space

The theory of operators and and operator algebras generated by them are developing quickly during the 20-30's of the last century, during functional analysis becoming a branch of mathematics, and accompanying the use deeply on dynamics and quantum physics, and it provides a new activate for the developing of the theory.As special class of operators, the reasons Toeplitz and Hankel operator studied by so many mathematicians is that they provide a model for the research of the general operator or algebra, as well as intervolve with many others branches of mathematics, such as: classical function theory, index theory, dynamics system, and so on. Moreover, they had provided some liable methods for the resolution of the open problems for the general operator theory, such as: the famous invariant subspace problem, it can be transformed as the study of the proportion of "saturated" for the invariant subspce lattice of the operator M_z on Bergman space. For that reason,the study of Toeplitz and Hankel operators are greatly activated recently, and it became an important constitute of the operator theory.Apart from Hardy space on the unit disk, Toeplitz and Hankel operators on Bergman space are studied until on the beginning of the 90's last century. In past years, much progress has been achieved in this field, e.g, the criteria for Toeplitz and Hankel operators being boundedness, compactness, Schatten classification and the spectrum of Toeplitz and Hankel operators on unit disk are obtainded([11]). At the same time, the boundedness and compactness of the products of some Toeplitz operators, the products of Toeplitz and Hankel opera-tors and different commutators([l,2,7,9,10]etc) are given.Based on the predecessor, the main result of this thesis is the proposition of the Toeplitz and Hankel operators on weighted Bergman space which expanded the result of the predecessor.In the first chapter of the thesis, some proposition of products of Toeplitz and Hankel operators on weighted Bergman space are introduced, and the sufficient and necessity condition are given.In the second chapter of the thesis, we will introduce the commutators [T°, T°] and [TJ, //￡], and obtainded the necessity and sufficient condition for the commutators compact as well as the necessity and sufficient condition for Toeplitz operator is essentially normal.