Study On Properties Of Toeplitz And Hankel Operators On The Harmonic Bergman Spaces

The operator theory in function spaces is a popular branch in the field of linear operators and has caused widespread concern, this is because many deep problems in operator theory can be modeled as the corresponding operator problems on specific function spaces, which derived from functions that have some special properties. Studies on these "specific" operators will reveal the intrinsic properties of the "abstract" operators. Toeplitz operator and Hankel operator on the Bergman spaces as important branches in operator theory get the attention of scholars in the recent half of century. On the one hand, they are closely related to many classical problems in operator theory and function theory such as the invariant subspace problem. On the other hand, they have very important applications in quantum mechanics, control theory, wavelet analysis and other disciplines. Studies on the two operators will produce positive effects to explore the structure of operator theory and even linear operators and its applications, and will also promote the integration of operator theory and algebra, geometry, topology and other fields. In this paper, we mainly study the product problem and commuting problem of Toeplitz operator and small Hankel operator with quasi-homogeneous symbols on the harmonic Bergman space of the unit disk. Meanwhile, the product problem and commutativity of quasi-homogeneous and separately quasi-homogeneous Toeplitz operators on the pluriharmonic Bergman space of the unit ball are characterized.In Chapter 1, we mainly introduces relevant background knowledge about Toeplitz operator and Hankel operator on Hardy space, Bergman space and harmonic Bergman space. Some research history and situation of boundedness, compactness, finite rank, product problem and commuting problem and others of Toeplitz operator and Hankel operator are reviewed.In Chapter 2, we mainly study some algebraic properties of Toeplitz operators and small Hankel operators with radial and quasi-homogeneous symbols on the harmonic Bergman space of the unit disk. We solve the product problem of quasi-homogeneous Toeplitz operator and quasi-homogeneous small Hankel operator. Meanwhile, we characterize the commutativity of quasi-homogeneous Toeplitz operator and quasi-homogeneous small Hankel operator.In Chapter 3, we mainly study some algebraic properties of Toeplitz operator with quasi-homogeneous or separately quasi-homogeneous symbol on the pluriharmonic Bergman space of the unit ball. Firstly, we determine when the product of two Toeplitz operators with cer-tain separately quasi-homogeneous symbols is a Toeplitz operator. Secondly, we discuss the zero-product problem for several Toeplitz operators, one of whose symbols is separately quasi-homogeneous and the others are quasi-homogeneous functions, and show that the zero-product problem for two Toeplitz operators has only a trivial solution if one of the symbols is separately quasi-homogeneous and the other is arbitrary. Finally, we also characterize the commutativity of certain quasi-homogeneous and separately quasi-homogeneous Toeplitz operators.