Toeplitz Operators And Hankel Operators On The Function Spaces

Toeplitz operators and Hankel operators are two types of important operators on the function spaces. By reason that they are closely related with other branches of mathematics, such as operator theory, function theory, Banach algebra, and have many important applications in physics, probability, control and so on, Toeplitz operators and Hankel operators become the hot problems for discussion in the field of operator theory, and have attracted multitudinous scholar's attention. Since 1950s, they have been studied in depth by many mathematicians and a lot of important achievements have been obtained. But so far there exist many unsolved problems about Toeplitz operators and Hankel operators, in particular, on the higher dimensional region. The main work of this thesis is to discuss the (essential) commutativity of k^th-order slant Toeplitz operators on the unit circle, the commutativity and hyponormality of Toeplitz operators on the weighted Bergman spaces of the unit disk, the boundedness of Toeplitz products and the boundedness and compactness of Hankel products on the weighted Bergman spaces of the unit ball.The second chapter studies the properties of k^th-ovder slant Toeplitz operators. We obtain the necessary and sufficient conditions for the (essential) commutativity of k^th-order slant Toeplitz operators and the compactness of products of such operators, in particular, the commutativity and the essential commutativity of such operators are in accord. Besides, through the study of essentially bounded functions on the unit circle, we get the necessary and sufficient conditions for the essential commutativity of k^th-order slant Toeplitz operators with special symbols.By mellin transform the properties of Toeplitz operators on the weighted Bergman spaces of the unit disk are studied in the third chapter. Firstly, we get the necessary and sufficient conditions for Toeplitz operators that commutes with another such operator whose symbol is a monomial, meanwhile, we completely characterize when the mellin transform of the bounded function on the interval [0,1) is a rational function. Secondly, by means of the properties of the mellin transform, we obtain some necessary and sufficient conditions for the hyponormality of Toeplitz operators with special symbols.Toeplitz and Hankel products on the weighted Bergman space of the unit ball are investigated in Chapter 4. Let-1<γ<∞,for f,g∈A_γ² satisfying sup_{ω∈Bn}B_γ[|f|²](ω)B_γ[|g|²)(ω)<∞, we get the operator T_f^αT_?^α:A_α²â†’A_α² determines a bounded linear operator for each α>γand the operator T_f^γT_?^γ:A_α²â†’A_α² is a bounded operator for-1<α<γ.Next, utilizing the expression of k_ω⁽(α)(?)k_ω⁽(α) we obtain the necessary condition and the sufficient condition for the boundedness of Hankel products, meanwhile, we get the necessary and sufficient condition for the compactness of Hankel products.