Algebraic Properties Of Toeplitz Operators In Function Spaces

Toeplitz and Hankel operators on Hardy or Bergman space is an active branch in operator theory. They are not only closely related with the other discipline of mathemat-ics, but also have many important applications in control, quantummechanics, probability and statistics, etc. It has been found that some classical problems in function and op-erator theory are equivalent to some problems of Toeplitz operators on Bergman space. For example, invariant subspace problem. On the other hand, many interesting analytic problems in complex analysis and differential equations are arose in the study of Bergman space and Toeplitz operators. These make scholars to pay more attention to the study of Toeplitz operators on Hardy and Bergman space.Since the twentieth century fifties, the study of Toeplitz and Hankel operators has a good development, especially on the Hardy and Bergman space, and many important results have been obtained. Those results have important applications in mathematics and engineering technology. On the Bergman space or several variables Hardy space, Toeplitz operators with harmonic symbols is well studied, but Toeplitz operators with general symbols seems quite challenging and is not fully understood unit now. and on vector valued Bergman space, few is known. In this paper, we mainly deal with the commutativity of Toeplitz operators and small Hankel operators on Hardy space of the bidisk, algebraic properties of Toeplitz operators with quasihomogeneous symbols on the Bergman space and dual Toeplitz operators on the valued Bergman space. This paper is organized as follows:In Chapter 1, some background information about Toeplitz operators, Hankel oper-ators, dual Toeplitz operators is reviewed, and the research course of algebraic properties of Toeplitz operators (such as products of Toeplitz operators, commutativity and so on) is also given.In Chapter 2, we characterize when the Toeplitz operator Tf and the small Hankel operator Hg commute on the Hardy space of the bidisk. At first, using theory of Fourier series, we give a necessary and sufficient condition on the symbols to guarantee Tf*Hg= HgTf*. In the following, using the function theory, we discuss the commutativity of Tf and Hg, where f=f+++f-In Chapter 3, commutativity of Toeplitz operators and small Hankel operators on the Bergman space is investigated. Using theory of Mellin transformation and Mellin con-volution, we give a necessary and sufficient condition on the general symbols to guarantee Tf and Hg are commuting, and give a good description for some special symbols. We also prove that if two small Hankel operators commute on the Bergman space and the symbol of one of them is quasihomogeneous, then the other one is also quasihomogeneous.In Chapter 4. we discuss some algebraic properties of Toeplitz operators on the Bergman space of the polydisk Dn. At first, we introduce Toeplitz operators with quasi-homogeneous symbols and property (P). Secondly, we study commutativity of certain quasihomogeneous Toeplitz operators and commutators of diagonal Toeplitz operators. Thirdly, we discuss finite rank semicommutators and commutators of Toeplitz operators with quasihomogeneous symbols. Finally, we solve the finite-rank product problem for Toeplitz operators on the polydisk.In Chapter 5, some algebraic properties of Toeplitz operators on the valued Bergman space is discussed. We characterize on the vector valued Bergman space when a block dual Toeplitz operator is bounded or compact, and give some necessary and sufficient conditions for the product of two block dual Toeplitz operators to be a block dual Toeplitz operator. At last, we characterize the commuting (semi-commuting) and the essentially commuting (semi-commuting) of block dual Toeplitz operators.