Function Space Of Toeplitz Operators On The Exchange,

In this thesis, we mainly concern the commutativity of Toeplitz operators with more general symbols on the Dirichlet space or harmonic Dirichlet space of the unit disk. It is well known that the Dirichlet space is one important class of function space and on which the function theory and operator theory have significant differences from the classical function spaces such as the Hardy space and Bergman space. So, on the Dirichlet space, the Toeplitz operator, as one most important class of special operators, plays great roles in function and operator theory.In Chapter One and Two, we discuss the commutativity of Toeplitz operators on the Dirichlet space D0 and D in the classical derivative sense. We introduce a class of more general symbols Lθ∞,1 and prove that there are two cases for the commutativity of Toeplitz operator:one case for symbols inΩis just the general-ization of harmonic symbols and the other for symbols in Lθ∞,1 -Ωis very complex just as the case of commutativity on the Bergman space. In the later case, we con-sider the commutativity of Toeplitz operators with the quasihomogeneous symbols and obtain some results which are very different from the case on the Hardy space and Bergman space. Furthermore, on the Dirichlet space (?), the commutativity of Toeplitz operators with symbols inΩis rather than the simple generalization of the case with harmonic symbols, and there exist some new cases, for which we give some examples.In Chapter Three, we discuss the commutativity of Toeplitz operators on the Dirichlet space D0 with the derivative in the distribution sense. We characterize not only the function classes of L∞,1 and L2,1, but also the boundedness and compactness of Toeplitz operator. On these basis, we obtain completely the commutativity of Toeplitz operators with general symbols on the Dirichlet space D0.In Chapter Four, we study some algebraic properties of Toeplitz operators on the harmonic Dirichlet space with the derivative in the distribution sense. We first give a characterization for boundedness and compactness of Toeplitz operators. Next we characterize commuting Toeplitz operators. At the same time, we study the prod-uct problem of when product of two Toeplitz operators is another Toeplitz operator. The corresponding problems for compactness are also studied. The commutativity we obtained for Toeplitz operators with general symbols are quietly different from that on the Hardy, Bergman and Dirichlet spaces.