Some Problems Of Toeplitz Operators On Dirichlet Spaces Of Several Variables

In this paper, we consider Toeplitz operators on Dirichlet spaces of several variables. Al-gebraic properties of Toeplitz operators on the Dirichlet space of the unit ball are given and reducing subspaces of some Toeplitz operators on the Dirichlet space of the bidisk are charac-terized.Chapter1briefly introduces the background and the development of Toeplitz operator the-ory on the topic. The definition of Dirichlet spaces and correlate notions are also presented.Chapter2studies the algebraic properties of Toeplitz operators on the Dirichlet space of the unit ball. We characterize pluriharmonic symbol for which the corresponding Toeplitz operator is normal or isometric. We also obtain the description of commuting Toeplitz operators with conjugate holomorphic symbols.Chapter3studies some algebraic properties of Toeplitz operators with quasihomogeneous symbols on the Dirichlet space of the unit ball. We describe the commutators of a radial Toeplitz operator and characterize commuting Toeplitz operators with quasihomogeneous symbols. Then we show that finite rank product of such operators only happens in the trivial case. Furthermore, a necessary and sufficient condition is given for the product of two quasihomogeneous Toeplitz operators to be another quasihomogeneous Toeplitz operator.Chapter4describes reducing subspaces of Toeplitz operators Tz1N (or Tz2N) and Tz1N z2N on the Dirichlet space of the bidisk. The result shows that for Tz1N (or Tz2N) the structure of reducing subspaces on the Dirichlet space of the bidisk is similar to the case on the Bergman space of the bidisk, while for Tz1N z2N the situation is very different.Chapter5describes the reducing subspaces of Toeplitz operators Tz1N (or Tz2N) and Tz1N z2N on the weighted Dirichlet space of the bidisk. The result shows that for Tz1N (or T z2N) and Tz1N z2N the structure of reducing subspaces on the weighted Dirichlet space of the bidisk is similar to the case on the weighted Bergman space of the bidisk.