Some Problems For Toeplitz Operators On The Function Spaces Of Several Complex Variables

In recent decades, Toeplitz operator and Hankel operator become an active branch of the function space operators theory, and attract the attention of many scholars. This is because they are closely related with operator theory、operator algebra、function the-ory、differential equations and so on. On the other hand, they have many important applications in control theory、quantum mechanics and wavelet analysis, etc.The research of scholars have obtained many important achievements for Toeplitz operator and Hankel operator on the Hardy space and Bergman space of unit disk. With the further development of research work and the pushing of Application background, people started to define corresponding Toeplitz operators and Hankel operators on Hardy space、Bergman and Dirichlet space of various domains. Meanwhile, the defined domain of these operators began to expand from analytic Bergman and Dirichlet space to harmonic Bergman and Dirichlet space. However, many basic problems is difficult to be solved for the reason that the property of function is very different in various function spaces. Thus. we need to use a series of new research tools and methods. In this paper, we mainly study some problems for Toeplitz operators on the pluriharmonic Bergman spaces of unit ball and Bergman spaces of polydisk.In Chapter1, we introduce some background knowledge about multiplication opera-tors、Toeplitz operators、Hankel operators. Some research situation for these operators about boundedness, compactness, algebraic properties, and the product properties and so on are reviewed.In Chapter2, We construct an operator R whose restriction onto weighted pluri-harmonic Bergman Space bμ2(Bn) is an isometric isomorphism between bμ2(Bn and l2#. Furthermore, using the operator R we prove that each Toeplitz operator Ta with radial symbols is unitary equivalence to the multication operator γa,μI acting on l2#. Meanwhile, the Wick function of a Toeplitz operator with radial symbol gives complete information about the operator, providing its spectral decomposition.In Chapter3, we completely characterize compact Toeplitz and Hankel operators on the pluriharmonic Bergman space. We establish the short exact sequences associated with the Toeplitz algebra and small Hankel algebra by using the criterion for compactness of Toeplitz operators.In Chapter4, We prove a reverse Holder inequality by using the cartesian product of dyadic rectangles and the dyadic cartesian product maximal function on Bergman space of polydisk. Next, we will furthemore describe when for which square integrable holomorphic functions f and g on the polydisk the densely defined products TfTg are bounded invertible Toeplitz operators.