Some Problems For Toeplitz Operators On Bergman Space Of The Unit Ball

Bergman space, Toeplitz operators and Hankel operators as active branch in operator theory not only closely link to many branchs of mathematics, but also closely to the other discipline, especially in wavelet analysis and signal analysis. Over the last decade, it has been found that some classic problem about function theory and operator theory are closely related to Bergman space. For example, invariant problem. Many interesting problem about complex analysis and differential equations have been arising in the study of Bergman space and Toeplitz operator. This makes scholars be more and more interested in studying the Toeplitz operators.We deal with commutativity of dual Toeplitz operators of the unit ball, such as the characterizations of commuting dual Toeplitz operators, essentially commuting dual Toeplitz operators and essentially semi-commuting dual Toeplitz operators. The following results are presented: by the property of mean value of the holomorphic functions and the property of the reproducing kernel k_ω^(α). we describe when two dual Toeplitz operators with bounded measurable functions symbol commute. As a consequence, we make a conclusion that the dual Toeplitz operator is normal if and only if the range of the symbol lies on a line. Moreover, we discuss the characterizations of essentially commuting dual Toeplitz operators and essentially semi-commuting dual Toeplitz operators.We discuss the compact operators on weighted Bergman space A_α^p(B_n), where p > 1. We prove that: "closely associated with function theory", the necessary and sufficient conditions for the operator be compact on weighted Bergman spaces of the unit ball under some integrable conditions.A Nehari-type theorem is investigated. We use the Atomic Decomposition of the Bergman space A_α^p to prove that if S is a bounded linear operator acting on the weightedBergman spaces A_α²(B_n) on the unit ball in Cⁿ such that (?)(i = 1.....n). whereT_zⁱ = z_if and (?), and where P is weighted Bergman projection, then S must be a Hankel operator.Products of Toeplitz operators and Hankel operators on the weighted Bergman space of the unit disk. We prove the fact that the invariant laplacian commutes with the Berezin transform on the weighted Bergman space A_α². secondly, we consider the question for which square integral functions f and g on A_α² the densely defined products H_fH_g^* are bounded and give a sufficient condition. Furthermore, we completely describe when Hankel products are compact. We also obtain similar results for the mixed Haplitz products H_g(?) = T_fH_g^*, where f and g are square integrable on the unit disk and f is analysis.