Dual Toeplitz Operators On The Polydisk

In this dissertation, we mainly study the dual Toeplitz operators on the orthogonal complement of the Bergman space of the unit polydisk, and focus on commutativity, essential commutativity and algebraic structure of the dual Toeplitz operators. In the research we take advantage of the tight relations between dual Toeplitz operators and Toeplitz operators, Hankel operators, as well as the function theory of several complex variables.In chapter 1, we discuss some related research ground, and give some basic definitions and symbols. At last, we show the research significance.In chapter 2, we characterize commuting dual Toeplitz operators with bounded pluriharmonic symbols. We prove that S_φS_ψ=S_ψS_φif and only ifφandψsatisfy one of the following conditions:(1) Bothφandψare analytic on Dⁿ.(2) Both (?) and ;xx, are analytic on D^.(3) There exist two constants A and B, not both zero, such that A_φ+B_ψis constant on Dⁿ.In chapter 3, we characterize essentially commuting dual Toeplitz operators with bounded measurable symbols and bounded pluriharmonic symbols respectively. In the former case, we get that T_fT_g-T_gT_f is compact if and only if‖(H_gk_ω)(?)(H_?-K_ω)(?)(H_?k_ω)‖→0,ω→αDⁿ。In chapter 4, we prove the dual Toeplitz algebra I(C(D)) contain the idealΚof compact operators as its semicommutator ideal, and study its algebraic structure. The following short exact sequence is established:(0)→semiI(L^∞(Dⁿ))→I(L^∞(Dⁿ))→L^∞(Dⁿ)→(0).