Dual Toeplitz Operators On Harmonic Hardy Space Over Polydisc

In the current field of mathematics, there are only few theories of dual Toeplitz oper-ators on harmonic Hardy space, since it is always about Hardy space,Bergman space,even harmonic Bergman space. The harmonic Hardy space on T" is introduced in this paper. This paper is divided into four parts.In the first part, we introduce the current situation about the research. Then it focuses on the definition of the harmonic Hardy space h~2(T~n)= H~2(T~n)+H~2(T~n), and the dual Toeplitz operator: where Q= I-P, and P is the projection from L2(T~n) onto h~2(T~n). We also get an equation by the multiplication operator Mφ on L2(T~n): This equation is used frequently in this paper.The most important theory in the second part is the spectral inclusion theorem on h~2(T~n):Ifφ∈ L∞(T~n), then R(φ) (?) σ(Sφ). There are some corollaries introduced after the proof of the spectral inclusion theorem. For example, the operator Sφ is self-adjoint, if and only if φ is real function.We start an investigation of the commuting dual Toeplitz operator Sφ on h~2(T~n) in the third part. Through a simple example, we can know the importance of the harmonicity. So we can not get a theory for all operators. In this paper, we only study the situation that n= 2, and the symbol function φ with following form: where f, g ∈ H∞(D2), z, w ∈ T, mi ni ∈ N, i= 1,2.In the last part, the study of the semi-commuting operator Sφ is in the same situation. Finally we get the conclusion by the discussion of the parameters m_i, n_i.