Spectral Properties Of Dual Toeplitz Operators On The Unit Polydisk

Although the spectral properties of operators have been studied by many scholars, there are few scholars to research the dual Toeplitz operators. The topic of this article is to study dual Toeplitz Operators. Inspired by the case of the unit sphere H~2(Sn) in the research written by Guediri, this paper considers the spectral properties of the dual Toeplitz operators on H~2(Tn), the Hardy space for the unit polydisk. This paper has four parts, and the object of this paper is the dual Toeplitz operators on the Hardy space of the unit polydisk H~2(Tn).In the first section of this article, we give a introduction of the operator theory, and then show a detailed description of the jobs about the dual Toeplitz operators and the Hardy Spaces by mathematical workers. This has a history of one hundred years. Then a brief introduction about the structure of this article is shown.In the second part, we define the Hardy space of the unit polydisk H~2(Tn), and we get some basic properties. Next, we give the reproducing kernel and the Poisson kernel of the Hardy space of the unit polydisk H~2(Tn). Then we introduce the dual Toeplitz operator on the Hardy space H~2(Tn):Suppose f?L?(Tn), the definition of dual Toeplitz operator is the following:Sf?:= (I-P)(f?)?H~2(Tn))(?),??H~2(Tn))(?).In the third part, we introduce the algebraic properties of the dual Toeplitz operators on the Hardy space of the unit polydisk H~2(Tn), including the commutativity and the product.In the last part of this article, we obtain a result that a dual Toeplitz operators on H~2(Tn) is quasinormal when its symbol is analytic or co-analytic. Then we get an important result which is called the spectral inclusion theorem.