Hyponormality Of Toeplitz Operators In Function Spaces

The operator theory in function spaces plays a very important role in operator theory, not only because a lot of instructive examples can be provided in the function spaces, but also because many problems from the abstract spaces can be changed into specific problems in function spaces.For example,the normal operator can be characterized clearly by the multiplication operator in some function space.In particular,the theory of Toeplitz operators in function spaces has received widespread attention because of the demand of their own theory development and the applications from quantum mechanics, control theory and so on.In 1950s,Paul R. Halmos introduced the concepts of subnormal operator and hy-ponormal operator based on the unilateral shift operator.In 1970,he raised the question in the paper "Ten problems in Hilbert space":"Is every subnormal Toeplitz operator on the Hardy space either normal or analytic?" Although this question was answered negatively, it leaded to the research on hyponormal Toeplitz operators.Carl C.Cowen firstly gave a complete characterization of the hyponormality of Toeplitz operators on the Hardy space. After that, many scholars obtained more delicate results about the hy-ponormality of Toeplitz operators with symbols in the class of some special symbols such as trigonometric polynomials.Moreover,the equivalent conditions for the hyponormality of some block Toeplitz operators and generalized Toeplitz operators were achieved. The hyponormality of the Toeplitz operators on the Bergman space has also been studied. Even though there have been some research findings,it is quite difficult to characterize the hyponormality of Toeplitz operators on the Bergman space completely.In this paper,we mainly study the hyponormality of Toeplitz operators in several function spaces.In Chapter 1,we recall some known results about the hyponormal Toeplitz operators.In Chapter 2,we focus on the hyponormality of the Toeplitz opera-tors on the weighted Bergman space Aα2(D).First of all, we consider the limiting behavior of some sequences induced by Hankel operators,and get that the limit of each sequence is not related with a.Then we give some properties of the hyponormal Toeplitz operators with harmonic symbols.In particular,we study the hyponormality of Toeplitz opera-tor on the Bergman space A2(D)with symbol in the class of trigonometric polynomial functions,and obtain a clear characterization about the hyponormality of Tf+g by the de-rived function of f and g.Moreover,we also discuss the hyponormality of some Toeplitz operators induced by some continuous non-harmonic functions and get some sufficient and necessary conditions.In Chapter 3,we completely characterize the hyponormality of Toeplitz operators on the Hardy space of the polydisk by the block matrix represen-tations of Hankel operators and the vector-valued functions.As an application, we give some examples to demonstrate the complexity of the hyponormal Toeplitz operators on the Hardy space of the polydisk.