Operator Theory On Function Spaces

Research on operator theory in function spaces is always an important topic of the functional analysis which is in close contact with many areas of mathematics. As a branch of modern mathematics, it has formed a series of theoretical system after a long time of study.Properties of operator are studied in operator theory, including boundedness, compactness, Fredholm properties, spectral properties and algebraic properties. Operator theory in classic Bergman, Hardy, Dirichlet spaces of one variable has formed its fruitful and complete theoretical system. After A.Brown and P.R.Halmos expressed Toeplitz matrix in the form of Toeplitz operators, it has aroused extensive attention. R.G.Douglas, for the first time, introduced systematically boundedness, compactness, Fredholm properties, spectral properties of Toeplitz operators on Hardy space H2(T) with all kinds of symbols. Furthermore, A.Bottcher introduced systematically the properties of Toeplitz operators and Hankel operators on Hardy space HP(T)(1＜p＜∞). K.H.Zhu discussed Toeplitz operators, Hankel operators and composition operators on the classic Bergman space. Cao studied the Fredholm and spectral properties of Toeplitz operators on Dirichlet space.Because the structure of function spaces of several variables is more complex than that of one variable, we lost many powerful tools to study the properties of operators on these function spaces, and we can't generalize the results about one variable to several variables, but it also makes the operator theory of several variables become an active field. In this paper, we focus on the properties of Toeplitz operators and Hankel operators on the high-dimensional Bergman space and Hardy space. The paper is divided into following sections:(1) The trace Toeplitz operators with unbounded symbols on the weighted Bergman space of unit ball in Cn.(2) The trace Toeplitz operators with unbounded symbols on Bergman space of polydisc.(3) Fredholm properties of Toeplitz operators on Hardy space H1(Sn). (4) Boundedness of Hankel operators on Hardy space H1(Sn).Firstly, we construct a class of unbounded functions on the weighted Bergman space of unit ball in Cn such that the Toeplitz operator with these functions as symbol are compact, and we also construct the trace of Toeplitz operator on this basis. Sercondly, we use the similar method to construct the trace of Toeplitz operators with unbounded symbols on Bergman space of polydisc. Thirdly, we study the Fredholm properties of Toeplitz operators on Hardy space H1(Sn), and get the index formula. Finally, we discuss boundedness of Hankel operators on Hardy space H1(Sn).