Products And Commutativity Of Toeplitz Operators And Hankel Operators On Function Spaces

Operator theory in function spaces is an important component of the operator theory, its core problem is how to describe the properties of operators by the properties of their symbol functions. In this paper, we study products and commutativity of Toeplitz operators, Hankel operators and dual Toeplitz operators on function spaces. The paper is organized as follows:In the first chapter, we review the background of operator theory in function spaces, and its development process.In Chapter 2, we first determine on the Hardy space of the unit disk when is the sum of products of Hankel and Toeplitz operators equal to zero, then we characterize when the product of a Toeplitz operator and a Hankel operator is a compact perturbation of a Hankel operator or a Toeplitz operator and when it is a finite rank perturbation of a Toeplitz operator.Chapter 3 deals with the commuting problem of Toeplitz operators on the Hardy space of the polydisk. By using induction on the dimension of the domain, we proved:Two Toeplitz operators can commute if and only if the Berezin transform of their commutator is n-harmonic.In Chapter 4, we consider Toeplitz operators on weighted pluriharmonic Bergman space on the unit ball. We characterize the Toeplitz operators commuting with Toeplitz operators whose symbols are certain separately radial functions or holomorphic monomials, and then give a partial answer to the finite-rank product problem of Toeplitz operators.Chapter 5 studies products of dual Toeplitz operators. We first construct the structure the-orem for dual Toeplitz algebra on the orthogonal complement of the Bergman space on the unit ball, as an application, we give an equivalent condition for the sum of finite products of dual Toeplitz operators to be a compact perturbation of a dual Toeplitz operator. Furthermore, we characterize when is the sum of products of two dual Toeplitz operators equal to a dual Toeplitz operator. At last, we obtain a similar conclusion on the orthogonal complement of the Hardy space on the sphere.