The Product Property Of Toeplitz Operator And Hankel Operator

The has been a hundred years of research history on operator theory in function space.In recent decades,due to the scholars' research on their painstaking attainments,it has gradually become a hot research topic.The core issue of its theoretical research is how to describe the properties of operators in different function spaces through the properties of symbolic functions.This paper is based on Hardy space.The product properties of Toeplitz operators and Hankel operators on Hardy spaces are mainly studied.Since these two types of operators are closely related to mathematical branches such as function theory,Banach algebra theory and operator theory,and also have important applications in physics,information cybernetics,engineering technology,probability theory and other related fields,the theoretical research on Toeplitz operators and Hankel operators has attracted the attention of many scholars and gradually become the focus of scholars.This paper mainly focuses on the Hardy space of the unit disk.The algebraic properties of Toeplitz operators and Hankel operators are mainly studied as follows : the commutativity of the product of Toeplitz operators,the commutativity of the product of Toeplitz operators after modular removal of a compact operator,the conditions under which the product of three Toeplitz operators is a Toeplitz operator and Hankel operator,and the boundedness and finite rank properties of small Hankel operators.The main structure of this paper is as follows :Firstly,we briefly introduce the related background knowledge of Toeplitz operators and Hankel operators,as well as the research background,development history and dynamics of the product of Toeplitz operators and Hankel operators.Secondly,the commutativity of the product of a Toeplitz operator is discussed on the Hardy space of the unit disk.By means of the Brown-Halmos theorem,the necessary and sufficient conditions for the commutativity of the product of a Toeplitz operator are characterized by using the Coburn lemma and other theories.Then,according to the Brown-Halmos theorem,the necessary and sufficient conditions for the recommutability of the product of a specific three Toeplitz operators are characterized,which paves the way for the of the recommutability of any three Toeplitz operators.Thirdly,the product problem of three and any number of Toeplitz operators is discussed on the Hardy space of the unit disk.By means of Brown-Halmos theorem and related theories,the necessary and sufficient conditions for the product of three Toeplitz operators to be equal to a Toeplitz operator are given,and the necessary and sufficient conditions for the product of three or any number of Toeplitz operators to be equal to a Hankel operator are characterized.Finally,we discuss the boundedness and finite rank properties of the small Hankel operator obtained by limiting the Hankel operator on the classical Hardy space,and then give the necessary and sufficient conditions for the product of the two small Hankel operators to be a finite rank operator.