Complex Symmetry Of Weighted Composition Operators And Toeplitz Operators

In this paper,we will study the complex symmetry of weighted composition operators on the classical Hardy space,Dirichlet space in the unit ball,and the complex symmetry of the monomial Toeplitz operator on the weighted Bergman space in the unit ball and the polydisk.This paper consists of six chapters and its structure is as follows:In the first chapter,we will introduce some basic symbols and concepts,and then we will introduce in detail the sources of the problems studied in this paper and the previous research results.In the second and third chapters,we will study the complex symmetry of weighted composition operators.In the second chapter,we will discuss the complex symmetry of weighted composition operators on the Hardy space in the unit ball.Compared with the unit disk,the situation will become more complicated,the research methods used will also be different from the unit disk.In the third chapter,we will study the complex symmetry of weighted composition operators on the Dirichlet space in the unit ball.We will give the concrete forms of two kinds of complex symmetric weighted composition operators on the Dirichlet space by constructing conjugate operator on the Dirichlet space.In addition,we will also characterize Hermitian weighted composition operators on the Dirichlet space in the unit ball.In the fourth chapter and fifth chapter,we will study the complex symmetry of monomial Toeplitz operator.In the fourth chapter,we will completely characterize when the monomial Toeplitz operator Tzpzq on the weighted Bergman space is JAsymmetric,where JAf(z)=f(Az)with the symmetric unitary n×n matrix A.Moreover,we will also determine all possible forms of A such that Tzpzq is JA-symmetric.As an application,we will construct the weakly closed unital commutative algebra generated by complex symmetric Toeplitz operators.In the fifth chapter,we will continue to discuss the complex symmetry of Toeplitz operator.We will completely describe the complex symmetry of Toeplitz operator Tzpzq on the Bergman space A2(Ω),where Ω is the 2-dimensional unit ball B2 or unit polydisk D2.We will prove that Tzpzq is complex symmetric on A2(Ω)if and only if p1=q2,p2=q1.Finally,in the last chapter,we will summarize the main contents of this paper and list some problems for further research.