M-complex Symmetric Toeplitz Operators On Hardy Spaces

Complex symmetric operators are of great significance in the study of operator theory.They exist in all directions of many disciplines.Moreover,the complex symmetry of Toeplitz operator also plays a major role in scientific application and occupies an important position in the theoretical research of some mathematical fields.This paper discusses the complex symmetry of Toeplitz operators with different symbols on Hardy spaces on the basis of previous studies.The m-complex symmetry of Toeplitz operator is an important theoretical property of operator.By consulting the literature resources,it is found that many problems about the symbolic characteristics of 2-complex symmetric Toeplitz operators and 3-complex symmetric Toeplitz operators on Hardy spaces have not been completely solved.In this paper,starting from the complex symmetric Toeplitz operators on Hardy spaces,we discuss the symbolic characteristics of 2-complex symmetric Toeplitz operators and3-complex symmetric Toeplitz operators on Hardy spaces,and study the necessary conditions for Toeplitz operators of some special symbols to be 2-complex symmetric,and describes the relations between 3-complex symmetric Toeplitz operator and the normal Toeplitz operator.This paper is divided into five parts,which respectively introduce the relevant knowledge background of the research work,the source of the topic,the required preparatory knowledge,the derivation process and the relevant conclusions,and finally summarize the conclusions and the prospect of the future.The first chapter introduces the origin of complex symmetric operators and m-complex symmetric operators,as well as the development of complex symmetric Toeplitz operators and m-complex symmetric Toeplitz operators on the Hardy space,the Bergman space and the Dirichlet space,the research status and the important research results so far.The second chapter introduces the preparatory knowledge related to this paper.In the third chapter,on the basis of previous studies,we characterize the symbolic function of the 2-complex symmetric Toeplitz operator.In the fourth chapter,we describe the symbolic function of the 3-complex symmetric Toeplitz operator.In the fifth chapter,the general conclusions of the m-complex symmetric Toeplitz operator and the summary and prospect of this paper are given.