Research On Symmetric Operators And Complex Symmetric Operators

The characterization of symmetric operators and complex symmetric operators are two hot topics in the direction of operator theory in recent years,which has aroused the research interest of many scholars.It has very important application value in the research of operator theory,and has attracted more and more attention in the fields of mathematics,quantum mechanics,physics,cybernetics and so on.This article focuses on using the operator blocking technique to appropriately block the operator according to the research content,and further proves our main conclusions.This paper is divided into three chapters.The main contents of each chapter are as follows:The first chapter mainly introduces the research background and some basic concepts and results involved in this paper,such as the concepts of symmetric operators,conjugations,anti-unitary operators,and the results of polar decomposition theorem for complex symmetric operators and C-normal operators.In the second chapter,we study the general explicit descriptions for two classes of symmetric operators involving of the idempotent operators.Let（39）be a separable complex Hilbert space,defined E as the idempotent operator in it,let（?）.Based on the block operator technique and Halmos’two projections theory,we give general explicit descriptions for all the symmetries inΓ_Eas well as in?_E.We also study the symmetryρ_E=（2E-I）|2E-I|^-1 in detail,and characterize the symmetries inΓ_E viaρ_E.Using this,the relationship between the symmetries inΓ_E and the symmetries in?_E is established.In the third chapter,we introduce the properties of complex symmetric operators and C-normal operators,and give a new method to prove the refined polar decomposition theorem of complex symmetric operators and C-normal operators with the help of the operator partition theory.The sufficient conditions for C-normal operators to become complex symmetric operators are also studied in this paper.